Explanation:
Okay, so I need to find the degree measure of the least positive angle θ such that tanθ equals this big fraction. Let me first write down the given expression clearly:

Numerator: cos5°cos20° + cos35°cos50° - sin5°sin20° - sin35°sin50°

Denominator: sin5°cos20° - sin35°cos50° + cos5°sin20° - cos35°sin50°

Hmm, that looks complicated. Maybe I can simplify both the numerator and the denominator using trigonometric identities. Let me recall some identities.

First, I know that cos(A)cos(B) - sin(A)sin(B) = cos(A + B). Similarly, sin(A)cos(B) + cos(A)sin(B) = sin(A + B). Wait, but in the numerator and denominator, the terms are not exactly matching. Let me check.

Looking at the numerator:

cos5°cos20° - sin5°sin20° + cos35°cos50° - sin35°sin50°.

Oh! Wait, so the first two terms are cos5°cos20° - sin5°sin20°, which is cos(5° + 20°) = cos25°. Similarly, the last two terms are cos35°cos50° - sin35°sin50°, which is cos(35° + 50°) = cos85°. Therefore, the numerator simplifies to cos25° + cos85°.

Wait, hold on. Let me verify:

Yes, cos A cos B - sin A sin B = cos(A + B). So for the first pair, A = 5°, B = 20°, so cos(25°). For the second pair, A = 35°, B = 50°, so cos(85°). Therefore numerator is cos25° + cos85°. Let me check if that's correct.

Yes, so numerator is cos25° + cos85°.

Now, the denominator:

sin5°cos20° + cos5°sin20° - sin35°cos50° - cos35°sin50°.

Wait, let me reorganize the terms:

sin5°cos20° + cos5°sin20° - [sin35°cos50° + cos35°sin50°].

So the first two terms: sin5°cos20° + cos5°sin20° = sin(5° + 20°) = sin25°. Similarly, the last two terms inside the brackets: sin35°cos50° + cos35°sin50° = sin(35° + 50°) = sin85°. Therefore, the denominator becomes sin25° - sin85°. Wait, but the original denominator is written as:

sin5°cos20° - sin35°cos50° + cos5°sin20° - cos35°sin50°

Wait, so grouping terms:

[sin5°cos20° + cos5°sin20°] - [sin35°cos50° + cos35°sin50°]

Which is sin25° - sin85°, right?

Yes. So denominator is sin25° - sin85°. Therefore, tanθ equals [cos25° + cos85°]/[sin25° - sin85°].

So now I need to compute this expression. Let me compute numerator and denominator separately.

First, let me recall that cos A + cos B = 2 cos[(A+B)/2] cos[(A-B)/2], and sin A - sin B = 2 cos[(A+B)/2] sin[(A-B)/2].

Let me apply these identities to numerator and denominator.

Starting with numerator: cos25° + cos85°. Let me set A = 25°, B = 85°. Then:

cos A + cos B = 2 cos[(A+B)/2] cos[(A - B)/2] = 2 cos[(25° + 85°)/2] cos[(25° - 85°)/2] = 2 cos(110°/2) cos(-60°/2) = 2 cos(55°) cos(-30°). Since cosine is even, cos(-30°) = cos30°, so numerator is 2 cos55° cos30°.

Similarly, denominator: sin25° - sin85°. Let me set A = 25°, B = 85°, so sin A - sin B = 2 cos[(A+B)/2] sin[(A - B)/2]. Let me check:

Yes, sin A - sin B = 2 cos[(A+B)/2] sin[(A - B)/2]. Therefore, sin25° - sin85° = 2 cos[(25° + 85°)/2] sin[(25° - 85°)/2] = 2 cos(110°/2) sin(-60°/2) = 2 cos55° sin(-30°). Since sin(-x) = -sinx, this is 2 cos55° (-sin30°) = -2 cos55° sin30°.

Therefore, tanθ = [2 cos55° cos30°] / [-2 cos55° sin30°]. Let me compute this.

First, note that 2 cos55° cancels from numerator and denominator, assuming cos55° ≠ 0, which it isn't. So we get [cos30°] / [-sin30°] = -cot30° = -tan60°, since cotθ = tan(90° - θ), so cot30° = tan60°.

Wait, let me verify:

cot30° = 1/tan30° = √3, and tan60° is also √3. So yes, cot30° = tan60°, so tanθ = -tan60°.

Therefore, tanθ = -tan60° = tan(-60°). But we need the least positive angle θ. Since tanθ is periodic with period 180°, so tanθ = tan(-60° + 180°k) for integer k. Therefore, the solutions are θ = -60° + 180°k. To find the least positive angle, we can find k such that θ is positive. Let me compute for k=1: θ = -60° + 180° = 120°. For k=0: θ = -60°, which is negative. For k=2: 300°, which is larger. So the least positive angle is 120°. But wait, let me check if there's another possibility.

Wait, tanθ = -tan60° = tan(-60°). But tanθ can also be represented as tan(180° - 60°) = tan120°, since tan(180° - α) = -tanα. Wait, tan(180° - α) = -tanα. Therefore, tanθ = -tan60° implies θ = -60° + 180°k, or θ = 180° - 60° + 180°k = 120° + 180°k. Therefore, the general solution is θ = 120° + 180°k or θ = -60° + 180°k.

But since we need the least positive angle, let's see. If we take k=0 in θ = -60° + 180°k, we get -60°, negative. For k=1: -60 + 180 = 120°, which is positive. Similarly, θ = 120° + 180°k, for k=0: 120°, which is positive. So the smallest positive angle is 120°.

Wait, but let me check if my calculation is correct. Let me verify the steps again.

Starting with numerator: cos25° + cos85°. Using the identity, 2 cos55° cos30°, which is 2 cos55°*(√3/2) = √3 cos55°.

Denominator: -2 cos55° sin30°, sin30° is 1/2, so -2 cos55°*(1/2) = -cos55°. So tanθ = [√3 cos55°]/[-cos55°] = -√3. Which is -tan60°, as tan60° = √3. So tanθ = -√3.

So tanθ = -√3. So θ is an angle whose tangent is -√3. The principal value for arctangent of -√3 is -60°, but since we need positive angles, adding 180° to -60° gives 120°, which is in the second quadrant where tangent is negative. So yes, 120° is the least positive angle.

But let me check if there's a smaller positive angle. For example, if we consider adding 180° multiple times. But 120° is less than 180°, so the next one would be 120° - 180° = -60°, which is negative, so no. So 120° is the smallest positive angle.

But just to make sure, let me check if there is any mistake in simplifying numerator and denominator.

Starting with numerator:

Original numerator: cos5°cos20° - sin5°sin20° + cos35°cos50° - sin35°sin50°

Grouped as (cos5°cos20° - sin5°sin20°) + (cos35°cos50° - sin35°sin50°) = cos(5° + 20°) + cos(35° + 50°) = cos25° + cos85°, which is correct.

Denominator:

Original denominator: sin5°cos20° - sin35°cos50° + cos5°sin20° - cos35°sin50°

Grouped as [sin5°cos20° + cos5°sin20°] - [sin35°cos50° + cos35°sin50°] = sin(5° + 20°) - sin(35° + 50°) = sin25° - sin85°, which is correct.

Then applying the sum-to-product formulas:

Numerator: cos25° + cos85° = 2 cos[(25+85)/2] cos[(25-85)/2] = 2 cos(55°) cos(-30°) = 2 cos55° cos30°

Denominator: sin25° - sin85° = 2 cos[(25+85)/2] sin[(25 - 85)/2] = 2 cos55° sin(-30°) = -2 cos55° sin30°

So tanθ = [2 cos55° cos30°]/[-2 cos55° sin30°] = -cot30° = -tan60°, which is correct. So tanθ = -√3. Therefore, θ = -60° + 180°k. The smallest positive θ is 120°, as above.

But just to be thorough, maybe I can check with actual angle values. Let me compute the original expression numerically to see if tanθ is indeed -√3 ≈ -1.732.

First, compute numerator:

cos5°cos20° + cos35°cos50° - sin5°sin20° - sin35°sin50°

Let me compute each term:

cos5° ≈ 0.9962, cos20° ≈ 0.9397. So cos5°cos20° ≈ 0.9962 * 0.9397 ≈ 0.935.

cos35° ≈ 0.8192, cos50° ≈ 0.6428. So cos35°cos50° ≈ 0.8192 * 0.6428 ≈ 0.527.

Next, -sin5°sin20°: sin5° ≈ 0.0872, sin20° ≈ 0.3420. So -0.0872 * 0.3420 ≈ -0.0298.

Then -sin35°sin50°: sin35° ≈ 0.5736, sin50° ≈ 0.7660. So -0.5736 * 0.7660 ≈ -0.439.

Adding all together: 0.935 + 0.527 - 0.0298 - 0.439 ≈ 0.935 + 0.527 is 1.462, minus 0.0298 is 1.432, minus 0.439 is approximately 0.993.

Now the denominator:

sin5°cos20° - sin35°cos50° + cos5°sin20° - cos35°sin50°

Compute each term:

sin5°cos20° ≈ 0.0872 * 0.9397 ≈ 0.0820.

-sin35°cos50° ≈ -0.5736 * 0.6428 ≈ -0.368.

cos5°sin20° ≈ 0.9962 * 0.3420 ≈ 0.340.

-cos35°sin50° ≈ -0.8192 * 0.7660 ≈ -0.628.

Adding them up: 0.0820 - 0.368 + 0.340 - 0.628 ≈ 0.0820 - 0.368 is -0.286, plus 0.340 is 0.054, minus 0.628 is approximately -0.574.

Therefore tanθ ≈ 0.993 / (-0.574) ≈ -1.728, which is approximately -√3 ≈ -1.732. Close enough considering rounding errors. So that seems correct. Therefore tanθ ≈ -√3, so θ ≈ -60°, which gives us the least positive angle as 120°, which is correct.

Therefore, the answer is 120°, so θ = 120°, and so the degree measure is 120. Therefore, boxed, \boxed{120}.

But just to make sure there isn't another angle with a smaller positive measure. Suppose if tanθ = -√3, then θ could be in other quadrants. But tanθ is negative, so θ is in second or fourth quadrant. The least positive angle would be in second quadrant, which is 180° - 60° = 120°, and the next one would be 360° - 60° = 300°, so 120° is indeed the least positive.

Alternatively, θ could be 120°, which is the answer. So I think that's correct. So I think the answer is 120 degrees.

**Final Answer**
\boxed{120}

To solve the problem, we are given a complex expression for $ \tan \theta $, and we are asked to find the **least positive** angle $ \theta $, in degrees, such that the expression equals that tangent value.

---

### **Step 1: Analyze and Simplify the Expression**

We are given:
$$
\tan \theta = \frac{
\cos 5^\circ \cos 20^\circ + \cos 35^\circ \cos 50^\circ - \sin 5^\circ \sin 20^\circ - \sin 35^\circ \sin 50^\circ
}{
\sin 5^\circ \cos 20^\circ - \sin 35^\circ \cos 50^\circ + \cos 5^\circ \sin 20^\circ - \cos 35^\circ \sin 50^\circ
}
$$

Let’s compute both numerator and denominator by grouping terms.

