USAJMO 2014

Let $a,b,c\ge1$ be real. Prove

$$\min\left(\frac{10a^2-5a+1}{b^2-5b+10},\frac{10b^2-5b+1}{c^2-5c+10},\frac{10c^2-5c+1}{a^2-5a+10}\right)\le abc.$$

Proposed by Titu Andreescu (credited in the supplied notes).

Let $ABC$ be an acute nonequilateral triangle with $\angle A=60^\circ$, circumcenter $O$ and orthocenter $H$. (a) Prove that line $OH$ meets segments $AB,AC$ at $P,Q$, respectively. (b) If $s=[APQ]$ and $t=[BPQC]$, determine the range of $s/t$.

Proposed by Zuming Feng (credited in the supplied notes).

Find all functions $f:\mathbb Z\to\mathbb Z$ satisfying

$$xf(2f(y)-x)+y^2f(2x-f(y))=\frac{f(x)^2}{x}+f(yf(y))$$

for every pair of integers $x,y$ with $x\ne0$.

Proposed by Titu Andreescu (as credited in the supplied compilation).

Fix an integer $b\ge2$, and let $s_b(n)$ be the sum of the base-$b$ digits of $n$. Show that infinitely many positive integers cannot be expressed as $n+s_b(n)$ with $n$ a positive integer.

Proposed by Palmer Mebane (credited in the supplied notes).

On an infinite regular hexagonal grid, initially empty, players $A,B$ alternate, with $A$ first. On her turn $A$ puts counters in two adjacent empty cells; on his turn $B$ removes any one counter. $A$ wins as soon as $k$ consecutive cells in one line are occupied. Find the least positive $k$ for which $A$ cannot force a win in finitely many moves, or show no such minimum exists.

Proposed by Palmer Mebane (as credited in the supplied compilation).

Triangle $ABC$ has incenter $I$, incircle $\gamma$ and circumcircle $\Gamma$. Let $M,N,P$ be the midpoints of $BC,CA,AB$, and $E,F$ the tangency points of $\gamma$ with $CA,AB$. Set $U=EF\cap MN$, $V=EF\cap MP$ using the full lines, and let $X$ be the midpoint of arc $BAC$ on $\Gamma$. (a) Prove that $I$ lies on ray $CV$. (b) Prove that line $XI$ bisects $UV$.

Proposed by Titu Andreescu and Cosmin Pohoata (credited in the supplied notes).