Question bank USAMO 2020

An acute triangle $ABC$ is inscribed in circle $\omega$ with center $O$. A variable point $X$ lies on the minor arc $AB$, and $CX$ meets $AB$ at $D$. Let $O_1,O_2$ be the circumcenters of $ADX,BDX$. Determine all $X$ minimizing $[OO_1O_2]$.

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

Source video link: https://youtu.be/r7j0oRtpErA?t=2070

An empty $2020\times2020\times2020$ cube has a unit-square grid on each face. A beam is a $1\times1\times2020$ prism. Its two unit faces must occupy grid cells on opposite cube faces; beam interiors must be disjoint; and the interior of each of its four long faces must touch a cube face or the interior of a face of another beam. Find the smallest positive possible number of beams.

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

Source video link: https://youtu.be/r7j0oRtpErA?t=2440

Let $p$ be an odd prime. An integer $x$ is a quadratic non-residue if $x\not\equiv t^2\pmod p$ for every integer $t$. Let $A$ consist of integers $1\le a\lt p$ such that both $a$ and $4-a$ are quadratic non-residues. Find the remainder of $\prod_{a\in A}a$ modulo $p$.

Proposed by Richard Stong, Toni Bluher. Source: Evan Chen’s third-party solution notes.

Source video link: https://youtu.be/r7j0oRtpErA?t=3240

Let $(a_1,b_1),\ldots,(a_{100},b_{100})$ be distinct ordered pairs of nonnegative integers. Let $N$ count pairs $1\le i\lt j\le100$ for which $|a_ib_j-a_jb_i|=1$. Determine the largest possible value of $N$.

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

Source video link: https://youtu.be/r7j0oRtpErA?t=4530

A finite set $S$ of plane points is overdetermined if $|S|\ge2$ and a nonzero real polynomial $P$ of degree at most $|S|-2$ satisfies $P(x)=y$ at every $(x,y)\in S$. For each $n\ge2$, find the largest number of overdetermined subsets that an n-point set can have while the whole set is not overdetermined.

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

Source video link: https://youtu.be/r7j0oRtpErA?t=5420

Let $n\ge2$. Suppose $x_1\ge\cdots\ge x_n$ and $y_1\ge\cdots\ge y_n$ are real, with $\sum x_i=\sum y_i=0$ and $\sum x_i^2=\sum y_i^2=1$. Prove

$$\sum_{i=1}^n(x_iy_i-x_iy_{n+1-i})\ge\frac2{\sqrt{n-1}}.$$

Proposed by David Speyer, Kiran Kedlaya. Source: Evan Chen’s third-party solution notes.

Source video link: https://youtu.be/r7j0oRtpErA?t=6165