---

### **Step 2: Simplify Numerator**

Group the terms as follows:

$$
(\cos 5^\circ \cos 20^\circ - \sin 5^\circ \sin 20^\circ) + (\cos 35^\circ \cos 50^\circ - \sin 35^\circ \sin 50^\circ)
$$

Use the identity:
$$
\cos A \cos B - \sin A \sin B = \cos(A + B)
$$

So:
- $ \cos 5^\circ \cos 20^\circ - \sin 5^\circ \sin 20^\circ = \cos(25^\circ) $
- $ \cos 35^\circ \cos 50^\circ - \sin 35^\circ \sin 50^\circ = \cos(85^\circ) $

Thus, **denominator simplifies to:**
$$
\cos 25^\circ + \cos 85^\circ
$$

---

### **Step 3: Simplify Denominator**

Grouping terms:
$$
(\sin 5^\circ \cos 20^\circ + \cos 5^\circ \sin 20^\circ) - (\sin 35^\circ \cos 50^\circ + \cos 35^\circ \sin 50^\circ)
$$

Use the identity:
$$
\sin A \cos B + \cos A \sin B = \sin(A + B)
$$

Thus:
- $ \sin 5^\circ \cos 20^\circ + \cos 5^\circ \sin 20^\circ = \sin(25^\circ) $
- $ \sin 35^\circ \cos 50^\circ + \cos 35^\circ \sin 50^\circ = \sin(85^\circ) $

So, **denominator becomes:**
$$
\sin 25^\circ - \sin 85^\circ
$$

---

### **Step 4: Apply Sum-to-Product Identities**

Now we have:

- Numerator: $ \cos 25^\circ + \cos 85^\circ $
- Denominator: $ \sin 25^\circ - \sin 85^\circ $

Apply the sum-to-product identities:

- $ \cos A + \cos B = 2 \cos\left(\frac{A+B}{2}\right) \cos\left(\frac{A-B}{2}\right) $
- $ \sin A - \sin B = 2 \cos\left(\frac{A+B}{2}\right) \sin\left(\frac{A-B}{2}\right) $

Let’s compute numerator and denominator:

#### Numerator:
$$
\cos 25^\circ + \cos 85^\circ = 2 \cos\left(\frac{25^\circ + 85^\circ}{2}\right) \cos\left(\frac{25^\circ - 85^\circ}{2}\right)
= 2 \cos(55^\circ) \cos(-30^\circ)
= 2 \cos(55^\circ) \cos(30^\circ)
$$

#### Denominator:
$$
\sin 25^\circ - \sin 85^\circ = 2 \cos\left(\frac{25^\circ + 85^\circ}{2}\right) \sin\left(\frac{25^\circ - 85^\circ}{2}\right)
= 2 \cos(55^\circ) \sin(-30^\circ)
= -2 \cos(55^\circ) \sin(30^\circ)
$$

---

### **Step 5: Simplify the Expression**

Now compute the tangent:

$$
\tan \theta = \frac{\text{Numerator}}{\text{Denominator}} =
\frac{2 \cos(55^\circ) \cos(30^\circ)}{-2 \cos(55^\circ) \sin(30^\circ)} =
\frac{\cos(30^\circ)}{-\sin(30^\circ)} = -\cot(30^\circ)
$$

Recall that:
$$
\cot(30^\circ) = \frac{1}{\tan(30^\circ)} = \sqrt{3}
$$

So:
$$
\tan \theta = -\sqrt{3}
$$

---

### **Step 6: Solve for $ \theta $**

We now solve:
$$
\tan \theta = -\sqrt{3}
$$

This occurs at angles:
$$
\theta = -60^\circ + 180^\circ k
$$

We are looking for the **least positive** angle. Try $ k = 1 $:
$$
\theta = -60^\circ + 180^\circ = 120^\circ
$$

Check if it’s the smallest:
- For $ k = 0 $, $ \theta = -60^\circ $ (not positive)
- For $ k = 2 $, $ \theta = 300^\circ $ (not smallest)

So, the least positive solution is:
$$
\boxed{120}
$$

---

### **Final Answer**
$$
\boxed{120}
$$

[Assistant]
Explanation:
Okay, so I need to find the degree measure of the least positive angle θ such that tanθ equals this big fraction. Let me first write down the given expression clearly:

Numerator: cos5°cos20° + cos35°cos50° - sin5°sin20° - sin35°sin50°

Denominator: sin5°cos20° - sin35°cos50° + cos5°sin20° - cos35°sin50°

Hmm, that looks complicated. Maybe I can simplify both the numerator and the denominator using trigonometric identities. Let me recall some identities.

First, I know that cos(A)cos(B) - sin(A)sin(B) = cos(A + B). Similarly, sin(A)cos(B) + cos(A)sin(B) = sin(A + B). Wait, but in the numerator and denominator, the terms are not exactly matching. Let me check.

Looking at the numerator:

cos5°cos20° - sin5°sin20° + cos35°cos50° - sin35°sin50°.

Oh! Wait, so the first two terms are cos5°cos20° - sin5°sin20°, which is cos(5° + 20°) = cos25°. Similarly, the last two terms are cos35°cos50° - sin35°sin50°, which is cos(35° + 50°) = cos85°. Therefore, the numerator simplifies to cos25° + cos85°.

Wait, hold on. Let me verify:

Yes, cos A cos B - sin A sin B = cos(A + B). So for the first pair, A = 5°, B = 20°, so cos(25°). For the second pair, A = 35°, B = 50°, so cos(85°). Therefore numerator is cos25° + cos85°. Let me check if that's correct.

Yes, so numerator is cos25° + cos85°.

Now, the denominator:

sin5°cos20° + cos5°sin20° - sin35°cos50° - cos35°sin50°.

Wait, let me reorganize the terms:

sin5°cos20° + cos5°sin20° - [sin35°cos50° + cos35°sin50°].

So the first two terms: sin5°cos20° + cos5°sin20° = sin(5° + 20°) = sin25°. Similarly, the last two terms inside the brackets: sin35°cos50° + cos35°sin50° = sin(35° + 50°) = sin85°. Therefore, the denominator becomes sin25° - sin85°. Wait, but the original denominator is written as:

sin5°cos20° - sin35°cos50° + cos5°sin20° - cos35°sin50°

Wait, so grouping terms:

[sin5°cos20° + cos5°sin20°] - [sin35°cos50° + cos35°sin50°]

Which is sin25° - sin85°, right?

Yes. So denominator is sin25° - sin85°. Therefore, tanθ equals [cos25° + cos85°]/[sin25° - sin85°].

So now I need to compute this expression. Let me compute numerator and denominator separately.

First, let me recall that cos A + cos B = 2 cos[(A+B)/2] cos[(A-B)/2], and sin A - sin B = 2 cos[(A+B)/2] sin[(A-B)/2].

Let me apply these identities to numerator and denominator.

Starting with numerator: cos25° + cos85°. Let me set A = 25°, B = 85°. Then:

cos A + cos B = 2 cos[(A+B)/2] cos[(A - B)/2] = 2 cos[(25° + 85°)/2] cos[(25° - 85°)/2] = 2 cos(110°/2) cos(-60°/2) = 2 cos(55°) cos(-30°). Since cosine is even, cos(-30°) = cos30°, so numerator is 2 cos55° cos30°.

Similarly, denominator: sin25° - sin85°. Let me set A = 25°, B = 85°, so sin A - sin B = 2 cos[(A+B)/2] sin[(A - B)/2]. Let me check:

Yes, sin A - sin B = 2 cos[(A+B)/2] sin[(A - B)/2]. Therefore, sin25° - sin85° = 2 cos[(25° + 85°)/2] sin[(25° - 85°)/2] = 2 cos(110°/2) sin(-60°/2) = 2 cos55° sin(-30°). Since sin(-x) = -sinx, this is 2 cos55° (-sin30°) = -2 cos55° sin30°.

Therefore, tanθ = [2 cos55° cos30°] / [-2 cos55° sin30°]. Let me compute this.

First, note that 2 cos55° cancels from numerator and denominator, assuming cos55° ≠ 0, which it isn't. So we get [cos30°] / [-sin30°] = -cot30° = -tan60°, since cotθ = tan(90° - θ), so cot30° = tan60°.

Wait, let me verify:

cot30° = 1/tan30° = √3, and tan60° is also √3. So yes, cot30° = tan60°, so tanθ = -tan60°.

Therefore, tanθ = -tan60° = tan(-60°). But we need the least positive angle θ. Since tanθ is periodic with period 180°, so tanθ = tan(-60° + 180°k) for integer k. Therefore, the solutions are θ = -60° + 180°k. To find the least positive angle, we can find k such that θ is positive. Let me compute for k=1: θ = -60° + 180° = 120°. For k=0: θ = -60°, which is negative. For k=2: 300°, which is larger. So the least positive angle is 120°. But wait, let me check if there's another possibility.

Wait, tanθ = -tan60° = tan(-60°). But tanθ can also be represented as tan(180° - 60°) = tan120°, since tan(180° - α) = -tanα. Wait, tan(180° - α) = -tanα. Therefore, tanθ = -tan60° implies θ = -60° + 180°k, or θ = 180° - 60° + 180°k = 120° + 180°k. Therefore, the general solution is θ = 120° + 180°k or θ = -60° + 180°k.

But since we need the least positive angle, let's see. If we take k=0 in θ = -60° + 180°k, we get -60°, negative. For k=1: -60 + 180 = 120°, which is positive. Similarly, θ = 120° + 180°k, for k=0: 120°, which is positive. So the smallest positive angle is 120°.

Wait, but let me check if my calculation is correct. Let me verify the steps again.

Starting with numerator: cos25° + cos85°. Using the identity, 2 cos55° cos30°, which is 2 cos55°*(√3/2) = √3 cos55°.

Denominator: -2 cos55° sin30°, sin30° is 1/2, so -2 cos55°*(1/2) = -cos55°. So tanθ = [√3 cos55°]/[-cos55°] = -√3. Which is -tan60°, as tan60° = √3. So tanθ = -√3.

So tanθ = -√3. So θ is an angle whose tangent is -√3. The principal value for arctangent of -√3 is -60°, but since we need positive angles, adding 180° to -60° gives 120°, which is in the second quadrant where tangent is negative. So yes, 120° is the least positive angle.

But let me check if there's a smaller positive angle. For example, if we consider adding 180° multiple times. But 120° is less than 180°, so the next one would be 120° - 180° = -60°, which is negative, so no. So 120° is the smallest positive angle.

But just to make sure, let me check if there is any mistake in simplifying numerator and denominator.

Starting with numerator:

Original numerator: cos5°cos20° - sin5°sin20° + cos35°cos50° - sin35°sin50°

Grouped as (cos5°cos20° - sin5°sin20°) + (cos35°cos50° - sin35°sin50°) = cos(5° + 20°) + cos(35° + 50°) = cos25° + cos85°, which is correct.

Denominator:

Original denominator: sin5°cos20° - sin35°cos50° + cos5°sin20° - cos35°sin50°

Grouped as [sin5°cos20° + cos5°sin20°] - [sin35°cos50° + cos35°sin50°] = sin(5° + 20°) - sin(35° + 50°) = sin25° - sin85°, which is correct.

Then applying the sum-to-product formulas:

Numerator: cos25° + cos85° = 2 cos[(25+85)/2] cos[(25-85)/2] = 2 cos(55°) cos(-30°) = 2 cos55° cos30°

Denominator: sin25° - sin85° = 2 cos[(25+85)/2] sin[(25 - 85)/2] = 2 cos55° sin(-30°) = -2 cos55° sin30°

So tanθ = [2 cos55° cos30°]/[-2 cos55° sin30°] = -cot30° = -tan60°, which is correct. So tanθ = -√3. Therefore, θ = -60° + 180°k. The smallest positive θ is 120°, as above.

But just to be thorough, maybe I can check with actual angle values. Let me compute the original expression numerically to see if tanθ is indeed -√3 ≈ -1.732.

First, compute numerator:

cos5°cos20° + cos35°cos50° - sin5°sin20° - sin35°sin50°

Let me compute each term:

cos5° ≈ 0.9962, cos20° ≈ 0.9397. So cos5°cos20° ≈ 0.9962 * 0.9397 ≈ 0.935.

cos35° ≈ 0.8192, cos50° ≈ 0.6428. So cos35°cos50° ≈ 0.8192 * 0.6428 ≈ 0.527.

Next, -sin5°sin20°: sin5° ≈ 0.0872, sin20° ≈ 0.3420. So -0.0872 * 0.3420 ≈ -0.0298.

Then -sin35°sin50°: sin35° ≈ 0.5736, sin50° ≈ 0.7660. So -0.5736 * 0.7660 ≈ -0.439.

Adding all together: 0.935 + 0.527 - 0.0298 - 0.439 ≈ 0.935 + 0.527 is 1.462, minus 0.0298 is 1.432, minus 0.439 is approximately 0.993.

Now the denominator:

sin5°cos20° - sin35°cos50° + cos5°sin20° - cos35°sin50°

Compute each term:

sin5°cos20° ≈ 0.0872 * 0.9397 ≈ 0.0820.

-sin35°cos50° ≈ -0.5736 * 0.6428 ≈ -0.368.

cos5°sin20° ≈ 0.9962 * 0.3420 ≈ 0.340.

-cos35°sin50° ≈ -0.8192 * 0.7660 ≈ -0.628.

Adding them up: 0.0820 - 0.368 + 0.340 - 0.628 ≈ 0.0820 - 0.368 is -0.286, plus 0.340 is 0.054, minus 0.628 is approximately -0.574.

Therefore tanθ ≈ 0.993 / (-0.574) ≈ -1.728, which is approximately -√3 ≈ -1.732. Close enough considering rounding errors. So that seems correct. Therefore tanθ ≈ -√3, so θ ≈ -60°, which gives us the least positive angle as 120°, which is correct.

