USAMO 2017

Prove that there are infinitely many distinct pairs $(a, b)$ of relatively prime integers $a>1$ and $b>1$ such that $a^b+b^a$ is divisible by $a+b$ .

Let $m_1, m_2, \ldots, m_n$ be a collection of $n$ positive integers, not necessarily distinct. For any sequence of integers $A = (a_1, \ldots, a_n)$ and any permutation $w = w_1, \ldots, w_n$ of $m_1, \ldots, m_n$ , define an $A$ -inversion of $w$ to be a pair of entries $w_i, w_j$ with $i < j$ for which one of the following conditions holds:

• $a_i \ge w_i > w_j$
• $w_j > a_i \ge w_i$ , or
• $w_i > w_j > a_i$ .

Show that, for any two sequences of integers $A = (a_1, \ldots, a_n)$ and $B = (b_1, \ldots, b_n)$ , and for any positive integer $k$ , the number of permutations of $m_1, \ldots, m_n$ having exactly $k$ $A$ -inversions is equal to the number of permutations of $m_1, \ldots, m_n$ having exactly $k$ $B$ -inversions.

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

Proposed by Evan Chen

Let $P_1$ , $P_2$ , $\dots$ , $P_{2n}$ be $2n$ distinct points on the unit circle $x^2+y^2=1$ , other than $(1,0)$ . Each point is colored either red or blue, with exactly $n$ red points and $n$ blue points. Let $R_1$ , $R_2$ , $\dots$ , $R_n$ be any ordering of the red points. Let $B_1$ be the nearest blue point to $R_1$ traveling counterclockwise around the circle starting from $R_1$ . Then let $B_2$ be the nearest of the remaining blue points to $R_2$ travelling counterclockwise around the circle from $R_2$ , and so on, until we have labeled all of the blue points $B_1, \dots, B_n$ . Show that the number of counterclockwise arcs of the form $R_i \to B_i$ that contain the point $(1,0)$ is independent of the way we chose the ordering $R_1, \dots, R_n$ of the red points.

Let $\mathbf{Z}$ denote the set of all integers. Find all real numbers $c > 0$ such that there exists a labeling of the lattice points $( x, y ) \in \mathbf{Z}^2$ with positive integers for which:

• only finitely many distinct labels occur, and
• for each label $i$ , the distance between any two points labeled $i$ is at least $c^i$ .

Proposed by Ricky Liu

Find the minimum possible value of $$\frac{a}{b^3+4}+\frac{b}{c^3+4}+\frac{c}{d^3+4}+\frac{d}{a^3+4}$$ given that $a$ , $b$ , $c$ , $d$ are nonnegative real numbers such that $a+b+c+d=4$ .

Proposed by Titu Andreescu