USAJMO 2017

Prove that there are infinitely many distinct pairs $(a, b)$ of relatively prime integers $a>1$ and $b>1$ such that $a^b+b^a$ is divisible by $a+b$ .

Let $ABC$ be an equilateral triangle, and point $P$ on its circumcircle. Let $PA$ and $BC$ intersect at $D$ , $PB$ and $AC$ intersect at $E$ , and $PC$ and $AB$ intersect at $F$ . Prove that the area of $\triangle DEF$ is twice the area of $\triangle ABC$ .

Proposed by Titu Andreescu, Luis Gonzales, Cosmin Pohoata

Are there any triples $(a,b,c)$ of positive integers such that $(a-2)(b-2)(c-2)+12$ is a prime number that properly divides the positive number $a^2+b^2+c^2+abc-2017$ ?

Let $O$ and $H$ be the circumcenter and the orthocenter of an acute triangle $ABC$ . Points $M$ and $D$ lie on side $BC$ such that $BM=CM$ and $\angle BAD = \angle CAD$ . Ray $MO$ intersects the circumcircle of triangle $BHC$ in point $N$ . Prove that $\angle ADO = \angle HAN$ .

Let $P_1$ , $P_2$ , $\dots$ , $P_{2n}$ be $2n$ distinct points on the unit circle $x^2+y^2=1$ , other than $(1,0)$ . Each point is colored either red or blue, with exactly $n$ red points and $n$ blue points. Let $R_1$ , $R_2$ , $\dots$ , $R_n$ be any ordering of the red points. Let $B_1$ be the nearest blue point to $R_1$ traveling counterclockwise around the circle starting from $R_1$ . Then let $B_2$ be the nearest of the remaining blue points to $R_2$ travelling counterclockwise around the circle from $R_2$ , and so on, until we have labeled all of the blue points $B_1, \dots, B_n$ . Show that the number of counterclockwise arcs of the form $R_i \to B_i$ that contain the point $(1,0)$ is independent of the way we chose the ordering $R_1, \dots, R_n$ of the red points.