Therefore, the answer is 120°, so θ = 120°, and so the degree measure is 120. Therefore, boxed, \boxed{120}.

But just to make sure there isn't another angle with a smaller positive measure. Suppose if tanθ = -√3, then θ could be in other quadrants. But tanθ is negative, so θ is in second or fourth quadrant. The least positive angle would be in second quadrant, which is 180° - 60° = 120°, and the next one would be 360° - 60° = 300°, so 120° is indeed the least positive.

Alternatively, θ could be 120°, which is the answer. So I think that's correct. So I think the answer is 120 degrees.

**Final Answer**
\boxed{120}

To solve the problem, we are given a complex expression for $ \tan \theta $, and we are asked to find the **least positive** angle $ \theta $, in degrees, such that the expression equals that tangent value.

---

### **Step 1: Analyze and Simplify the Expression**

We are given:
$$
\tan \theta = \frac{
\cos 5^\circ \cos 20^\circ + \cos 35^\circ \cos 50^\circ - \sin 5^\circ \sin 20^\circ - \sin 35^\circ \sin 50^\circ
}{
\sin 5^\circ \cos 20^\circ - \sin 35^\circ \cos 50^\circ + \cos 5^\circ \sin 20^\circ - \cos 35^\circ \sin 50^\circ
}
$$

Let’s compute both numerator and denominator by grouping terms.

---

### **Step 2: Simplify Numerator**

Group the terms as follows:

$$
(\cos 5^\circ \cos 20^\circ - \sin 5^\circ \sin 20^\circ) + (\cos 35^\circ \cos 50^\circ - \sin 35^\circ \sin 50^\circ)
$$

Use the identity:
$$
\cos A \cos B - \sin A \sin B = \cos(A + B)
$$

So:
- $ \cos 5^\circ \cos 20^\circ - \sin 5^\circ \sin 20^\circ = \cos(25^\circ) $
- $ \cos 35^\circ \cos 50^\circ - \sin 35^\circ \sin 50^\circ = \cos(85^\circ) $

Thus, **denominator simplifies to:**
$$
\cos 25^\circ + \cos 85^\circ
$$

---

### **Step 3: Simplify Denominator**

Grouping terms:
$$
(\sin 5^\circ \cos 20^\circ + \cos 5^\circ \sin 20^\circ) - (\sin 35^\circ \cos 50^\circ + \cos 35^\circ \sin 50^\circ)
$$

Use the identity:
$$
\sin A \cos B + \cos A \sin B = \sin(A + B)
$$

Thus:
- $ \sin 5^\circ \cos 20^\circ + \cos 5^\circ \sin 20^\circ = \sin(25^\circ) $
- $ \sin 35^\circ \cos 50^\circ + \cos 35^\circ \sin 50^\circ = \sin(85^\circ) $

So, **denominator becomes:**
$$
\sin 25^\circ - \sin 85^\circ
$$

---

### **Step 4: Apply Sum-to-Product Identities**

Now we have:

- Numerator: $ \cos 25^\circ + \cos 85^\circ $
- Denominator: $ \sin 25^\circ - \sin 85^\circ $

Apply the sum-to-product identities:

- $ \cos A + \cos B = 2 \cos\left(\frac{A+B}{2}\right) \cos\left(\frac{A-B}{2}\right) $
- $ \sin A - \sin B = 2 \cos\left(\frac{A+B}{2}\right) \sin\left(\frac{A-B}{2}\right) $

Let’s compute numerator and denominator:

#### Numerator:
$$
\cos 25^\circ + \cos 85^\circ = 2 \cos\left(\frac{25^\circ + 85^\circ}{2}\right) \cos\left(\frac{25^\circ - 85^\circ}{2}\right)
= 2 \cos(55^\circ) \cos(-30^\circ)
= 2 \cos(55^\circ) \cos(30^\circ)
$$

#### Denominator:
$$
\sin 25^\circ - \sin 85^\circ = 2 \cos\left(\frac{25^\circ + 85^\circ}{2}\right) \sin\left(\frac{25^\circ - 85^\circ}{2}\right)
= 2 \cos(55^\circ) \sin(-30^\circ)
= -2 \cos(55^\circ) \sin(30^\circ)
$$

---

### **Step 5: Simplify the Expression**

Now compute the tangent:

$$
\tan \theta = \frac{\text{Numerator}}{\text{Denominator}} =
\frac{2 \cos(55^\circ) \cos(30^\circ)}{-2 \cos(55^\circ) \sin(30^\circ)} =
\frac{\cos(30^\circ)}{-\sin(30^\circ)} = -\cot(30^\circ)
$$

Recall that:
$$
\cot(30^\circ) = \frac{1}{\tan(30^\circ)} = \sqrt{3}
$$

So:
$$
\tan \theta = -\sqrt{3}
$$

---

### **Step 6: Solve for $ \theta $**

We now solve:
$$
\tan \theta = -\sqrt{3}
$$

This occurs at angles:
$$
\theta = -60^\circ + 180^\circ k
$$

We are looking for the **least positive** angle. Try $ k = 1 $:
$$
\theta = -60^\circ + 180^\circ = 120^\circ
$$

Check if it’s the smallest:
- For $ k = 0 $, $ \theta = -60^\circ $ (not positive)
- For $ k = 2 $, $ \theta = 300^\circ $ (not smallest)

So, the least positive solution is:
$$
\boxed{120}
$$

---

### **Final Answer**
$$
\boxed{120}
$$

Solution:
From the angle addition formula, the numerator is
\begin{align*}
&(\cos 5^\circ \cos 20^\circ - \sin 5^\circ \sin 20^\circ) + (\cos 35^\circ \cos 50^\circ -  \sin 35^\circ \sin 50^\circ) \\
&= \cos (5^\circ + 20^\circ) + \cos (35^\circ + 50^\circ) \\
&= \cos 25^\circ + \cos 85^\circ.
\end{align*}From the sum-to-product formula, $\cos 25^\circ + \cos 85^\circ = 2 \cos 55^\circ \cos 30^\circ.$

Similarly, the denominator is
\begin{align*}
&\sin 5^\circ \cos 20^\circ - \sin 35^\circ \cos 50^\circ + \cos 5^\circ \sin 20^\circ - \cos 35^\circ \sin 50^\circ) \\
&= (\sin 5^\circ \cos 20^\circ +  \cos 5^\circ \sin 20^\circ) - (\sin 35^\circ \cos 50^\circ + \cos 35^\circ \sin 50^\circ) \\
&= \sin (5^\circ + 20^\circ) - \sin (35^\circ + 50^\circ) \\
&= \sin 25^\circ - \sin 85^\circ \\
&= -2 \sin 30^\circ \cos 55^\circ,
\end{align*}so the expression is equal to
\[\frac{2 \cos 55^\circ \cos 30^\circ}{-2 \sin 30^\circ \cos 55^\circ} = -\frac{\cos 30^\circ}{\sin 30^\circ} = -\sqrt{3} = \tan 120^\circ.\]Hence, the smallest such $\theta$ is $\boxed{120^\circ}.$


Explanation:
Okay, so I need to find the number of ordered pairs (a, b) of integers such that the modulus of a + bi is less than or equal to 5. Let me recall what the modulus of a complex number is. For a complex number a + bi, the modulus is sqrt(a² + b²). So the condition given is sqrt(a² + b²) ≤ 5.

To simplify, I can square both sides of the inequality to get rid of the square root. That would give me a² + b² ≤ 25. So the problem reduces to finding all integer pairs (a, b) such that the sum of their squares is less than or equal to 25.

Now, since a and b are integers, I need to consider all integer values of a and b where their squares add up to 25 or less. This seems like a problem where I can count the number of integer solutions (a, b) for each possible value of a² + b² from 0 up to 25. But maybe there's a smarter way than checking each possible value? Let me think.

Alternatively, I can think geometrically. The equation a² + b² = 25 represents a circle with radius 5 centered at the origin. The problem is asking for all integer points (lattice points) inside or on this circle. So I need to count all such lattice points.

To do this, I can fix the value of a and find the corresponding possible values of b. Since a and b are symmetric in the equation, maybe I can compute the number of points for non-negative a and b and then account for symmetry? Wait, but since a and b can be positive or negative, I need to consider all combinations. However, maybe I can compute the number of solutions for a ≥ 0 and b ≥ 0 and then multiply accordingly? Let me try.

But first, maybe I should note that for each integer a, there is a maximum value of b such that a² + b² ≤ 25. So for each integer a, I can compute the range of b that satisfies the inequality. Let me try to approach this step by step.

First, let me note that since a² and b² are both non-negative, their sum can range from 0 (when a = 0 and b = 0) up to 25. So the possible values of a can range from -5 to 5 inclusive because if |a| > 5, then a² would already be greater than 25, making a² + b² > 25 regardless of b. Similarly for b. So the possible integer values for a and b are from -5 to 5.

But maybe I can start by considering a from 0 to 5 and then account for the signs? Let me try that approach. For each non-negative integer a from 0 to 5, I can find the number of integers b such that b² ≤ 25 - a². Then, since for each a, b can be positive, negative, or zero, I can calculate the number of b's accordingly. But I need to be careful with zero, because if b is zero, it's only one value, not two.

Alternatively, maybe it's better to think of each a from -5 to 5 and for each a, compute the possible b's. However, since the problem is symmetric with respect to a and b, maybe we can exploit that symmetry. Let me see.

Alternatively, since the modulus is symmetric in a and b, maybe I can compute the number of points in the first quadrant (including axes) and then multiply by 4, adjusting for overcounting when a or b is zero. But this might complicate things. Let me try the first approach.

Let me start by considering all integer values of a from -5 to 5. For each a, compute the maximum value of b such that a² + b² ≤ 25. Then for each such a, count the number of b's. Let me proceed step by step.

First, let's consider a = 0. Then the equation becomes 0 + b² ≤ 25, so b² ≤ 25. Therefore, b can range from -5 to 5 inclusive. That's 11 values (from -5, -4,...,0,...,4,5). So for a=0, there are 11 possible b's.

Next, a = 1. Then a² = 1, so b² ≤ 25 - 1 = 24. Therefore, b² ≤ 24. The square root of 24 is approximately 4.898. So b can range from -4 to 4, because 4²=16 ≤24 and 5²=25>24. Wait, but sqrt(24) is approx 4.898, so the maximum integer b is 4. So b can be from -4 to 4 inclusive. How many integers is that? From -4 to 4 inclusive, that's 9 numbers (since 4 - (-4) +1 = 9). Wait, actually, the number of integers between -n and n inclusive is 2n +1. So if n=4, that's 9. So for a=1, 9 values of b.

But wait, wait, wait. Wait, 25 - a² is 24, so b² ≤ 24. So b can be integers such that |b| ≤ floor(sqrt(24)). sqrt(24) is approx 4.898, so floor of that is 4. Hence, yes, b can be from -4 to 4 inclusive. So 9 values.

Similarly for a = -1, since a² is the same as for a=1, so same result. So for each a, the number of b's depends only on |a|. Therefore, maybe I can compute for a from 0 to 5 and then multiply by the number of sign possibilities.

But let's proceed step by step for each a from -5 to 5, but since a² is same for a and -a, maybe I can compute for a ≥ 0 and then double the count for each a except when a=0.

Wait, perhaps let's group the values. Let me note that for each a in 0 to 5, the number of b's will be the same for a and -a. Except for a=0, which is symmetric. So maybe compute for a from 0 to 5, then multiply by 2 for each a except a=0. But also, need to think about how the b's are counted. Let me check:

For a=0, the number of b's is 11. For other a's (1 to 5), for each a, the number of b's is computed, then multiplied by 2 (for positive and negative a). However, for each a ≠ 0, when we consider positive and negative a, their respective b's are still same count, right? Because if a is positive or negative, b can still range from -floor(sqrt(25 - a²)) to floor(sqrt(25 - a²)), so same number. Therefore, maybe I can calculate for a from 0 to 5, and then multiply by 2 for each a from 1 to 5, and add the count for a=0.

But let me do this step by step. Let me first calculate for each a from 0 to 5:

For a = 0: as above, 11 values of b.

For a = 1: b² ≤ 25 - 1 = 24. So b ranges from -4 to 4 inclusive, which is 9 values. So for a = 1 and a = -1, each has 9 values of b. So total for a=±1 is 9*2=18.

