Let $n\ge2$. Carl has $n$ books, with distinct heights and distinct widths, initially arranged from left to right in increasing height. A move swaps two adjacent books when the left book is wider and shorter than the right book. He continues until no move is possible. Prove that every sequence terminates and the final order is increasing in width.
Proposed by Milan Haiman. Evan Chen’s third-party compilation; source discussion https://aops.com/community/p15952780. Source video https://youtu.be/r7j0oRtpErA?t=1190.
Let $\omega$ be the incircle of a fixed equilateral triangle $ABC$. A variable tangent line $\ell$ to $\omega$ meets the interiors of $BC,CA$ at $P,Q$, respectively. Choose $R$ with $PR=PA$ and $QR=QB$. Find all possible locations of $R$.
Proposed by Titu Andreescu, Waldemar Pompe. Evan Chen’s third-party compilation; source discussion https://aops.com/community/p15952801. Source video https://youtu.be/r7j0oRtpErA?t=1575.
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 $ABCD$ be a convex cyclic quadrilateral with $DA\lt AB=BC\lt CD$. Choose $E$ on $CD$ and $F$ on $AB$ such that $BE\perp AC$ and $EF\parallel BC$. Prove $FB=FD$.
Proposed by Milan Haiman. Evan Chen’s third-party compilation; source discussion https://aops.com/community/p15952890. Source video https://youtu.be/r7j0oRtpErA?t=3960.
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
Let $n\ge2$, and let $P(x_1,\ldots,x_n)$ be a nonconstant polynomial with real coefficients. Suppose $P$ vanishes whenever two of its arguments are equal. Prove that $P$ has at least $n!$ terms.
Proposed by Ankan Bhattacharya. Evan Chen’s third-party compilation; source discussion https://aops.com/community/p15952921. Source video https://youtu.be/r7j0oRtpErA?t=4905.