US-TST 2001

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 $ABCD$ 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

$$\prod_{x\in B} x = \sum_{x\in B} x^2.$$