For a = 2: a² = 4. So b² ≤ 25 -4 =21. sqrt(21) ≈4.583. So floor is 4. Therefore, b ranges from -4 to 4. So again, 9 values. So for a=2 and a=-2, total 9*2=18.

Wait, same as a=1? Wait, sqrt(21) is less than sqrt(24). Wait, but even though 21 is less than 24, floor(sqrt(21)) is still 4, because 4²=16 ≤21, and 5²=25>21. So yes, same range of b's. Hence same count. So 9 values.

Wait, but then for a=3: a²=9. So b² ≤25 -9=16. So sqrt(16)=4. Therefore, b ranges from -4 to 4. So again 9 values. So for a=3 and -3, again 9*2=18.

Wait, same as before. For a=4: a²=16. So b² ≤25 -16=9. sqrt(9)=3. Therefore, b ranges from -3 to 3 inclusive. That's 7 values. Therefore, for a=4 and a=-4, total 7*2=14.

For a=5: a²=25. So b² ≤25 -25=0. Therefore, b²=0, so b=0. Therefore, only one value. Therefore, for a=5 and a=-5, total 1*2=2.

Wait, but for a=0, we have 11 b's. Let me check again:

Total number of ordered pairs would be:

For a=0: 11

For a=±1: 9*2=18

For a=±2: 9*2=18

For a=±3: 9*2=18

For a=±4: 7*2=14

For a=±5: 1*2=2

Wait, adding these up:

11 + 18 + 18 + 18 +14 + 2. Let me compute:

Start with 11 + 18 = 29

29 + 18 = 47

47 + 18 = 65

65 + 14 = 79

79 + 2 = 81.

Wait, so total number of ordered pairs is 81? But wait, let me check again. Because maybe I made a mistake here.

Wait, hold on. Let me check again for each a:

For a=0: 11 (b from -5 to 5)

For a=1: 9 (b from -4 to 4)

But when I multiply by 2 for a=1 and a=-1, that would be 9*2=18

Similarly for a=2: same as 1? Wait, but when a=2, b ranges from -4 to 4? Wait, but a² + b² ≤25. For a=2, b² ≤21, so b can be -4, -3,..., 3, 4. Which is 9 values. So yes, same as a=1. So yes, same count.

Similarly for a=3: b² ≤25 -9=16. So b from -4 to 4? Wait, no. Wait sqrt(16)=4. So b can be from -4 to 4 inclusive. Wait, but 4²=16. So 2² + 4² = 4 + 16 = 20 ≤25. Yes, that's okay. So yes, same as before. So for a=3, same 9 values for b. So same count.

For a=4: b² ≤25 - 16 =9. So b from -3 to 3 inclusive. Which is 7 values. So 7*2=14 for a=±4.

For a=5: b² ≤0. Only b=0. So 1*2=2 for a=±5.

Adding up:

a=0: 11

a=±1: 18

a=±2: 18

a=±3: 18

a=±4:14

a=±5:2

Total: 11 + 18 + 18 +18 +14 +2 = let's compute:

Start with 11 + 18 = 29

29 + 18 = 47

47 + 18 = 65

65 +14=79

79 +2=81.

So total 81 ordered pairs. Wait, but I'm supposed to find the number of ordered pairs (a,b) where |a + bi| ≤5, which is equivalent to a² + b² ≤25. So according to this calculation, the answer is 81. But let me check if this is correct. Let me verify with some known values.

First, when a and b are both zero, that's one point. For a=0, we have 11 points. Let me check if that's correct. For a=0, b ranges from -5 to 5, inclusive, which is 11 values. Correct.

Now, for example, when a=1, b ranges from -4 to 4. Let me check if 1² +4² =1 +16=17≤25. Yes. And 1² +5²=1 +25=26>25. So that's correct. Hence, b can only go up to 4. Similarly for negative b.

Similarly for a=2: 2² + 4²=4 +16=20≤25. 2² +5²=4 +25=29>25. So yes, same thing.

For a=3: 3² + 4²=9 +16=25. Which is equal to 25. So 3² +4²=25. Therefore, b=4 is allowed. So for a=3, b can be from -4 to 4. Therefore, same count as before.

For a=4: 4² + b² ≤25. So b² ≤9, so b from -3 to 3. Which is 7 values. Therefore, 4² +3²=16 +9=25. So that's okay. So that is correct.

For a=5: only b=0. So (5,0), (-5, 0). So 2 points. Correct.

So adding up the numbers seems right. But let me check another way. Maybe compute the number of points in each quadrant and multiply accordingly. Let's try that approach.

First, consider the first quadrant where a ≥0 and b ≥0. Then multiply by 4, and then subtract overlaps. But maybe that's more complex. Let me try.

In the first quadrant (including axes), the number of points is the number of (a,b) with a ≥0, b ≥0, and a² +b² ≤25. Then total number of points is 4*(number in first quadrant) - overlaps. Wait, actually, the standard formula is that the total number is 4*(number in first quadrant excluding axes) + 2*(number on axes excluding origin) +1 (origin). Wait, maybe this is getting too complicated. Alternatively, since my previous calculation gave 81, perhaps I can check if that seems reasonable.

Alternatively, maybe I can compute the number of points in the first quadrant (including axes) and then multiply by 4, adjusting for the axes. Let me try.

Let me compute the number of non-negative integer solutions (a,b) with a ≥0, b ≥0, and a² +b² ≤25. Let me compute that.

So for a from 0 to 5, and for each a, compute the number of b's from 0 to floor(sqrt(25 -a²)).

So:

For a=0: b can be from 0 to 5 (since 0² +5²=25). So 6 values (0,1,2,3,4,5).

For a=1: b² ≤25 -1=24. So b ≤4. So b from 0 to 4: 5 values.

For a=2: b² ≤21. So b ≤4 (since 4²=16 ≤21, 5²=25>21). So b from 0 to 4: 5 values.

For a=3: b² ≤16. So b from 0 to 4 (since 4²=16). Wait, sqrt(16)=4. So b can be 0 to 4. So 5 values.

For a=4: b² ≤25 -16=9. So b from 0 to 3. Because 3²=9, 4²=16>9. So 4 values (0,1,2,3).

For a=5: b² ≤25 -25=0. So b=0. 1 value.

Adding these up:

a=0: 6

a=1:5

a=2:5

a=3:5

a=4:4

a=5:1

Total for first quadrant (including axes): 6+5+5+5+4+1= 26.

Now, how does this relate to the total number of ordered pairs?

The total number of points in all quadrants can be calculated as follows:

- The origin (0,0): counted once in first quadrant.

- Points on the axes (excluding origin): For example, points where a=0, b≠0 or b=0, a≠0. These are counted once in the first quadrant, but need to be multiplied by 2 (for positive and negative). Except for when a or b is zero.

Wait, maybe a better way is:

The total number of points is equal to:

4*(number of points in first quadrant excluding axes) + 2*(number of points on axes excluding origin) + 1*(origin).

So let me compute that.

First, number of points in first quadrant excluding axes: this would be points where a>0 and b>0.

So in the first quadrant, total points are 26. From this, subtract the points on the axes.

Points on the axes in first quadrant:

- On the a-axis (b=0): for a from 0 to5, which is 6 points (a=0 to 5, b=0).

- On the b-axis (a=0): for b from 0 to5, which is 6 points (a=0, b=0 to5).

But the origin is counted in both, so total points on axes in first quadrant: 6 +6 -1=11.

Therefore, points in first quadrant excluding axes: 26 -11=15.

So total number of points:

4*15 (for the four quadrants) + 2*11 (for the axes excluding origin) +1 (origin). Wait, no. Wait, if we have 15 points in the first quadrant excluding axes, then each of those points has counterparts in other quadrants. So 4*15.

Then, the points on the axes excluding origin: each of these points is on an axis, so they have symmetric counterparts. For example, (a,0) where a≠0 has a counterpart (-a, 0). Similarly for (0,b) with b≠0. So the number of points on the axes excluding origin is 11 (as computed: 6 on a-axis including origin, 6 on b-axis including origin, subtract 1 for origin, gives 11). But each of these points has a mirror image except for when they are on both axes (i.e., the origin, which is already counted). Wait, actually, the points on the axes excluding origin: for example, (a,0) where a≠0. Each such point has a mirror image (-a,0). Similarly for (0,b) where b≠0. So total number of such points is 2*(number of points on a-axis excluding origin) + 2*(number of points on b-axis excluding origin)? Wait no, because the points on the axes are already considered. Wait, maybe better:

Number of points on the a-axis (excluding origin) is 5 (a=±1, ±2, ±3, ±4, ±5). Wait, no, in the first quadrant, we had 6 points on a-axis (from 0 to5). So excluding origin, it's 5 points on a-axis (a=1,...,5). Similarly, 5 points on b-axis (b=1,...,5). So total points on axes excluding origin: 5 +5=10. But in the first quadrant, the axes include origin, so when we calculated first quadrant points on axes as 11 (including origin), so excluding origin is 10. Therefore, total number of points on the axes (excluding origin) is 10. But each of these points has a mirror image on the other side. Therefore, the total number of points on the axes (excluding origin) is 10, but when counting total points, these points are in four quadrants? No, because they are on axes. For example, the point (1,0) is only on the a-axis. Its mirror image is (-1, 0). So total number of such points is 2*number of points on a-axis excluding origin + 2*number of points on b-axis excluding origin? Wait, no. For each point on the a-axis (excluding origin), there is a corresponding point on the negative a-axis. Similarly for b-axis. Therefore, total number of points on the axes (excluding origin) is 2*(number on positive a-axis) + 2*(number on positive b-axis). Wait, if we have 5 points on positive a-axis (a=1,...,5), then negative a-axis also has 5 points, so total 10 points on a-axis. Similarly for b-axis: 10 points. Wait, but that would be 20 points. But actually, points on a-axis are (±1,0), (±2,0), ..., (±5,0): total 10 points. Similarly, points on b-axis are (0, ±1), (0,±2), ..., (0, ±5): another 10 points. So total of 20 points on the axes excluding origin. But in the first quadrant, we accounted for 5 points on a-axis (excluding origin) and 5 points on b-axis (excluding origin). So total of 10 points on axes excluding origin in first quadrant.

So going back, if we have in total:

Total number of points = 4*(points in first quadrant excluding axes) + 2*(points on axes excluding origin) + 1*(origin). Wait, no. Wait, if we consider that points in first quadrant excluding axes are counted once, but each of them has 3 symmetric counterparts (but actually, for each point (a,b) with a>0 and b>0, there are 3 other quadrants it can be in: (a,-b), (-a,b), (-a,-b). So 4 points total, but since we have already counted the first quadrant ones, multiplying by 4 gives all four quadrants.

But the points on the axes (excluding origin) are already on the axes, so each such point has only one symmetric counterpart. For example, (a,0) with a>0 has (-a,0). Similarly, (0,b) with b>0 has (0,-b). Therefore, each of these points (on axes) has one mirror image. So the total number of such points is 2*(number of points on positive a-axis) + 2*(number of points on positive b-axis). Wait, no. The total number of points on the axes excluding origin is 20, as above. But in the first quadrant, we have 10 points (5 on a-axis, 5 on b-axis). Then, if we multiply by 2 for the other quadrants? Wait, this is getting confusing. Maybe an alternative approach:

Total number of points = points in first quadrant (including axes) * 4 - points on the axes (excluding origin)*2 - origin*3? Wait, no. Maybe it's easier to use inclusion-exclusion.

Alternatively, since we have already calculated the number of points in first quadrant as 26, and we can calculate the total number of points as follows:

Each point not on an axis is counted 4 times in the first quadrant. Each point on the axis (excluding origin) is counted 2 times in the first quadrant. And the origin is counted once.

So, let me denote:

Let N be the total number of points. Let me denote:

Let A be the number of points in first quadrant excluding axes: 15 (as above)

Let B be the number of points on axes excluding origin in first quadrant: 10 (5 on a-axis, 5 on b-axis)

Let C be the origin:1

So total N = 4*A + 2*B + C

Therefore:

N = 4*15 + 2*10 +1 = 60 +20 +1=81. Which matches the previous total of 81. So that's good, this cross-verification shows that 81 is correct.

Therefore, the answer is 81. But just to be thorough, let me check with another approach.

Alternatively, I can compute for each radius from 0 to 5, the number of integer points on the circle of radius r, but since we are looking for all points inside or on the circle, it's the sum over r from 0 to 5 of the number of integer points on the circle of radius r. Wait, no, actually, that would not be correct because the points inside the circle are not just the points on the circles of radius 0 to 5, but all points with a² + b² ≤25. Maybe another way is to use Gauss circle problem, but that's approximate. However, since we can compute exactly here, maybe not necessary.

