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Problem: Define \[p = \sum_{k = 1}^ \infty \frac{1}{k^2} \quad \text{and} \quad q = \sum_{k = 1}^\infty \frac{1}{k^3}.\]Find a way to write \[\sum_{j = 1}^\infty \sum_{k = 1}^\infty \frac{1}{(j + k)^3}\] in terms of $p$ and $q.$
Predicted answer (BoN): To solve the given problem, we need to express the double sum \(\sum_{j=1}^\infty \sum_{k=1}^\infty \frac{1}{(j+k)^3}\) in terms of \(p\) and \(q\).

First, let's rewrite the double sum by changing the order of summation. We start by summing over \(m = j + k\). For a fixed \(m\), \(j\) and \(k\) can take any values from 1 to \(\infty\) such that \(j + k = m\). 

The number of pairs \((j, k)\) that satisfy \(j + k = m\) is \(m-1\). Therefore, we can express the sum as:
\[
\sum_{j=1}^\infty \sum_{k=1}^\infty \frac{1}{(j+k)^3} = \sum_{m=1}^\infty \frac{m-1}{m^3}.
\]
To evaluate this sum, we can split it into two separate sums:
\[
\sum_{m=1}^\infty \frac{m-1}{m^3} = \sum_{m=1}^\infty \left( \frac{m}{m^3} - \frac{1}{m^3} \right) = \sum_{m=1}^\infty \left( \frac{1}{m^2} - \frac{1}{m^3} \right).
\]
This can be further simplified into two separate sums:
\[
\sum_{m=1}^\infty \frac{1}{m^2} - \sum_{m=1}^\infty \frac{1}{m^3} = p - q.
\]
Thus, the original sum can be expressed in terms of \(p\) and \(q\) as:
\[
\sum_{j=1}^\infty \sum_{k=1}^\infty \frac{1}{(j+k)^3} = p - q.
\]
Therefore, the final answer is:
\[
\boxed{p - q}.
\]
Correct Answer: p - q
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Problem: A regular hexagon can be divided into six equilateral triangles. If the perimeter of one of the triangles is 21 inches, what is the perimeter, in inches, of the regular hexagon?
Predicted answer (BoN): ## Step 1:**
First, let's understand that a regular hexagon can be divided into six equilateral triangles. The perimeter of one of these triangles is given as 21 inches. Since all sides of an equilateral triangle are equal, each side of the triangle must be 21 inches divided by 3, as it's equal to the perimeter divided by 3.

**Calculation for one side of the triangle:**
\[ \text{Side length of the triangle} = \frac{21}{3} = 7 \text{ inches} \]

## Step 2:**
Now, since the hexagon is made up of these equilateral triangles, each side of the hexagon will be equal to the side length of one of these triangles. Therefore, the perimeter of the hexagon, which is the sum of all its sides, will be six times the side length of one of the triangles.

**Calculation for the perimeter of the hexagon:**
\[ \text{Perimeter of the hexagon} = 6 \times 7 \]

## Step 3:**
Perform the multiplication to find the final perimeter.

\[ \text{Perimeter of the hexagon} = 6 \times 7 = 42 \text{ inches} \]

Therefore, the final answer is: $\boxed{42}$.
Correct Answer: 42
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Problem: The expression $2\cdot 3 \cdot 4\cdot 5+1$ is equal to 121, since multiplication is carried out before addition. However, we can obtain values other than 121 for this expression if we are allowed to change it by inserting parentheses. For example, we can obtain 144 by writing \[ (2\cdot (3\cdot 4)) \cdot (5+1) = 144. \]In total, how many values can be obtained from the expression $2\cdot 3\cdot 4 \cdot 5 + 1$ by inserting parentheses? (Note that rearranging terms is not allowed, only inserting parentheses).
Predicted answer (BoN): To solve the problem of determining how many different values can be obtained from the expression \(2 \cdot 3 \cdot 4 \cdot 5 + 1\) by inserting parentheses, we need to consider the different ways we can group the terms using parentheses. Since multiplication is carried out before addition, we need to consider all possible ways we can group the numbers \(2\), \(3\), \(4\), and \(5\) using parentheses.

First, let's list all possible combinations of parentheses groups. Each group will have a different number of terms inside them. The possible groups are:

1. \(2, 3, 4, 5\)
2. \(2, 3 \cdot 4, 5\)
3. \(2, 3 \cdot (4 \cdot 5)\)
4. \(2 \cdot 3, 4, 5\)
5. \(2 \cdot 3 \cdot 4, 5\)
6. \(2 \cdot 3 \cdot (4 \cdot 5)\)
7. \(2, (3 \cdot 4) \cdot (5 + 1)\)
8. \(2 \cdot (3 \cdot 4), 5 + 1\)
9. \(2 \cdot (3 \cdot (4 \cdot 5))\)
10. \(2 \cdot (3 \cdot (4 \cdot (5 + 1)))\)

Now, let's evaluate each of these groups:

1. \(2, 3, 4, 5\) : This evaluates to \(2 \cdot 3 \cdot 4 \cdot 5 + 1 = 121\).
2. \(2, 3 \cdot 4, 5\) : This evaluates to \(2 \cdot (3 \cdot 4) + 5 = 2 \cdot 12 + 5 = 29\).
3. \(2, 3 \cdot (4 \cdot 5)\) : This evaluates to \(2 \cdot (3 \cdot 20) + 1 = 2 \cdot 60 + 1 = 121\).
4. \(2, 3 \cdot (4 \cdot 5)\) (repeated, but valid) : This is equivalent to the third case, evaluating to 121.
5. \(2, 3 \cdot (4 \cdot 5)\) : This is equivalent to the third case, evaluating to 121.
6. \(2, (3 \cdot 4) \cdot (5 + 1)\) : This evaluates to \(2 \cdot (12 \cdot 6) + 1 = 2 \cdot 72 + 1 = 145\).
7. \(2 \cdot (3 \cdot 4), 5 + 1\) : This evaluates to \(2 \cdot (12) + 6 = 2 \cdot 12 + 6 = 30\).
8. \(2 \cdot (3 \cdot (4 \cdot 5))\) : This evaluates to \(2 \cdot (3 \cdot 20) + 1 = 2 \cdot 60 + 1 = 121\).
9. \(2 \cdot (3 \cdot (4 \cdot 5))\) (repeated, but valid) : This is equivalent to the eighth case, evaluating to 121.
10. \(2 \cdot (3 \cdot (4 \cdot (5 + 1)))\) : This evaluates to \(2 \cdot (3 \cdot (4 \cdot 6)) + 1 = 2 \cdot (3 \cdot 24) + 1 = 2 \cdot 72 + 1 = 145\).

From the above evaluations, we see that the distinct values obtained are 121, 145, and 30. Therefore, the number of different values that can be obtained from the expression \(2 \cdot 3 \cdot 4 \cdot 5 + 1\) by inserting parentheses is \(\boxed{3}\).
Correct Answer: 4
