Input:
**User**:
Twelve friends met for dinner at Oscar's Overstuffed Oyster House, and each ordered one meal. The portions were so large, there was enough food for 18 people. If they share, how many meals should they have ordered to have just enough food for the 12 of them?
Please reason step by step, and put your final answer within \boxed{}.

To determine how many meals the twelve friends should have ordered to have just enough food for all of them, we begin by analyzing the relationship between the number of meals and the amount of food they provided.

---

### Step 1: Understand the Given Information

- **12 meals** were ordered.
- These meals were **enough for 18 people**.
- This implies that the **total food** from 12 meals is equivalent to the food needed for 18 people.

---

### Step 2: Determine the Food per Meal

We can calculate how much food each meal provides by dividing the total number of people the food can feed by the number of meals:

$$
\text{Food per meal} = \frac{18 \text{ people}}{12 \text{ meals}} = 1.5 \text{ people per meal}
$$

So, **each meal provides enough food for 1.5 people**.

---

### Step 3: Calculate the Number of Meals Needed for 12 People

Now, we want to find out how many meals are needed to provide food for **12 people**, given that each meal provides food for 1.5 people:

$$
\text{Number of meals} = \frac{12 \text{ people}}{1.5 \text{ people per meal}} = 8 \text{ meals}
$$

---

### Step 4: Verify the Result

To double-check, we can use a proportion:

$$
\frac{12 \text{ meals}}{18 \text{ people}} = \frac{x \text{ meals}}{12 \text{ people}}
$$

Cross-multiplying:

$$
12x = 18 \times 12 \Rightarrow x = \frac{18 \times 12}{12} = 18 \div 1.5 = 8
$$

This confirms that **8 meals** are sufficient for 12 people.

---

### Final Answer

$$
\boxed{8}
$$

Solution: If 12 people order $\frac{18}{12}=1\frac{1}{2}$ times too much food, they should have ordered $\frac{12}{\frac{3}{2}}=\frac{2}{3}\times 12=\boxed{8}$ meals.


Input:
**User**:
Find the curve defined by the equation
\[r^2 \cos 2 \theta = 4.\](A) Line
(B) Circle
(C) Parabola
(D) Ellipse
(E) Hyperbola

Enter the letter of the correct option.
Please reason step by step, and put your final answer within \boxed{}.

To determine the type of curve defined by the polar equation

$$
r^2 \cos 2\theta = 4,
$$

we begin by converting it into Cartesian coordinates, which is often more intuitive for identifying conic sections.

---

### Step 1: Use Trigonometric Identity

Recall the identity:

$$
\cos 2\theta = \frac{x^2 - y^2}{r^2}.
$$

This identity comes from the fact that:

$$
x = r \cos \theta, \quad y = r \sin \theta \Rightarrow x^2 + y^2 = r^2.
$$

So,

$$
x^2 - y^2 = r^2 (\cos^2 \theta - \sin^2 \theta) = r^2 \cos 2\theta.
$$

Therefore, we can substitute this into the original equation:

$$
r^2 \cos 2\theta = 4 \Rightarrow r^2 \cdot \frac{x^2 - y^2}{r^2} = 4.
$$

Simplifying:

$$
x^2 - y^2 = 4.
$$

---

### Step 2: Identify the Curve

The equation $ x^2 - y^2 = 4 $ is a standard form of a **hyperbola**. To see this more clearly, divide both sides by 4:

$$
\frac{x^2}{4} - \frac{y^2}{4} = 1.
$$

This is the standard form of a hyperbola centered at the origin, opening along the x-axis, with:

- $ a^2 = 4 \Rightarrow a = 2 $,
- $ b^2 = 4 \Rightarrow b = 2 $.

---

### Step 3: Confirm the Nature of the Curve

The original polar equation is defined only when $ \cos 2\theta > 0 $, since $ r^2 $ must be non-negative. This restricts the values of $ \theta $ to regions where $ \cos 2\theta > 0 $, which corresponds to the branches of the hyperbola in the Cartesian plane.