Alternatively, let me check for some smaller radii and see if counts make sense. For example, when radius is 0: only (0,0). So 1. For radius 1: all points with a² + b² ≤1. Which are (0,0), (0,±1), (±1,0). So total 1 + 4 = 5. According to formula above, if we do the same method:

First quadrant points:

a=0: b=0,1 => 2 points

a=1: b=0 => 1 point

Total first quadrant: 2 +1=3

Then total points: 4*(3 - 2) + 2*(number on axes excluding origin) +1. Wait, maybe not. Wait, in this case:

Using the formula N = 4*A + 2*B + C where A = points in first quadrant excluding axes. For radius 1, first quadrant points (including axes): 3. Points on axes in first quadrant: 2 (a=0,b=0,1 and a=1,b=0). So points on axes excluding origin: 2 -1=1 (i.e., (0,1) and (1,0) minus origin). Wait, perhaps not. Let me recast:

For radius 1, first quadrant (including axes) points:

a=0: b=0,1 (2 points)

a=1: b=0 (since 1² +0²=1 ≤1, b=0. b=1 would be 1+1=2>1. So only b=0. So total 1 point.

Total first quadrant points: 2 +1=3.

Points on axes in first quadrant:

a-axis: (0,0), (0,1), (1,0). Wait, no. Wait for a ≥0, b ≥0:

Points on a-axis (b=0): (0,0), (1,0). Similarly, points on b-axis (a=0): (0,0), (0,1). So total points on axes in first quadrant: 3 ( (0,0), (1,0), (0,1)).

Therefore, points on axes excluding origin: 2 points: (1,0) and (0,1).

Therefore:

A (points in first quadrant excluding axes): total first quadrant points (3) minus points on axes (3) plus origin? Wait, no. Wait, points in first quadrant excluding axes would be points where a>0 and b>0. In this case, with radius 1, a and b can be 0 or 1. The only point with a>0 and b>0 would be (1,1), but 1² +1²=2>1, which is outside. So A=0. Therefore:

N = 4*A + 2*B + C = 4*0 + 2*2 +1= 0 +4 +1=5. Which is correct. So 5 points total. Therefore, my formula works here.

So applying this formula for the original problem gives 81. Hence, this cross-check seems to validate that the count is correct.

But to be absolutely sure, maybe let me check another small radius. Let's take radius sqrt(2), but in integers, the points with a² + b² ≤2.

Points are:

(0,0), (0,±1), (0,±2), (±1,0), (±1,±1), (±2,0). Let's count:

(0,0): 1

(0,±1): 2

(0,±2): 2

(±1,0): 2

(±1,±1): 4

(±2,0): 2

Total: 1 + 2 +2 +2 +4 +2=13? Wait:

Wait, let's count properly:

- (0,0): 1

- (0,1), (0,-1): 2

- (0,2), (0,-2): 2

- (1,0), (-1,0): 2

- (1,1), (1,-1), (-1,1), (-1,-1): 4

- (2,0), (-2,0): 2

Total: 1 +2+2+2+4+2= 13. Now, using our formula:

First quadrant points (including axes):

For a=0: b=0,1,2 => 3

For a=1: b=0,1 (since 1 + b² ≤2 => b² ≤1 => b=0,1). So 2

For a=2: b=0 (since 4 +0 ≤2? No, 4 >2. So b must be 0. Wait 2² +0=4>2. So b has to be 0? No, a=2 can't be because 2²=4>2. Hence, for a=2, there are no b's. Therefore, first quadrant points are:

a=0: 3

a=1:2

a=2:0

Total first quadrant points: 5

Points on axes in first quadrant:

a-axis (b=0): a=0,1,2. But b=0. Wait, a ranges from 0 to... in first quadrant, a and b non-negative. So for a-axis (b=0): a can be 0,1,2? Wait a² +0 ≤2 => a² ≤2 => a can be 0,1. Because 2²=4>2. So a=0,1. Similarly for b-axis. So points on axes in first quadrant: (0,0), (0,1), (0,2)? Wait no, (0,2) would have b=2, which is allowed because 0² +2²=4>2. Wait no, original radius squared is 2. So a² +b² ≤2. For (0,2): 0+4=4>2. So not allowed. Therefore, points on axes in first quadrant:

a-axis: (0,0), (1,0)

b-axis: (0,0), (0,1)

So total points on axes in first quadrant: 3 ( (0,0), (1,0), (0,1) )

Therefore, points on axes excluding origin: 2 ( (1,0), (0,1) )

Hence, A (points in first quadrant excluding axes): total first quadrant points (5) minus points on axes in first quadrant (3) = 2? Wait no. Wait points in first quadrant excluding axes are those where a>0 and b>0. Which would be (1,1). But (1,1) has a² +b²=2, which is allowed. So that's 1 point. So A=1. Then:

N=4*A + 2*B + C = 4*1 + 2*2 +1= 4 +4 +1=9? But earlier count was 13. Wait discrepancy here. So something is wrong.

Wait, according to my previous calculation, we had 13 points, but according to formula N=4*A +2*B + C= 4*1 + 2*2 +1=9. Which is wrong. So where is the mistake?

Let me recast. Let me redo the formula:

Total points = 4*(points in first quadrant excluding axes) + 2*(points on axes excluding origin) + 1*(origin).

But in this case, points in first quadrant excluding axes: points where a>0 and b>0. For our case with radius squared 2, that's (1,1). So 1 point. Points on axes excluding origin: (1,0), (0,1). So 2 points. Origin: 1. Therefore:

Total points = 4*1 + 2*2 +1 = 4 +4 +1=9. But actual count was 13. So clearly something is wrong with the formula.

Wait, what is happening here. Maybe my formula is incorrect. Let me think again.

Alternative approach:

Each point not on an axis is counted 4 times in the first quadrant. Each point on an axis but not origin is counted 2 times in the first quadrant. The origin is counted once.

But in the first quadrant, the total number of points is:

Number of points in first quadrant = number of points not on axes + number of points on axes in first quadrant.

Therefore:

Total points not on axes =

Number of points in first quadrant - number of points on axes in first quadrant.

Then, total points in all quadrants:

Total points not on axes: 4*(number in first quadrant - number on axes in first quadrant)

Plus total points on axes (excluding origin): 2*(number on positive axes)

Plus origin: 1

Wait, let me define:

Let’s denote:

- Let Q be the number of points in the first quadrant (including axes).

- Let A be the number of points on the axes in the first quadrant (including origin).

- Let B be the number of points on the positive axes (excluding origin). So A = 1 + 2*B? Wait, no. For example, in the case of radius 2:

In first quadrant, points on axes: (0,0), (1,0), (0,1). So A=3. The number of points on positive axes (excluding origin) is 2: (1,0) and (0,1). So B=2. Then:

Total points not on axes in first quadrant: Q - A = 5 -3=2. But in reality, there is only 1 point not on axes: (1,1). So discrepancy here.

Wait, maybe my definitions are conflicting. Let me try again.

Let me define:

In the first quadrant (including axes):

- The number of points not on the axes: let's call this C.

- The number of points on the axes (including origin): let's call this D.

Then, Q = C + D.

Total points in all quadrants:

The points not on the axes: each such point has been counted once in first quadrant, and there are 4 such points in each quadrant. So total points not on axes: 4*C.

The points on the axes (excluding origin): each such point has been counted once in first quadrant, and each has a mirror image on the opposite axis. For example, (1,0) has mirror (-1,0), so total 2 points. Similarly for (0,1) and (0,-1). Therefore, total points on axes (excluding origin): 2*D' where D' is the number of points on positive axes (excluding origin). Wait, if D includes the origin, then D' = D -1. For example, in our radius 2 example:

D = number of points on axes in first quadrant: 3 (origin, (1,0), (0,1))

So D' = D -1 = 2 (points on positive axes excluding origin).

Therefore, total points on axes excluding origin: 2*D' (because each point on positive axis has a mirror on negative axis). So 2*2=4.

Therefore, total number of points:

Total = 4*C + 2*D' +1 (origin)

In the case of radius 2 example, Q = 5, D =3, D'=2, C=Q - D=5-3=2.

But in reality, the number of points not on the axes in first quadrant is 1 ((1,1)), but according to C=2. Where is the mistake?

Ah! Wait, Q counts all points in first quadrant, including axes. So if Q = 5, then D = number of points on axes in first quadrant. Wait in our case, with radius squared 2:

First quadrant points (including axes):

a=0: b=0,1 (since 0² + b² ≤2 => b² ≤2 => b=0,1. Wait (0,1) gives 1, which is ≤2. But also (0, sqrt(2)), but b must be integer. So yes, (0,0), (0,1). For a=0: 2 points.

a=1: b=0,1 (because 1 + b² ≤2 => b² ≤1 => b=0,1). So (1,0), (1,1). So 2 points.

a=2: b² ≤2 -4= -2? No, a=2, a²=4>2, so no points.

Total first quadrant points: 2 + 2 =4. Wait earlier I thought it was 5. Wait let me recount:

For a=0: b=0,1 => 2 points

For a=1: b=0,1 => 2 points

For a=2: 0 points

Total first quadrant points: 4. So earlier mistake was in thinking it was 5. So Q=4.

Then D is the number of points on axes in first quadrant. For axes:

On a-axis: (0,0), (1,0). On b-axis: (0,0), (0,1). So total points on axes in first quadrant: 3. So D=3. Hence, C=Q - D=4 -3=1. Which is correct: point (1,1).

Therefore, total points:

Total = 4*C + 2*(D -1) +1 = 4*1 + 2*(3 -1) +1=4 + 4 +1=9. Which matches the formula. However, when I counted manually earlier, I had 13 points. Wait, let me recount manually.

For a² +b² ≤2:

Possible points:

(0,0): 1

(0, ±1): 2

(0, ±2): but 0 + 4 =4>2, so no.

(±1, 0): 2

(±1, ±1): 4 points (1,1), (1,-1), (-1,1), (-1,-1)

(±2, 0): 2 points (2,0), (-2,0), but 4>2, so no. Wait (2,0) gives 4>2, so invalid. So only (±1,0), (±1, ±1), (0,±1), and (0,0).

Therefore total points:

(0,0):1

(±1,0):2

(0, ±1):2

(±1, ±1):4

Total: 1 +2 +2 +4=9. Which matches the formula. Earlier, I had mistakenly included (0, ±2) and (±2,0), which are invalid. So my manual count was wrong before. Therefore, formula works.

Therefore, this gives me more confidence that the formula is correct. Thus, returning to our original problem, where the total was 81, seems correct.

But just to be thorough, let me check for another radius. For example, radius 1, as before:

First quadrant points: for a² + b² ≤1:

a=0: b=0,1 => 2 points

a=1: b=0 => 1 point

Total Q=3

D: points on axes in first quadrant: (0,0), (1,0), (0,1). D=3. Hence, C= Q - D= 0.

Total points: 4*0 + 2*(D -1) +1=0 + 2*(3 -1) +1= 4 +1=5, which is correct.

So the formula works. Hence, returning to our original problem, where we calculated via two methods:

First method: 11 + 18 + 18 +18 +14 +2 =81.

Second method: using first quadrant points: total of 26, which led to N=81.

Therefore, confident that 81 is correct.

But just for absolute surety, let me check for radius 5. For example, how many points are there with a² +b² ≤25?

Alternatively, I can think of the number of points as follows:

For each a from -5 to 5, compute the number of b's.

As I did earlier:

a=-5: b=0 => 1

a=-4: b=-3,...,3 =>7

a=-3: b=-4,...,4 =>9

a=-2: same as +2 =>9

a=-1: same as +1 =>9

a=0: 11

a=1:9

a=2:9

a=3:9

a=4:7

a=5:1

So adding these up:

For a=-5:1

a=-4:7

a=-3:9

a=-2:9

a=-1:9

a=0:11

a=1:9

a=2:9

a=3:9

a=4:7

a=5:1

Total:

Let me add them sequentially:

Start with a=-5:1

a=-4:1+7=8

a=-3:8+9=17

a=-2:17+9=26

a=-1:26+9=35

a=0:35+11=46

a=1:46+9=55

a=2:55+9=64

a=3:64+9=73

a=4:73+7=80

a=5:80+1=81.

Yes, same total. Hence, 81.

