USAMO 2013 — Days 1 and 2

In triangle $ABC$, points $P,Q,R$ lie on $BC,CA,AB$, respectively. Let $\omega_A,\omega_B,\omega_C$ be the circumcircles of $AQR,BRP,CPQ$. Segment $AP$ meets these circles again at $X,Y,Z$, respectively. Prove $YX/XZ=BP/PC$.

Proposed by Zuming Feng (as credited in the supplied compilation).

For $n\ge3$, place $n$ equally spaced points around a circle, label one $A$, and start a marker there. Each move goes clockwise to the next point or the point after that, giving $2n$ distinct directed moves. Let $a_n$ count the ways to go around exactly twice, starting and ending at $A$, without repeating a move. Prove $a_{n-1}+a_n=2^n$ for every $n\ge4$.

Proposed by Kiran Kedlaya (as credited in the supplied compilation).

Arrange $n(n+1)/2$ black-and-white tokens in an equilateral triangular array with $n$ tokens in the largest row, initially all white. An operation flips every token on a line parallel to a side. A configuration is admissible if finitely many operations produce it. For admissible $C$, let $f(C)$ be the minimum number of operations producing $C$. Determine $\max_C f(C)$ for every positive integer $n$.

Proposed by Warut Suksompong (as credited in the supplied compilation).

Find all real $x,y,z\ge1$ such that
$$\min\bigl(\sqrt{x+xyz},\sqrt{y+xyz},\sqrt{z+xyz}\bigr)=\sqrt{x-1}+\sqrt{y-1}+\sqrt{z-1}.$$

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

For positive integers $m,n$, prove that some positive integer $c$ makes $cm$ and $cn$ have the same nonzero decimal digits.

Proposed by Richard Stong (as credited in the supplied compilation).

Let $ABC$ be a triangle. Find all $P$ on segment $BC$ with the following property. If $X,Y$ are the intersections of line $PA$ with the common external tangent lines of the circumcircles of $PAB,PAC$, then
$$\left(\frac{PA}{XY}\right)^2+\frac{PB\cdot PC}{AB\cdot AC}=1.$$

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