Thus, the curve is a **hyperbola**, and the correct answer is:

$$
\boxed{E}
$$

Solution:
From $r^2 \cos 2 \theta = 4,$
\[r^2 (\cos^2 \theta - \sin^2 \theta) = r^2 \cos^2 \theta - r^2 \sin^2 \theta = 4.\]Then $x^2 - y^2 = 4,$ or
\[\frac{x^2}{4} - \frac{y^2}{4} = 1.\]Thus, the graph represents a hyperbola.  The answer is $\boxed{\text{(E)}}.$

[asy]
unitsize(0.5 cm);

pair moo (real t) {
  real r = sqrt(4/Cos(2*t));
  return (r*Cos(t), r*Sin(t));
}

path foo = moo(-44);
real t;

for (t = -44; t <= 44; t = t + 0.1) {
  foo = foo--moo(t);
}

draw(foo,red);
draw(reflect((0,0),(0,1))*(foo),red);

draw((-4,0)--(4,0));
draw((0,-4)--(0,4));

limits((-4,-4),(4,4),Crop);

label("$r^2 \cos 2 \theta = 4$", (6.5,1.5), red);
[/asy]


Input:
**User**:
What is the 2003rd term of the sequence of odd numbers 1, 3, 5, 7, $\dots$?
Please reason step by step, and put your final answer within \boxed{}.

To determine the 2003rd term of the sequence of odd numbers:
$$
1, 3, 5, 7, \dots
$$
we recognize that this is an **arithmetic sequence** with the following properties:

- **First term** $ a_1 = 1 $
- **Common difference** $ d = 2 $

---

### Step 1: Use the General Formula for the $ n $-th Term of an Arithmetic Sequence

The general formula for the $ n $-th term of an arithmetic sequence is:

$$
a_n = a_1 + (n - 1)d
$$

Substituting the known values:

$$
a_{2003} = 1 + (2003 - 1) \cdot 2 = 1 + 2002 \cdot 2
$$

$$
a_{2003} = 1 + 4004 = 4005
$$

---

### Step 2: Confirm with an Alternative Formula

There is a well-known formula for the $ n $-th odd number:

$$
a_n = 2n - 1
$$

Applying this directly:

$$
a_{2003} = 2 \cdot 2003 - 1 = 4006 - 1 = 4005
$$

This confirms the result.

---

### Final Answer

$$
\boxed{4005}
$$

Solution: The sequence of odd numbers 1, 3, 5, 7, and so on, is an arithmetic sequence, with common difference 2.  Therefore, the $2003^{\text{rd}}$ term is $1+2002\cdot2=\boxed{4005}$.


Input:
**User**:
Let $f(x) = 2x-3$ and $g(x) = x+1$. What is the value of $g(f(5)-1)$?
Please reason step by step, and put your final answer within \boxed{}.

To solve the expression $ g(f(5) - 1) $, we proceed step by step, evaluating the inner function first and then applying the outer function.

---

### Step 1: Evaluate $ f(5) $

We are given the function:

$$
f(x) = 2x - 3
$$

Substitute $ x = 5 $:

$$
f(5) = 2(5) - 3 = 10 - 3 = 7
$$

---

### Step 2: Compute $ f(5) - 1 $

Now subtract 1 from the result:

$$
f(5) - 1 = 7 - 1 = 6
$$

---

### Step 3: Evaluate $ g(6) $

We are given the function:

$$
g(x) = x + 1
$$

Substitute $ x = 6 $:

$$
g(6) = 6 + 1 = 7
$$

---

### Final Answer

Thus, the value of $ g(f(5) - 1) $ is:

$$
\boxed{7}
$$

Solution: We have $f(5) = 2(5) -3 = 7$, so $g(f(5)-1) = g(7-1) = g(6) = 6+1 = \boxed{7}$.