Therefore, after multiple approaches and cross-verifications, I believe the correct answer is 81.

**Final Answer**
\boxed{81}

To find the number of ordered pairs $(a, b)$ of integers such that $ |a + bi| \leq 5 $, we proceed by analyzing the condition that gives rise to this inequality.

---

### Step 1: Understanding the Inequality

For a complex number $ z = a + bi $, the modulus is defined as:

$$
|z| = \sqrt{a^2 + b^2}
$$

So the inequality $ |a + bi| \leq 5 $ becomes:

$$
\sqrt{a^2 + b^2} \leq 5
\Rightarrow a^2 + b^2 \leq 25
$$

We are to count all integer pairs $(a, b)$ such that $ a^2 + b^2 \leq 25 $.

---

### Step 2: Counting Lattice Points Inside a Circle

The inequality $ a^2 + b^2 \leq 25 $ represents all integer lattice points (points with integer coordinates) located inside or on the circle of radius 5 centered at the origin.

To count such points, we can iterate through all possible integer values of $ a $, and for each, determine the range of possible values of $ b $ that satisfy the inequality.

---

### Step 3: Iterating Through All Integer Values of $ a $

We observe that:

$$
a^2 \leq 25 \Rightarrow -5 \leq a \leq 5
$$

So we evaluate the number of valid $ b $ values for each $ a $ from $-5$ to $ 5 $, and sum up the counts.

For each $ a $:

- Compute the maximum value of $ b $ such that $ b^2 \leq 25 - a^2 $
- The number of possible $ b $ values corresponding to that range is:

$$
\text{Number of valid } b = 2 \times \lfloor \sqrt{25 - a^2} \rfloor + 1
$$

Let’s go through this:

| $ a $ | $ a^2 $ | $ 25 - a^2 $ | $ \sqrt{25 - a^2} $ | $ \lfloor \sqrt{25 - a^2} \rfloor $ | $ \text{count of } b $ |
|--------|----------|------------------|------------------------|--------------------------------------|-------------------------------|
| -5     | 25       | 0                | 0                      | 0                                   | 1                             |
| -4     | 16       | 9                | 3                      | 3                                   | 7                             |
| -3     | 9        | 16               | 4                      | 4                                   | 9                             |
| -2     | 4        | 21               | 4.57                   | 4                                   | 9                             |
| -1     | 1        | 24               | 4.89                   | 4                                   | 9                             |
| 0      | 0        | 25               | 5                      | 5                                   | 11                            |
| 1      | 1        | 24               | 4.89                   | 4                                   | 9                             |
| 2      | 4        | 21               | 4.57                   | 4                                   | 9                             |
| 3      | 9        | 16               | 4                      | 4                                   | 9                             |
| 4      | 16       | 9                | 3                      | 3                                   | 7                             |
| 5      | 25       | 0                | 0                      | 0                                   | 1                             |

Now sum up the counts:

$$
1 + 7 + 9 + 9 + 9 + 11 + 9 + 9 + 9 + 7 + 1 = 81
$$

---

### Conclusion

Therefore, the number of ordered pairs $(a, b)$ of integers such that $ |a + bi| \leq 5 $ is:

$$
\boxed{81}
$$

[Assistant]
Explanation:
Okay, so I need to find the number of ordered pairs (a, b) of integers such that the modulus of a + bi is less than or equal to 5. Let me recall what the modulus of a complex number is. For a complex number a + bi, the modulus is sqrt(a² + b²). So the condition given is sqrt(a² + b²) ≤ 5.

To simplify, I can square both sides of the inequality to get rid of the square root. That would give me a² + b² ≤ 25. So the problem reduces to finding all integer pairs (a, b) such that the sum of their squares is less than or equal to 25.

Now, since a and b are integers, I need to consider all integer values of a and b where their squares add up to 25 or less. This seems like a problem where I can count the number of integer solutions (a, b) for each possible value of a² + b² from 0 up to 25. But maybe there's a smarter way than checking each possible value? Let me think.

Alternatively, I can think geometrically. The equation a² + b² = 25 represents a circle with radius 5 centered at the origin. The problem is asking for all integer points (lattice points) inside or on this circle. So I need to count all such lattice points.

To do this, I can fix the value of a and find the corresponding possible values of b. Since a and b are symmetric in the equation, maybe I can compute the number of points for non-negative a and b and then account for symmetry? Wait, but since a and b can be positive or negative, I need to consider all combinations. However, maybe I can compute the number of solutions for a ≥ 0 and b ≥ 0 and then multiply accordingly? Let me try.

But first, maybe I should note that for each integer a, there is a maximum value of b such that a² + b² ≤ 25. So for each integer a, I can compute the range of b that satisfies the inequality. Let me try to approach this step by step.

First, let me note that since a² and b² are both non-negative, their sum can range from 0 (when a = 0 and b = 0) up to 25. So the possible values of a can range from -5 to 5 inclusive because if |a| > 5, then a² would already be greater than 25, making a² + b² > 25 regardless of b. Similarly for b. So the possible integer values for a and b are from -5 to 5.

But maybe I can start by considering a from 0 to 5 and then account for the signs? Let me try that approach. For each non-negative integer a from 0 to 5, I can find the number of integers b such that b² ≤ 25 - a². Then, since for each a, b can be positive, negative, or zero, I can calculate the number of b's accordingly. But I need to be careful with zero, because if b is zero, it's only one value, not two.

Alternatively, maybe it's better to think of each a from -5 to 5 and for each a, compute the possible b's. However, since the problem is symmetric with respect to a and b, maybe we can exploit that symmetry. Let me see.

Alternatively, since the modulus is symmetric in a and b, maybe I can compute the number of points in the first quadrant (including axes) and then multiply by 4, adjusting for overcounting when a or b is zero. But this might complicate things. Let me try the first approach.

Let me start by considering all integer values of a from -5 to 5. For each a, compute the maximum value of b such that a² + b² ≤ 25. Then for each such a, count the number of b's. Let me proceed step by step.

First, let's consider a = 0. Then the equation becomes 0 + b² ≤ 25, so b² ≤ 25. Therefore, b can range from -5 to 5 inclusive. That's 11 values (from -5, -4,...,0,...,4,5). So for a=0, there are 11 possible b's.

Next, a = 1. Then a² = 1, so b² ≤ 25 - 1 = 24. Therefore, b² ≤ 24. The square root of 24 is approximately 4.898. So b can range from -4 to 4, because 4²=16 ≤24 and 5²=25>24. Wait, but sqrt(24) is approx 4.898, so the maximum integer b is 4. So b can be from -4 to 4 inclusive. How many integers is that? From -4 to 4 inclusive, that's 9 numbers (since 4 - (-4) +1 = 9). Wait, actually, the number of integers between -n and n inclusive is 2n +1. So if n=4, that's 9. So for a=1, 9 values of b.

But wait, wait, wait. Wait, 25 - a² is 24, so b² ≤ 24. So b can be integers such that |b| ≤ floor(sqrt(24)). sqrt(24) is approx 4.898, so floor of that is 4. Hence, yes, b can be from -4 to 4 inclusive. So 9 values.

Similarly for a = -1, since a² is the same as for a=1, so same result. So for each a, the number of b's depends only on |a|. Therefore, maybe I can compute for a from 0 to 5 and then multiply by the number of sign possibilities.

But let's proceed step by step for each a from -5 to 5, but since a² is same for a and -a, maybe I can compute for a ≥ 0 and then double the count for each a except when a=0.

Wait, perhaps let's group the values. Let me note that for each a in 0 to 5, the number of b's will be the same for a and -a. Except for a=0, which is symmetric. So maybe compute for a from 0 to 5, then multiply by 2 for each a except a=0. But also, need to think about how the b's are counted. Let me check:

For a=0, the number of b's is 11. For other a's (1 to 5), for each a, the number of b's is computed, then multiplied by 2 (for positive and negative a). However, for each a ≠ 0, when we consider positive and negative a, their respective b's are still same count, right? Because if a is positive or negative, b can still range from -floor(sqrt(25 - a²)) to floor(sqrt(25 - a²)), so same number. Therefore, maybe I can calculate for a from 0 to 5, and then multiply by 2 for each a from 1 to 5, and add the count for a=0.

But let me do this step by step. Let me first calculate for each a from 0 to 5:

For a = 0: as above, 11 values of b.

For a = 1: b² ≤ 25 - 1 = 24. So b ranges from -4 to 4 inclusive, which is 9 values. So for a = 1 and a = -1, each has 9 values of b. So total for a=±1 is 9*2=18.

For a = 2: a² = 4. So b² ≤ 25 -4 =21. sqrt(21) ≈4.583. So floor is 4. Therefore, b ranges from -4 to 4. So again, 9 values. So for a=2 and a=-2, total 9*2=18.

Wait, same as a=1? Wait, sqrt(21) is less than sqrt(24). Wait, but even though 21 is less than 24, floor(sqrt(21)) is still 4, because 4²=16 ≤21, and 5²=25>21. So yes, same range of b's. Hence same count. So 9 values.

Wait, but then for a=3: a²=9. So b² ≤25 -9=16. So sqrt(16)=4. Therefore, b ranges from -4 to 4. So again 9 values. So for a=3 and -3, again 9*2=18.

Wait, same as before. For a=4: a²=16. So b² ≤25 -16=9. sqrt(9)=3. Therefore, b ranges from -3 to 3 inclusive. That's 7 values. Therefore, for a=4 and a=-4, total 7*2=14.

For a=5: a²=25. So b² ≤25 -25=0. Therefore, b²=0, so b=0. Therefore, only one value. Therefore, for a=5 and a=-5, total 1*2=2.

Wait, but for a=0, we have 11 b's. Let me check again:

Total number of ordered pairs would be:

For a=0: 11

For a=±1: 9*2=18

For a=±2: 9*2=18

For a=±3: 9*2=18

For a=±4: 7*2=14

For a=±5: 1*2=2

Wait, adding these up:

11 + 18 + 18 + 18 +14 + 2. Let me compute:

Start with 11 + 18 = 29

29 + 18 = 47

47 + 18 = 65

65 + 14 = 79

79 + 2 = 81.

Wait, so total number of ordered pairs is 81? But wait, let me check again. Because maybe I made a mistake here.

Wait, hold on. Let me check again for each a:

For a=0: 11 (b from -5 to 5)

For a=1: 9 (b from -4 to 4)

But when I multiply by 2 for a=1 and a=-1, that would be 9*2=18

Similarly for a=2: same as 1? Wait, but when a=2, b ranges from -4 to 4? Wait, but a² + b² ≤25. For a=2, b² ≤21, so b can be -4, -3,..., 3, 4. Which is 9 values. So yes, same as a=1. So yes, same count.

Similarly for a=3: b² ≤25 -9=16. So b from -4 to 4? Wait, no. Wait sqrt(16)=4. So b can be from -4 to 4 inclusive. Wait, but 4²=16. So 2² + 4² = 4 + 16 = 20 ≤25. Yes, that's okay. So yes, same as before. So for a=3, same 9 values for b. So same count.

For a=4: b² ≤25 - 16 =9. So b from -3 to 3 inclusive. Which is 7 values. So 7*2=14 for a=±4.

For a=5: b² ≤0. Only b=0. So 1*2=2 for a=±5.

Adding up:

a=0: 11

a=±1: 18

a=±2: 18

a=±3: 18

a=±4:14

a=±5:2

Total: 11 + 18 + 18 +18 +14 +2 = let's compute:

Start with 11 + 18 = 29

29 + 18 = 47

47 + 18 = 65

65 +14=79

79 +2=81.

So total 81 ordered pairs. Wait, but I'm supposed to find the number of ordered pairs (a,b) where |a + bi| ≤5, which is equivalent to a² + b² ≤25. So according to this calculation, the answer is 81. But let me check if this is correct. Let me verify with some known values.

First, when a and b are both zero, that's one point. For a=0, we have 11 points. Let me check if that's correct. For a=0, b ranges from -5 to 5, inclusive, which is 11 values. Correct.

Now, for example, when a=1, b ranges from -4 to 4. Let me check if 1² +4² =1 +16=17≤25. Yes. And 1² +5²=1 +25=26>25. So that's correct. Hence, b can only go up to 4. Similarly for negative b.

Similarly for a=2: 2² + 4²=4 +16=20≤25. 2² +5²=4 +25=29>25. So yes, same thing.

For a=3: 3² + 4²=9 +16=25. Which is equal to 25. So 3² +4²=25. Therefore, b=4 is allowed. So for a=3, b can be from -4 to 4. Therefore, same count as before.

