USAMO 2017 — Days 1 and 2

Prove that infinitely many pairs of relatively prime positive integers $a,b\gt 1$ satisfy $a+b\mid a^b+b^a$.

Proposed by Gregory Galperin. Source: Evan Chen’s third-party compilation, not an official publication. One editorial point; manual grading only.

Let $m_1,\ldots,m_n$ be positive integers, possibly repeated. For an integer sequence $A=(a_1,\ldots,a_n)$ and a permutation $w$ of this multiset, an $A$-inversion is a pair $i\lt j$ satisfying one of

$$a_i\ge w_i\gt w_j,\qquad w_j\gt a_i\ge w_i,\qquad w_i\gt w_j\gt a_i.$$

Prove that for any integer sequences $A,B$ and every positive integer $k$, the numbers of permutations having exactly $k$ $A$-inversions and exactly $k$ $B$-inversions are equal.

Proposed by Maria Monks. Source: Evan Chen’s third-party compilation, not an official publication. One editorial point; manual grading only.

Let $ABC$ be a scalene triangle with circumcircle $\Omega$ and incenter $I$. Ray $AI$ meets segment $BC$ at $D$ and $\Omega$ again at $M$. The circle with diameter $DM$ meets $\Omega$ again at $K$. Lines $MK$ and $BC$ meet at $S$, and $N$ is the midpoint of $IS$. The circumcircles of $KID$ and $MAN$ intersect at $L_1,L_2$. Prove that $\Omega$ passes through the midpoint of either $IL_1$ or $IL_2$.

Proposed by Evan Chen. Source: Evan Chen’s third-party compilation, not an official publication. One editorial point; manual grading only.

There are $2n$ distinct points on the unit circle, none equal to $O=(1,0)$, colored with $n$ red and $n$ blue points. Order the red points arbitrarily as $R_1,\ldots,R_n$. Match $R_1$ to the first blue point encountered counterclockwise from it, call this $B_1$, then match $R_2$ to the first remaining blue point, and continue. Prove that the number of counterclockwise arcs $R_i\to B_i$ containing $O$ does not depend on the ordering of the red points.

Proposed by Maria Monks. Source: Evan Chen’s third-party compilation, not an official publication. One editorial point; manual grading only.

Find all real $c\gt 0$ for which the lattice points of $\mathbb Z^2$ can be labeled with positive integers so that only finitely many labels occur and any two distinct points with label $i$ are at distance at least $c^i$.

Proposed by Ricky Liu. Source: Evan Chen’s third-party compilation, not an official publication. One editorial point; manual grading only.

Find the minimum of

$$\frac{a}{b^3+4}+\frac{b}{c^3+4}+\frac{c}{d^3+4}+\frac{d}{a^3+4}$$

for $a,b,c,d\ge0$ with $a+b+c+d=4$.

Proposed by Titu Andreescu. Source: Evan Chen’s third-party compilation, not an official publication. One editorial point; manual grading only.