ELMO 2009

Let $a,b,c$ be positive integers such that $a^2 - bc$ is a square. Prove that $2a + b + c$ is not prime.

Evan o'Dorney

Let $ABC$ be a triangle such that $AB < AC$ . Let $P$ lie on a line through $A$ parallel to line $BC$ such that $C$ and $P$ are on the same side of line $AB$ . Let $M$ be the midpoint of segment $BC$ . Define $D$ on segment $BC$ such that $\angle BAD = \angle CAM$ , and define $T$ on the extension of ray $CB$ beyond $B$ so that $\angle BAT = \angle CAP$ . Given that lines $PC$ and $AD$ intersect at $Q$ , that lines $PD$ and $AB$ intersect at $R$ , and that $S$ is the midpoint of segment $DT$ , prove that if $A$ , $P$ , $Q$ , and $R$ lie on a circle, then $Q$ , $R$ , and $S$ are collinear.

David Rush

Let $a,b,c$ be nonnegative real numbers. Prove that $$a(a - b)(a - 2b) + b(b - c)(b - 2c) + c(c - a)(c - 2a) \geq 0.$$Wenyu Cao

Let $n$ be a positive integer. Given $n^2$ points in a unit square, prove that there exists a broken line of length $2n + 1$ that passes through all the points.

Allen Yuan

Let $ABCDEFG$ be a regular heptagon with center $O$ . Let $M$ be the centroid of $\triangle ABD$ . Prove that $\cos^2(\angle GOM)$ is rational and determine its value.

Evan o'Dorney

Let $p$ be an odd prime and $x$ be an integer such that $p \mid x^3 - 1$ but $p \nmid x - 1$ . Prove that $$p \mid (p - 1)!\left(x - \frac {x^2}{2} + \frac {x^3}{3} - \cdots - \frac {x^{p - 1}}{p - 1}\right).$$John Berman