For a=4: 4² + b² ≤25. So b² ≤9, so b from -3 to 3. Which is 7 values. Therefore, 4² +3²=16 +9=25. So that's okay. So that is correct.

For a=5: only b=0. So (5,0), (-5, 0). So 2 points. Correct.

So adding up the numbers seems right. But let me check another way. Maybe compute the number of points in each quadrant and multiply accordingly. Let's try that approach.

First, consider the first quadrant where a ≥0 and b ≥0. Then multiply by 4, and then subtract overlaps. But maybe that's more complex. Let me try.

In the first quadrant (including axes), the number of points is the number of (a,b) with a ≥0, b ≥0, and a² +b² ≤25. Then total number of points is 4*(number in first quadrant) - overlaps. Wait, actually, the standard formula is that the total number is 4*(number in first quadrant excluding axes) + 2*(number on axes excluding origin) +1 (origin). Wait, maybe this is getting too complicated. Alternatively, since my previous calculation gave 81, perhaps I can check if that seems reasonable.

Alternatively, maybe I can compute the number of points in the first quadrant (including axes) and then multiply by 4, adjusting for the axes. Let me try.

Let me compute the number of non-negative integer solutions (a,b) with a ≥0, b ≥0, and a² +b² ≤25. Let me compute that.

So for a from 0 to 5, and for each a, compute the number of b's from 0 to floor(sqrt(25 -a²)).

So:

For a=0: b can be from 0 to 5 (since 0² +5²=25). So 6 values (0,1,2,3,4,5).

For a=1: b² ≤25 -1=24. So b ≤4. So b from 0 to 4: 5 values.

For a=2: b² ≤21. So b ≤4 (since 4²=16 ≤21, 5²=25>21). So b from 0 to 4: 5 values.

For a=3: b² ≤16. So b from 0 to 4 (since 4²=16). Wait, sqrt(16)=4. So b can be 0 to 4. So 5 values.

For a=4: b² ≤25 -16=9. So b from 0 to 3. Because 3²=9, 4²=16>9. So 4 values (0,1,2,3).

For a=5: b² ≤25 -25=0. So b=0. 1 value.

Adding these up:

a=0: 6

a=1:5

a=2:5

a=3:5

a=4:4

a=5:1

Total for first quadrant (including axes): 6+5+5+5+4+1= 26.

Now, how does this relate to the total number of ordered pairs?

The total number of points in all quadrants can be calculated as follows:

- The origin (0,0): counted once in first quadrant.

- Points on the axes (excluding origin): For example, points where a=0, b≠0 or b=0, a≠0. These are counted once in the first quadrant, but need to be multiplied by 2 (for positive and negative). Except for when a or b is zero.

Wait, maybe a better way is:

The total number of points is equal to:

4*(number of points in first quadrant excluding axes) + 2*(number of points on axes excluding origin) + 1*(origin).

So let me compute that.

First, number of points in first quadrant excluding axes: this would be points where a>0 and b>0.

So in the first quadrant, total points are 26. From this, subtract the points on the axes.

Points on the axes in first quadrant:

- On the a-axis (b=0): for a from 0 to5, which is 6 points (a=0 to 5, b=0).

- On the b-axis (a=0): for b from 0 to5, which is 6 points (a=0, b=0 to5).

But the origin is counted in both, so total points on axes in first quadrant: 6 +6 -1=11.

Therefore, points in first quadrant excluding axes: 26 -11=15.

So total number of points:

4*15 (for the four quadrants) + 2*11 (for the axes excluding origin) +1 (origin). Wait, no. Wait, if we have 15 points in the first quadrant excluding axes, then each of those points has counterparts in other quadrants. So 4*15.

Then, the points on the axes excluding origin: each of these points is on an axis, so they have symmetric counterparts. For example, (a,0) where a≠0 has a counterpart (-a, 0). Similarly for (0,b) with b≠0. So the number of points on the axes excluding origin is 11 (as computed: 6 on a-axis including origin, 6 on b-axis including origin, subtract 1 for origin, gives 11). But each of these points has a mirror image except for when they are on both axes (i.e., the origin, which is already counted). Wait, actually, the points on the axes excluding origin: for example, (a,0) where a≠0. Each such point has a mirror image (-a,0). Similarly for (0,b) where b≠0. So total number of such points is 2*(number of points on a-axis excluding origin) + 2*(number of points on b-axis excluding origin)? Wait no, because the points on the axes are already considered. Wait, maybe better:

Number of points on the a-axis (excluding origin) is 5 (a=±1, ±2, ±3, ±4, ±5). Wait, no, in the first quadrant, we had 6 points on a-axis (from 0 to5). So excluding origin, it's 5 points on a-axis (a=1,...,5). Similarly, 5 points on b-axis (b=1,...,5). So total points on axes excluding origin: 5 +5=10. But in the first quadrant, the axes include origin, so when we calculated first quadrant points on axes as 11 (including origin), so excluding origin is 10. Therefore, total number of points on the axes (excluding origin) is 10. But each of these points has a mirror image on the other side. Therefore, the total number of points on the axes (excluding origin) is 10, but when counting total points, these points are in four quadrants? No, because they are on axes. For example, the point (1,0) is only on the a-axis. Its mirror image is (-1, 0). So total number of such points is 2*number of points on a-axis excluding origin + 2*number of points on b-axis excluding origin? Wait, no. For each point on the a-axis (excluding origin), there is a corresponding point on the negative a-axis. Similarly for b-axis. Therefore, total number of points on the axes (excluding origin) is 2*(number on positive a-axis) + 2*(number on positive b-axis). Wait, if we have 5 points on positive a-axis (a=1,...,5), then negative a-axis also has 5 points, so total 10 points on a-axis. Similarly for b-axis: 10 points. Wait, but that would be 20 points. But actually, points on a-axis are (±1,0), (±2,0), ..., (±5,0): total 10 points. Similarly, points on b-axis are (0, ±1), (0,±2), ..., (0, ±5): another 10 points. So total of 20 points on the axes excluding origin. But in the first quadrant, we accounted for 5 points on a-axis (excluding origin) and 5 points on b-axis (excluding origin). So total of 10 points on axes excluding origin in first quadrant.

So going back, if we have in total:

Total number of points = 4*(points in first quadrant excluding axes) + 2*(points on axes excluding origin) + 1*(origin). Wait, no. Wait, if we consider that points in first quadrant excluding axes are counted once, but each of them has 3 symmetric counterparts (but actually, for each point (a,b) with a>0 and b>0, there are 3 other quadrants it can be in: (a,-b), (-a,b), (-a,-b). So 4 points total, but since we have already counted the first quadrant ones, multiplying by 4 gives all four quadrants.

But the points on the axes (excluding origin) are already on the axes, so each such point has only one symmetric counterpart. For example, (a,0) with a>0 has (-a,0). Similarly, (0,b) with b>0 has (0,-b). Therefore, each of these points (on axes) has one mirror image. So the total number of such points is 2*(number of points on positive a-axis) + 2*(number of points on positive b-axis). Wait, no. The total number of points on the axes excluding origin is 20, as above. But in the first quadrant, we have 10 points (5 on a-axis, 5 on b-axis). Then, if we multiply by 2 for the other quadrants? Wait, this is getting confusing. Maybe an alternative approach:

Total number of points = points in first quadrant (including axes) * 4 - points on the axes (excluding origin)*2 - origin*3? Wait, no. Maybe it's easier to use inclusion-exclusion.

Alternatively, since we have already calculated the number of points in first quadrant as 26, and we can calculate the total number of points as follows:

Each point not on an axis is counted 4 times in the first quadrant. Each point on the axis (excluding origin) is counted 2 times in the first quadrant. And the origin is counted once.

So, let me denote:

Let N be the total number of points. Let me denote:

Let A be the number of points in first quadrant excluding axes: 15 (as above)

Let B be the number of points on axes excluding origin in first quadrant: 10 (5 on a-axis, 5 on b-axis)

Let C be the origin:1

So total N = 4*A + 2*B + C

Therefore:

N = 4*15 + 2*10 +1 = 60 +20 +1=81. Which matches the previous total of 81. So that's good, this cross-verification shows that 81 is correct.

Therefore, the answer is 81. But just to be thorough, let me check with another approach.

Alternatively, I can compute for each radius from 0 to 5, the number of integer points on the circle of radius r, but since we are looking for all points inside or on the circle, it's the sum over r from 0 to 5 of the number of integer points on the circle of radius r. Wait, no, actually, that would not be correct because the points inside the circle are not just the points on the circles of radius 0 to 5, but all points with a² + b² ≤25. Maybe another way is to use Gauss circle problem, but that's approximate. However, since we can compute exactly here, maybe not necessary.

Alternatively, let me check for some smaller radii and see if counts make sense. For example, when radius is 0: only (0,0). So 1. For radius 1: all points with a² + b² ≤1. Which are (0,0), (0,±1), (±1,0). So total 1 + 4 = 5. According to formula above, if we do the same method:

First quadrant points:

a=0: b=0,1 => 2 points

a=1: b=0 => 1 point

Total first quadrant: 2 +1=3

Then total points: 4*(3 - 2) + 2*(number on axes excluding origin) +1. Wait, maybe not. Wait, in this case:

Using the formula N = 4*A + 2*B + C where A = points in first quadrant excluding axes. For radius 1, first quadrant points (including axes): 3. Points on axes in first quadrant: 2 (a=0,b=0,1 and a=1,b=0). So points on axes excluding origin: 2 -1=1 (i.e., (0,1) and (1,0) minus origin). Wait, perhaps not. Let me recast:

For radius 1, first quadrant (including axes) points:

a=0: b=0,1 (2 points)

a=1: b=0 (since 1² +0²=1 ≤1, b=0. b=1 would be 1+1=2>1. So only b=0. So total 1 point.

Total first quadrant points: 2 +1=3.

Points on axes in first quadrant:

a-axis: (0,0), (0,1), (1,0). Wait, no. Wait for a ≥0, b ≥0:

Points on a-axis (b=0): (0,0), (1,0). Similarly, points on b-axis (a=0): (0,0), (0,1). So total points on axes in first quadrant: 3 ( (0,0), (1,0), (0,1)).

Therefore, points on axes excluding origin: 2 points: (1,0) and (0,1).

Therefore:

A (points in first quadrant excluding axes): total first quadrant points (3) minus points on axes (3) plus origin? Wait, no. Wait, points in first quadrant excluding axes would be points where a>0 and b>0. In this case, with radius 1, a and b can be 0 or 1. The only point with a>0 and b>0 would be (1,1), but 1² +1²=2>1, which is outside. So A=0. Therefore:

N = 4*A + 2*B + C = 4*0 + 2*2 +1= 0 +4 +1=5. Which is correct. So 5 points total. Therefore, my formula works here.

So applying this formula for the original problem gives 81. Hence, this cross-check seems to validate that the count is correct.

But to be absolutely sure, maybe let me check another small radius. Let's take radius sqrt(2), but in integers, the points with a² + b² ≤2.

Points are:

(0,0), (0,±1), (0,±2), (±1,0), (±1,±1), (±2,0). Let's count:

(0,0): 1

(0,±1): 2

(0,±2): 2

(±1,0): 2

(±1,±1): 4

(±2,0): 2

Total: 1 + 2 +2 +2 +4 +2=13? Wait:

Wait, let's count properly:

- (0,0): 1

- (0,1), (0,-1): 2

- (0,2), (0,-2): 2

- (1,0), (-1,0): 2

- (1,1), (1,-1), (-1,1), (-1,-1): 4

- (2,0), (-2,0): 2

Total: 1 +2+2+2+4+2= 13. Now, using our formula:

First quadrant points (including axes):

For a=0: b=0,1,2 => 3

For a=1: b=0,1 (since 1 + b² ≤2 => b² ≤1 => b=0,1). So 2

For a=2: b=0 (since 4 +0 ≤2? No, 4 >2. So b must be 0. Wait 2² +0=4>2. So b has to be 0? No, a=2 can't be because 2²=4>2. Hence, for a=2, there are no b's. Therefore, first quadrant points are:

a=0: 3

a=1:2

a=2:0

Total first quadrant points: 5

Points on axes in first quadrant:

a-axis (b=0): a=0,1,2. But b=0. Wait, a ranges from 0 to... in first quadrant, a and b non-negative. So for a-axis (b=0): a can be 0,1,2? Wait a² +0 ≤2 => a² ≤2 => a can be 0,1. Because 2²=4>2. So a=0,1. Similarly for b-axis. So points on axes in first quadrant: (0,0), (0,1), (0,2)? Wait no, (0,2) would have b=2, which is allowed because 0² +2²=4>2. Wait no, original radius squared is 2. So a² +b² ≤2. For (0,2): 0+4=4>2. So not allowed. Therefore, points on axes in first quadrant:

a-axis: (0,0), (1,0)

b-axis: (0,0), (0,1)

So total points on axes in first quadrant: 3 ( (0,0), (1,0), (0,1) )

Therefore, points on axes excluding origin: 2 ( (1,0), (0,1) )

Hence, A (points in first quadrant excluding axes): total first quadrant points (5) minus points on axes in first quadrant (3) = 2? Wait no. Wait points in first quadrant excluding axes are those where a>0 and b>0. Which would be (1,1). But (1,1) has a² +b²=2, which is allowed. So that's 1 point. So A=1. Then:

N=4*A + 2*B + C = 4*1 + 2*2 +1= 4 +4 +1=9? But earlier count was 13. Wait discrepancy here. So something is wrong.

