Found problems: 233
Let $P_1P_2 \ldots P_8$ be a convex octagon. An integer $i$ is chosen uniformly at random from $1$ to $7$, inclusive. For each vertex of the octagon, the line between that vertex and the vertex $i$ vertices to the right is painted red. What is the expected number times two red lines intersect at a point that is not one of the vertices, given that no three diagonals are concurrent?
Compute the value of $1^{25}+2^{24}+3^{23}+\ldots+24^2+25^1$. If your answer is $A$ and the correct answer is $C$, then your score on this problem will be $\left\lfloor25\min\left(\left(\frac AC\right)^2,\left(\frac CA\right)^2\right)\right\rfloor$.
Given $a$, $b$, and $c$ are complex numbers satisfying
\[ a^2+ab+b^2=1+i \]
\[ b^2+bc+c^2=-2 \]
\[ c^2+ca+a^2=1, \]
compute $(ab+bc+ca)^2$. (Here, $i=\sqrt{-1}$)
Prove that for all positive integers $n$, all complex roots $r$ of the polynomial
\[P(x) = (2n)x^{2n} + (2n-1)x^{2n-1} + \dots + (n+1)x^{n+1} + nx^n + (n+1)x^{n-1} + \dots + (2n-1)x + 2n\]
lie on the unit circle (i.e. $|r| = 1$).
Given $a,b,c \in\mathbb{R}^+$, and that $a^2+b^2+c^2=3$. Prove that
\[ \frac{1}{a^3+2}+\frac{1}{b^3+2}+\frac{1}{c^3+2}\ge 1 \]
How many distinct permutations of the letters in the word REDDER are there that do not contain a palindromic substring of length at least two? (A [i]substring[/i] is a continuous block of letters that is part of the string. A string is [i]palindromic[/i] if it is the same when read backwards.)
Let $A = \{1,2,\ldots,2011\}$. Find the number of functions $f$ from $A$ to $A$ that satisfy $f(n) \le n$ for all $n$ in $A$ and attain exactly $2010$ distinct values.
Let $f(x) = x^2 + 6x + c$ for all real number s$x$, where $c$ is some real number. For what values of $c$ does $f(f(x))$ have exactly $3$ distinct real roots?