[b]p1.[/b] What is $6\times 7 + 4 \times 7 + 6\times 3 + 4\times 3$?


[b]p2.[/b] How many integers $n$ have exactly $\sqrt{n}$ factors?


[b]p3.[/b] A triangle has distinct angles $3x+10$, $2x+20$, and $x+30$. What is the value of $x$?


[b]p4.[/b] If $4$ people of the Math Club are randomly chosen to be captains, and Henry is one of the $30$ people eligible to be chosen, what is the probability that he is not chosen to be captain?


[b]p5.[/b] $a, b, c, d$ is an arithmetic sequence with difference $x$ such that $a, c, d$ is a geometric sequence. If $b$ is $12$, what is $x$? (Note: the difference of an aritmetic sequence can be positive or negative, but not $0$)


[b]p6.[/b] What is the smallest positive integer that contains only $0$s and $5$s that is a multiple of $24$.


[b]p7.[/b] If $ABC$ is a triangle with side lengths $13$, $14$, and $15$, what is the area of the triangle made by connecting the points at the midpoints of its sides?


[b]p8.[/b] How many ways are there to order the numbers $1,2,3,4,5,6,7,8$ such that $1$ and $8$ are not adjacent?


[b]p9.[/b] Find all ordered triples of nonnegative integers $(x, y, z)$ such that $x + y + z = xyz$.


[b]p10.[/b] Noah inscribes equilateral triangle $ABC$ with area $\sqrt3$ in a cricle. If $BR$ is a diameter of the circle, then what is the arc length of Noah's $ARC$?


[b]p11.[/b] Today, $4/12/14$, is a palindromic date, because the number without slashes $41214$ is a palindrome. What is the last palindromic date before the year $3000$?


[b]p12.[/b] Every other vertex of a regular hexagon is connected to form an equilateral triangle. What is the ratio of the area of the triangle to that of the hexagon?


[b]p13.[/b] How many ways are there to pick four cards from a deck, none of which are the same suit or number as another, if order is not important?


[b]p14.[/b] Find all functions $f$ from $R \to R$ such that $f(x + y) + f(x - y) = x^2 + y^2$.


[b]p15.[/b] What are the last four digits of $1(1!) + 2(2!) + 3(3!) + ... + 2013(2013!)$/


[b]p16.[/b] In how many distinct ways can a regular octagon be divided up into $6$ non-overlapping triangles?


[b]p17.[/b] Find the sum of the solutions to the equation $\frac{1}{x-3} + \frac{1}{x-5} + \frac{1}{x-7} + \frac{1}{x-9} = 2014$ .


[b]p18.[/b] How many integers $n$ have the property that $(n+1)(n+2)(n+3)(n+4)$ is a perfect square of an integer?


[b]p19.[/b] A quadrilateral is inscribed in a unit circle, and another one is circumscribed. What is the minimum possible area in between the two quadrilaterals?


[b]p20.[/b] In blindfolded solitary tic-tac-toe, a player starts with a blank $3$-by-$3$ tic-tac-toe board. On each turn, he randomly places an "$X$" in one of the open spaces on the board. The game ends when the player gets $3$ $X$s in a row, in a column, or in a diagonal as per normal tic-tac-toe rules. (Note that only $X$s are used, not $O$s). What fraction of games will run the maximum $7$ amount of moves?


PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
1. Plugging in \( y = 0 \), we have:
   \[
   f(x + 0) + f(x - 0) = x^2 + 0^2
   \]
   Simplifying, we get:
   \[
   f(x) + f(x) = x^2
   \]
   Therefore:
   \[
   2f(x) = x^2 \implies f(x) = \frac{x^2}{2}
   \]

2. To verify that \( f(x) = \frac{x^2}{2} \) satisfies the original functional equation, we substitute \( f(x) = \frac{x^2}{2} \) back into the equation:
   \[
   f(x + y) + f(x - y) = \frac{(x + y)^2}{2} + \frac{(x - y)^2}{2}
   \]
   Expanding the squares, we get:
   \[
   \frac{x^2 + 2xy + y^2}{2} + \frac{x^2 - 2xy + y^2}{2}
   \]
   Combining the terms, we have:
   \[
   \frac{x^2 + 2xy + y^2 + x^2 - 2xy + y^2}{2} = \frac{2x^2 + 2y^2}{2} = x^2 + y^2
   \]
   This matches the right-hand side of the original equation, confirming that \( f(x) = \frac{x^2}{2} \) is indeed a solution.

\(\blacksquare\)

The final answer is \( \boxed{ f(x) = \frac{x^2}{2} } \)