Wait, according to my previous calculation, we had 13 points, but according to formula N=4*A +2*B + C= 4*1 + 2*2 +1=9. Which is wrong. So where is the mistake?

Let me recast. Let me redo the formula:

Total points = 4*(points in first quadrant excluding axes) + 2*(points on axes excluding origin) + 1*(origin).

But in this case, points in first quadrant excluding axes: points where a>0 and b>0. For our case with radius squared 2, that's (1,1). So 1 point. Points on axes excluding origin: (1,0), (0,1). So 2 points. Origin: 1. Therefore:

Total points = 4*1 + 2*2 +1 = 4 +4 +1=9. But actual count was 13. So clearly something is wrong with the formula.

Wait, what is happening here. Maybe my formula is incorrect. Let me think again.

Alternative approach:

Each point not on an axis is counted 4 times in the first quadrant. Each point on an axis but not origin is counted 2 times in the first quadrant. The origin is counted once.

But in the first quadrant, the total number of points is:

Number of points in first quadrant = number of points not on axes + number of points on axes in first quadrant.

Therefore:

Total points not on axes =

Number of points in first quadrant - number of points on axes in first quadrant.

Then, total points in all quadrants:

Total points not on axes: 4*(number in first quadrant - number on axes in first quadrant)

Plus total points on axes (excluding origin): 2*(number on positive axes)

Plus origin: 1

Wait, let me define:

Let’s denote:

- Let Q be the number of points in the first quadrant (including axes).

- Let A be the number of points on the axes in the first quadrant (including origin).

- Let B be the number of points on the positive axes (excluding origin). So A = 1 + 2*B? Wait, no. For example, in the case of radius 2:

In first quadrant, points on axes: (0,0), (1,0), (0,1). So A=3. The number of points on positive axes (excluding origin) is 2: (1,0) and (0,1). So B=2. Then:

Total points not on axes in first quadrant: Q - A = 5 -3=2. But in reality, there is only 1 point not on axes: (1,1). So discrepancy here.

Wait, maybe my definitions are conflicting. Let me try again.

Let me define:

In the first quadrant (including axes):

- The number of points not on the axes: let's call this C.

- The number of points on the axes (including origin): let's call this D.

Then, Q = C + D.

Total points in all quadrants:

The points not on the axes: each such point has been counted once in first quadrant, and there are 4 such points in each quadrant. So total points not on axes: 4*C.

The points on the axes (excluding origin): each such point has been counted once in first quadrant, and each has a mirror image on the opposite axis. For example, (1,0) has mirror (-1,0), so total 2 points. Similarly for (0,1) and (0,-1). Therefore, total points on axes (excluding origin): 2*D' where D' is the number of points on positive axes (excluding origin). Wait, if D includes the origin, then D' = D -1. For example, in our radius 2 example:

D = number of points on axes in first quadrant: 3 (origin, (1,0), (0,1))

So D' = D -1 = 2 (points on positive axes excluding origin).

Therefore, total points on axes excluding origin: 2*D' (because each point on positive axis has a mirror on negative axis). So 2*2=4.

Therefore, total number of points:

Total = 4*C + 2*D' +1 (origin)

In the case of radius 2 example, Q = 5, D =3, D'=2, C=Q - D=5-3=2.

But in reality, the number of points not on the axes in first quadrant is 1 ((1,1)), but according to C=2. Where is the mistake?

Ah! Wait, Q counts all points in first quadrant, including axes. So if Q = 5, then D = number of points on axes in first quadrant. Wait in our case, with radius squared 2:

First quadrant points (including axes):

a=0: b=0,1 (since 0² + b² ≤2 => b² ≤2 => b=0,1. Wait (0,1) gives 1, which is ≤2. But also (0, sqrt(2)), but b must be integer. So yes, (0,0), (0,1). For a=0: 2 points.

a=1: b=0,1 (because 1 + b² ≤2 => b² ≤1 => b=0,1). So (1,0), (1,1). So 2 points.

a=2: b² ≤2 -4= -2? No, a=2, a²=4>2, so no points.

Total first quadrant points: 2 + 2 =4. Wait earlier I thought it was 5. Wait let me recount:

For a=0: b=0,1 => 2 points

For a=1: b=0,1 => 2 points

For a=2: 0 points

Total first quadrant points: 4. So earlier mistake was in thinking it was 5. So Q=4.

Then D is the number of points on axes in first quadrant. For axes:

On a-axis: (0,0), (1,0). On b-axis: (0,0), (0,1). So total points on axes in first quadrant: 3. So D=3. Hence, C=Q - D=4 -3=1. Which is correct: point (1,1).

Therefore, total points:

Total = 4*C + 2*(D -1) +1 = 4*1 + 2*(3 -1) +1=4 + 4 +1=9. Which matches the formula. However, when I counted manually earlier, I had 13 points. Wait, let me recount manually.

For a² +b² ≤2:

Possible points:

(0,0): 1

(0, ±1): 2

(0, ±2): but 0 + 4 =4>2, so no.

(±1, 0): 2

(±1, ±1): 4 points (1,1), (1,-1), (-1,1), (-1,-1)

(±2, 0): 2 points (2,0), (-2,0), but 4>2, so no. Wait (2,0) gives 4>2, so invalid. So only (±1,0), (±1, ±1), (0,±1), and (0,0).

Therefore total points:

(0,0):1

(±1,0):2

(0, ±1):2

(±1, ±1):4

Total: 1 +2 +2 +4=9. Which matches the formula. Earlier, I had mistakenly included (0, ±2) and (±2,0), which are invalid. So my manual count was wrong before. Therefore, formula works.

Therefore, this gives me more confidence that the formula is correct. Thus, returning to our original problem, where the total was 81, seems correct.

But just to be thorough, let me check for another radius. For example, radius 1, as before:

First quadrant points: for a² + b² ≤1:

a=0: b=0,1 => 2 points

a=1: b=0 => 1 point

Total Q=3

D: points on axes in first quadrant: (0,0), (1,0), (0,1). D=3. Hence, C= Q - D= 0.

Total points: 4*0 + 2*(D -1) +1=0 + 2*(3 -1) +1= 4 +1=5, which is correct.

So the formula works. Hence, returning to our original problem, where we calculated via two methods:

First method: 11 + 18 + 18 +18 +14 +2 =81.

Second method: using first quadrant points: total of 26, which led to N=81.

Therefore, confident that 81 is correct.

But just for absolute surety, let me check for radius 5. For example, how many points are there with a² +b² ≤25?

Alternatively, I can think of the number of points as follows:

For each a from -5 to 5, compute the number of b's.

As I did earlier:

a=-5: b=0 => 1

a=-4: b=-3,...,3 =>7

a=-3: b=-4,...,4 =>9

a=-2: same as +2 =>9

a=-1: same as +1 =>9

a=0: 11

a=1:9

a=2:9

a=3:9

a=4:7

a=5:1

So adding these up:

For a=-5:1

a=-4:7

a=-3:9

a=-2:9

a=-1:9

a=0:11

a=1:9

a=2:9

a=3:9

a=4:7

a=5:1

Total:

Let me add them sequentially:

Start with a=-5:1

a=-4:1+7=8

a=-3:8+9=17

a=-2:17+9=26

a=-1:26+9=35

a=0:35+11=46

a=1:46+9=55

a=2:55+9=64

a=3:64+9=73

a=4:73+7=80

a=5:80+1=81.

Yes, same total. Hence, 81.

Therefore, after multiple approaches and cross-verifications, I believe the correct answer is 81.

**Final Answer**
\boxed{81}

To find the number of ordered pairs $(a, b)$ of integers such that $ |a + bi| \leq 5 $, we proceed by analyzing the condition that gives rise to this inequality.

---

### Step 1: Understanding the Inequality

For a complex number $ z = a + bi $, the modulus is defined as:

$$
|z| = \sqrt{a^2 + b^2}
$$

So the inequality $ |a + bi| \leq 5 $ becomes:

$$
\sqrt{a^2 + b^2} \leq 5
\Rightarrow a^2 + b^2 \leq 25
$$

We are to count all integer pairs $(a, b)$ such that $ a^2 + b^2 \leq 25 $.

---

### Step 2: Counting Lattice Points Inside a Circle

The inequality $ a^2 + b^2 \leq 25 $ represents all integer lattice points (points with integer coordinates) located inside or on the circle of radius 5 centered at the origin.

To count such points, we can iterate through all possible integer values of $ a $, and for each, determine the range of possible values of $ b $ that satisfy the inequality.

---

### Step 3: Iterating Through All Integer Values of $ a $

We observe that:

$$
a^2 \leq 25 \Rightarrow -5 \leq a \leq 5
$$

So we evaluate the number of valid $ b $ values for each $ a $ from $-5$ to $ 5 $, and sum up the counts.

For each $ a $:

- Compute the maximum value of $ b $ such that $ b^2 \leq 25 - a^2 $
- The number of possible $ b $ values corresponding to that range is:

$$
\text{Number of valid } b = 2 \times \lfloor \sqrt{25 - a^2} \rfloor + 1
$$

Let’s go through this:

| $ a $ | $ a^2 $ | $ 25 - a^2 $ | $ \sqrt{25 - a^2} $ | $ \lfloor \sqrt{25 - a^2} \rfloor $ | $ \text{count of } b $ |
|--------|----------|------------------|------------------------|--------------------------------------|-------------------------------|
| -5     | 25       | 0                | 0                      | 0                                   | 1                             |
| -4     | 16       | 9                | 3                      | 3                                   | 7                             |
| -3     | 9        | 16               | 4                      | 4                                   | 9                             |
| -2     | 4        | 21               | 4.57                   | 4                                   | 9                             |
| -1     | 1        | 24               | 4.89                   | 4                                   | 9                             |
| 0      | 0        | 25               | 5                      | 5                                   | 11                            |
| 1      | 1        | 24               | 4.89                   | 4                                   | 9                             |
| 2      | 4        | 21               | 4.57                   | 4                                   | 9                             |
| 3      | 9        | 16               | 4                      | 4                                   | 9                             |
| 4      | 16       | 9                | 3                      | 3                                   | 7                             |
| 5      | 25       | 0                | 0                      | 0                                   | 1                             |

Now sum up the counts:

$$
1 + 7 + 9 + 9 + 9 + 11 + 9 + 9 + 9 + 7 + 1 = 81
$$

---

### Conclusion

Therefore, the number of ordered pairs $(a, b)$ of integers such that $ |a + bi| \leq 5 $ is:

$$
\boxed{81}
$$

Solution:
The problem asks us to count the number of complex numbers that lie in or on the circle of radius 5 centered at the origin, with integer real and imaginary parts.

[asy]
unitsize(0.5 cm);

int i, j;

draw((-5,0)--(5,0));
draw((0,-5)--(0,5));
draw(Circle((0,0),5));

for (i = -5; i <= 5; ++i) {
for (j = -5; j <= 5; ++j) {
  if (i^2 + j^2 > 25) {dot((i,j));}
  if (i^2 + j^2 <= 25) {dot((i,j),red);}
}}
[/asy]

We can count that there are 15 such complex numbers in the first quadrant (not including the axes).  Then there are 5 complex on the positive real axis, the negative real axis, the positive imaginary axis, and negative imaginary axis.  Finally, there is the origin itself, which gives us $4 \cdot 15 + 4 \cdot 5 + 1 = \boxed{81}$ complex numbers.