USAMO 2019 — Days 1 and 2

A function $f:\mathbb N\to\mathbb N$ on the positive integers satisfies $f^{f(n)}(n)=n^2/f(f(n))$ for every $n\ge1$, where the exponent denotes iteration. Determine all possible values of $f(1000)$.

Proposed by Evan Chen. Source: Evan Chen’s third-party solution notes.

Let $ABCD$ be a cyclic quadrilateral with $AD^2+BC^2=AB^2$. Its diagonals meet at $E$. A point $P$ on side $AB$ satisfies $\angle APD=\angle BPC$. Prove that line $PE$ passes through the midpoint of $CD$.

Proposed by Ankan Bhattacharya. Source: Evan Chen’s third-party solution notes.

Let $K$ be the positive integers whose decimal representations contain no digit $7$. Determine all polynomials $f$ with nonnegative coefficients such that $f(x)\in K$ for every $x\in K$.

Proposed by Titu Andreescu, Vlad Matei, Cosmin Pohoata. Source: Evan Chen’s third-party solution notes.

Let $n\ge0$ be an integer. Count families of sets $S_{ij}\subseteq\{1,\ldots,2n\}$ for $0\le i,j\le n$ such that $|S_{ij}|=i+j$ and $S_{ij}\subseteq S_{kl}$ whenever $i\le k$ and $j\le l$. The sets need not be distinct.

Proposed by Ricky Liu. Source: Evan Chen’s third-party solution notes.

Let $m,n$ be coprime positive integers. Initially $m/n$ and $n/m$ are on a blackboard. At each step Evan may choose two numbers $x,y$ on the board and add their arithmetic mean $(x+y)/2$ or their harmonic mean $2xy/(x+y)$. For which pairs $(m,n)$ can he write $1$ after finitely many steps?

Proposed by Yannick Yao. Source: Evan Chen’s third-party solution notes.

Find all real-coefficient polynomials $P$ such that

$$\frac{P(x)}{yz}+\frac{P(y)}{zx}+\frac{P(z)}{xy}=P(x-y)+P(y-z)+P(z-x)$$

for all nonzero real $x,y,z$ satisfying $2xyz=x+y+z$.

Proposed by Titu Andreescu, Gabriel Dospinescu. Source: Evan Chen’s third-party solution notes.