Question bank USAJMO 2013

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

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

Every cell of an $m\times n$ board is filled with a nonnegative integer. Two cells are adjacent if they share a side. A filling is a garden if (i) any two adjacent numbers differ by $0$ or $1$, and (ii) a number less than or equal to all its adjacent numbers must equal $0$. Determine the number of gardens in terms of $m,n$.

Proposed by Sungyoon Kim (credited in the supplied notes).

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).

Let $f(n)$ count the ordered expressions of $n$ as a sum of powers of $2$. For example $f(4)=6$, corresponding to $4$, $2+2$, $2+1+1$, $1+2+1$, $1+1+2$, $1+1+1+1$. Find the smallest $n>2013$ for which $f(n)$ is odd.

Proposed by Kiran Kedlaya (credited in the supplied notes).

Quadrilateral $XABY$ is inscribed in a semicircle with diameter $XY$. Segments $AY$ and $BX$ meet at $P$. Let $Z$ be the perpendicular projection of $P$ onto line $XY$. Point $C$ lies on the semicircle with $XC\perp AZ$, and $Q=AY\cap XC$. Prove
$$\frac{BY}{XP}+\frac{CY}{XQ}=\frac{AY}{AX}.$$

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

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).