[u]Round 1[/u]


[b]p1.[/b] Every angle of a regular polygon has degree measure $179.99$ degrees. How many sides does it have?


[b]p2.[/b] What is $\frac{1}{20} + \frac{1}{1}+ \frac{1}{5}$ ?


[b]p3.[/b] If the area bounded by the lines $y = 0$, $x = 0$, and $x = 3$ and the curve $y = f(x)$ is $10$ units, what is the area bounded by $y = 0$, $x = 0$, $x = 6$, and $y = f(x/2)$?


[u]Round 2[/u]


[b]p4.[/b] How many ways can $42$ be expressed as the sum of $2$ or more consecutive positive integers?


[b]p5.[/b] How many integers less than or equal to $2015$ can be expressed as the sum of $2$ (not necessarily distinct) powers of two?


[b]p6.[/b] $p,q$, and $q^2 - p^2$ are all prime. What is $pq$?


[u]Round 3[/u]


[b]p7.[/b] Let $p(x) = x^2 + ax + a$ be a polynomial with integer roots, where $a$ is an integer. What are all the possible values of $a$?


[b]p8.[/b] In a given right triangle, the perimeter is $30$ and the sum of the squares of the sides is $338$. Find the lengths of the three sides.


[b]p9.[/b] Each of the $6$ main diagonals of a regular hexagon is drawn, resulting in $6$ triangles. Each of those triangles is then split into $4$ equilateral triangles by connecting the midpoints of the $3$ sides. How many triangles are in the resulting figure?


[u]Round 4[/u]


[b]p10.[/b] Let $f = 5x+3y$, where $x$ and $y$ are positive real numbers such that $xy$ is $100$. Find the minimum possible value of $f$.


[b]p11.[/b] An integer is called "Awesome" if its base $8$ expression contains the digit string $17$ at any point (i.e. if it ever has a $1$ followed immediately by a $7$). How many integers from $1$ to $500$ (base $10$) inclusive are Awesome?


[b]p12.[/b] A certain pool table is a rectangle measuring $15 \times 24$ feet, with $4$ holes, one at each vertex. When playing pool, Joe decides that a ball has to hit at least $2$ sides before getting into a hole or else the shot does not count. What is the minimum distance a ball can travel after being hit on this table if it was hit at a vertex (assume it only stops after going into a hole) such that the shot counts?


PS. You should use hide for answers. Rounds 5-8 have been posted [url=https://artofproblemsolving.com/community/c3h3157013p28696685]here [/url] and 9-12 [url=https://artofproblemsolving.com/community/c3h3158564p28715928]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
1. We are given that \( p, q \), and \( q^2 - p^2 \) are all prime numbers. We need to find the value of \( pq \).

2. First, consider the parity of \( p \) and \( q \). If both \( p \) and \( q \) are odd, then \( q^2 - p^2 \) will be even. The only even prime number is 2. Therefore, we would have:
   \[
   q^2 - p^2 = 2
   \]
   However, there are no two odd primes \( p \) and \( q \) such that \( q^2 - p^2 = 2 \). This leads to a contradiction.

3. If both \( p \) and \( q \) are even, then they must both be equal to 2 (since 2 is the only even prime). But then:
   \[
   q^2 - p^2 = 2^2 - 2^2 = 4 - 4 = 0
   \]
   which is not a prime number. This also leads to a contradiction.

4. Therefore, one of \( p \) or \( q \) must be 2 (the only even prime), and the other must be an odd prime. Without loss of generality, let \( p = 2 \). Then:
   \[
   q^2 - p^2 = q^2 - 2^2 = q^2 - 4
   \]
   Since \( q^2 - 4 \) is prime, we can write:
   \[
   q^2 - 4 = (q - 2)(q + 2)
   \]
   For this product to be prime, one of the factors must be 1 (since the only way to get a prime number from a product of two integers is if one of them is 1). Therefore, we have:
   \[
   q - 2 = 1 \quad \text{or} \quad q + 2 = 1
   \]
   Solving these, we get:
   \[
   q - 2 = 1 \implies q = 3
   \]
   \[
   q + 2 = 1 \implies q = -1 \quad (\text{not a prime})
   \]
   Thus, the only solution is \( q = 3 \).

5. Finally, we calculate \( pq \):
   \[
   pq = 2 \cdot 3 = 6
   \]

The final answer is \( \boxed{6} \).