Let $a,b$ be positive integers. Every cell of an $(a+b+1)\times(a+b+1)$ grid is amber or bronze, with at least $a^2+ab-b$ amber and at least $b^2+ab-a$ bronze cells. Prove that one can choose $a$ amber and $b$ bronze cells, no two in the same row or column.
Proposed by Ankan Bhattacharya. Source: Evan Chen’s third-party solution notes; https://aops.com/community/p24774812. One point is a nonofficial editorial normalization; this essay requires manual grading.
Let $b,w\ge2$ be integers and $n=b+w$. Assemble a regular $2n$-gon from $2b$ black and $2w$ white unit rods so that parallel sides have the same color. Translate the black rods to form a convex $2b$-gon $B$, and the white rods to form a convex $2w$-gon $W$. Prove that $[B]-[W]$ depends only on $b,w$, not the assembly. The source example has $b=3,w=2$.
Proposed by Ankan Bhattacharya. Source: Evan Chen’s third-party solution notes; https://aops.com/community/p24775345. One point is a nonofficial editorial normalization; this essay requires manual grading.
Solve for all functions $f:\mathbb R_{>0}\to\mathbb R_{>0}$ satisfying, for every $x,y>0$,
$$f(x)=f(f(f(x))+y)+f(xf(y))f(x+y).$$
Proposed by Hung-Hsun Hans Yu. Source: Evan Chen’s third-party solution notes; https://aops.com/community/p24774907. One point is a nonofficial editorial normalization; this essay requires manual grading.
Find all pairs of primes $(p,q)$ for which $p-q$ and $pq-q$ are both perfect squares.
Proposed by Holden Mui. Source: Evan Chen’s third-party solution notes; https://aops.com/community/p24774670. One point is a nonofficial editorial normalization; this essay requires manual grading.
A function $f:\mathbb R\to\mathbb R$ is essentially increasing if $f(s)\le f(t)$ whenever $s\le t$ and both values are nonzero. Find the smallest integer $k$ such that for any real numbers $x_1,\ldots,x_{2022}$ there are $k$ essentially increasing functions with
$$\sum_{i=1}^k f_i(n)=x_n\quad(1\le n\le2022).$$
Proposed by Gabriel Carroll. Source: Evan Chen’s third-party solution notes; https://aops.com/community/p24774633. One point is a nonofficial editorial normalization; this essay requires manual grading.
There are $2022$ users on Mathbook. Friendship is mutual and permanent. A new friendship may be formed only between users who have at least two friends in common. What is the minimum number of existing friendships that permits everyone eventually to become friends with everyone else?
Proposed by Yannick Yao. Source: Evan Chen’s third-party solution notes; https://aops.com/community/p24774626. One point is a nonofficial editorial normalization; this essay requires manual grading.
