Let $\{ a_n\}_{n \ge 0}$ be a sequence of real numbers such that $a_{n+1} \ge a_n^2 + \frac{1}{5}$ for all $n \ge 0$. Prove that $\sqrt{a_{n+5}} \ge a_{n-5}^2$ for all $n \ge 5$.

Express \[ \sum_{k=0}^n (-1)^k (n-k)!(n+k)! \] in closed form.

For a set $S$, let $|S|$ denote the number of elements in $S$. Let $A$ be a set of positive integers with $|A| = 2001$. Prove that there exists a set $B$ such that (i) $B \subseteq A$; (ii) $|B| \ge 668$; (iii) for any $u, v \in B$ (not necessarily distinct), $u+v \not\in B$.

There are 51 senators in a senate. The senate needs to be divided into $n$ committees so that each senator is on one committee. Each senator hates exactly three other senators. (If senator A hates senator B, then senator B does not necessarily hate senator A.) Find the smallest $n$ such that it is always possible to arrange the committees so that no senator hates another senator on his or her committee.

In triangle $ABC$, $\angle B = 2\angle C$. Let $P$ and $Q$ be points on the perpendicular bisector of segment $BC$ such that rays $AP$ and $AQ$ trisect $\angle A$. Prove that $PQ < AB$ if and only if $\angle B$ is obtuse.

Let $a,b,c$ be positive real numbers such that \[ a+b+c\geq abc. \] Prove that at least two of the inequalities \[ \frac{2}{a}+\frac{3}{b}+\frac{6}{c}\geq6,\;\;\;\;\;\frac{2}{b}+\frac{3}{c}+\frac{6}{a}\geq6,\;\;\;\;\;\frac{2}{c}+\frac{3}{a}+\frac{6}{b}\geq6 \] are true.

Let $ABCD$ be a convex quadrilateral such that $\angle ABC = \angle ADC = 135^{\circ}$ and \[AC^2\cdot BD^2 = 2\cdot AB\cdot BC\cdot CD\cdot DA.\]Prove that the diagonals of the quadrilateral $ABCD$ are perpendicular.

Find all pairs of nonnegative integers $(m,n)$ such that \[(m+n-5)^2=9mn.\]

Let $A$ be a finite set of positive integers. Prove that there exists a finite set $B$ of positive integers such that $A \subseteq B$ and \[\prod_{x\in B} x = \sum_{x\in B} x^2.\]

These problems are copyright $\copyright$ Mathematical Association of America.