USAJMO 2016

The isosceles triangle $ABC$, with $AB=AC$, is inscribed in circle $\omega$. Let $P$ vary on the arc $BC$ not containing $A$, and let $I_B,I_C$ be the incenters of $ABP,ACP$. Prove that the circumcircle of $PI_BI_C$ passes through a fixed point.

Proposed by Ivan Borsenco, Zuming Feng.

Prove that there exists a positive integer $n\lt 10^6$ such that $5^n$ has six consecutive zeros in its decimal representation.

Proposed by Evan Chen.

Let $X_1$, $X_2$, …, $X_{100}$ be a sequence of mutually distinct nonempty subsets of a set $S$. Any two sets $X_i$ and $X_{i+1}$ are disjoint and their union is not the whole set $S$, that is, $X_i \cap X_{i+1} = \emptyset$ and $X_i \cup X_{i+1} \neq S$, for all $i \in \{1, …, 99\}$. Find the smallest possible number of elements in $S$.

Proposed by Iurie Boreico.

Find, with proof, the least integer $N$ such that if any $2016$ elements are removed from $\{1,2,\ldots,N\}$, one can still find $2016$ distinct remaining numbers whose sum is $N$.

Proposed by Gregory Galperin.

Let $ABC$ be acute with circumcenter $O$. Let $H$ be the foot from $A$ to $BC$, and $P,Q$ the feet from $H$ to $AB,AC$. Given $AH^2=2AO^2$, prove that $O,P,Q$ are collinear.

Proposed by Zuming Feng, Jacek Fabrykowski.

Find all functions $f \colon \mathbb{R} \to \mathbb{R}$ such that for all real numbers $x$ and $y$, $$ (f(x)+xy) \cdot f(x-3y) + (f(y)+xy) \cdot f(3x-y) = (f(x+y))^2. $$

Proposed by Titu Andreescu.