Question bank USAJMO 2017

Prove that infinitely many pairs of relatively prime positive integers $a,b\gt 1$ satisfy $a+b\mid a^b+b^a$.

Proposed by Gregory Galperin.

Show that $(3x^3+xy^2)(x^2y+3y^3)=(x-y)^7$ has infinitely many solutions in positive integers, and characterize all of them.

Proposed by Titu Andreescu.

Let $ABC$ be equilateral and $P$ a point on its circumcircle. Set $D=PA\cap BC$, $E=PB\cap CA$, $F=PC\cap AB$. Prove that $[DEF]=2[ABC]$.

Proposed by Titu Andreescu, Luis Gonzalez, Cosmin Pohoata.

Are there 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 $a^2+b^2+c^2+abc-2017$?

Proposed by Titu Andreescu.

Let $O,H$ be the circumcenter and orthocenter of acute triangle $ABC$. Points $M,D$ lie on $BC$ with $BM=CM$ and $\angle BAD=\angle CAD$. Ray $MO$ meets the circumcircle of $BHC$ at $N$. Prove that $\angle ADO=\angle HAN$.

Proposed by Ivan Borsenco.

There are $2n$ distinct points on the unit circle, none equal to $O=(1,0)$, colored with $n$ red and $n$ blue points. Order the red points arbitrarily as $R_1,\ldots,R_n$. Match $R_1$ to the first blue point encountered counterclockwise from it, call this $B_1$, then match $R_2$ to the first remaining blue point, and continue. Prove that the number of counterclockwise arcs $R_i\to B_i$ containing $O$ does not depend on the ordering of the red points.

Proposed by Maria Monks.