USAJMO 2013

Are there integers $a$ and $b$ such that $a^5b+3$ and $ab^5+3$ are both perfect cubes of integers?

Each cell of an $m\times n$ board is filled with some nonnegative integer. Two numbers in the filling are said to be adjacent if their cells share a common side. (Note that two numbers in cells that share only a corner are not adjacent). The filling is called a garden if it satisfies the following two conditions:

(i) The difference between any two adjacent numbers is either $0$ or $1$ . (ii) If a number is less than or equal to all of its adjacent numbers, then it is equal to $0$ .

Determine the number of distinct gardens in terms of $m$ and $n$ .

In triangle $ABC$ , points $P$ , $Q$ , $R$ lie on sides $BC$ , $CA$ , $AB$ respectively. Let $\omega_A$ , $\omega_B$ , $\omega_C$ denote the circumcircles of triangles $AQR$ , $BRP$ , $CPQ$ , respectively. Given the fact that segment $AP$ intersects $\omega_A$ , $\omega_B$ , $\omega_C$ again at $X$ , $Y$ , $Z$ , respectively, prove that $YX/XZ=BP/PC$ .

Let $f(n)$ be the number of ways to write $n$ as a sum of powers of $2$ , where we keep track of the order of the summation. For example, $f(4)=6$ because $4$ can be written as $4$ , $2+2$ , $2+1+1$ , $1+2+1$ , $1+1+2$ , and $1+1+1+1$ . Find the smallest $n$ greater than $2013$ for which $f(n)$ is odd.

Quadrilateral $XABY$ is inscribed in the semicircle $\omega$ with diameter $XY$ . Segments $AY$ and $BX$ meet at $P$ . Point $Z$ is the foot of the perpendicular from $P$ to line $XY$ . Point $C$ lies on $\omega$ such that line $XC$ is perpendicular to line $AZ$ . Let $Q$ be the intersection of segments $AY$ and $XC$ . Prove that $$\dfrac{BY}{XP}+\dfrac{CY}{XQ}=\dfrac{AY}{AX}.$$

Find all real numbers $x,y,z\geq 1$ satisfying $$\min(\sqrt{x+xyz},\sqrt{y+xyz},\sqrt{z+xyz})=\sqrt{x-1}+\sqrt{y-1}+\sqrt{z-1}.$$