Here's a list of math problems I've written for various math olympiads and shortlists.
A line in the plane is called sunny if it is not parallel to any of the \(x\)-axis, the \(y\)-axis, and the line \(x+y=0\). Let \(n\ge 3\) be a given integer. Determine all nonnegative integers \(k\) such that there exist \(n\) distinct lines in the plane satisfying both of the following:
A pond has $2025$ lily pads arranged in a circle. Two frogs, Alice and Bob, begin on different lily pads. A frog jump is a jump which travels $2$, $3$, or $5$ positions clockwise. Alice and Bob each make a series of frog jumps, and each frog ends on the same lily pad that it started from. Given that each lily pad is the destination of exactly one jump, prove that each frog completes exactly two laps around the pond (i.e. travels $4050$ positions in total).
For positive integers $a$ and $b$, an $(a,b)$-shuffle of a deck of $a+b$ cards is any shuffle that preserves the relative order of the top $a$ cards and the relative order of the bottom $b$ cards. Let $n$, $k$, $a_1$, $a_2$, $\dots$, $a_k$, $b_1$, $b_2$, $\dots$, $b_k$ be fixed positive integers such that $a_i+b_i=n$ for all $1\leq i\leq k$. Big Bird has a deck of $n$ cards and will perform an $(a_i,b_i)$-shuffle for each $1\leq i\leq k$, in ascending order of $i$. Suppose that Big Bird can reverse the order of the deck. Prove that Big Bird can also achieve any of the $n!$ permutations of the cards.
Let $m,n,a_1,a_2,\dots,a_n$ be positive integers and $r$ be a real number. Prove that the equation
\[ \lfloor a_1x\rfloor + \lfloor a_2x\rfloor + \cdots + \lfloor a_nx\rfloor = sx + r \]
has exactly $ms$ solutions in $x$, where $s = a_1 + a_2 + \cdots + a_n + m$.
Allen and Alan play a game. A nonconstant polynomial $P(x,y)$ with real coefficients and a positive integer $d$ greater than the degree of $P$ are known to both Allen and Alan. Alan thinks of a polynomial $Q(x,y)$ with real coefficients and degree at most $d$ and keeps it secret. Allen can make queries of the form $(s,t)$, where $s$ and $t$ are real numbers such that $P(s,t)\ne 0$. Alan must respond with the value $Q(s,t)$. Allen’s goal is to determine whether $P$ divides $Q$. Find (in terms of $P$ and $d$) the smallest positive integer $g$ such that Allen can always achieve this goal making no more than $g$ queries.
Let $\mathbb{R}_{+}$ denote the set of positive real numbers. Find all functions $f:\mathbb{R}_{+}\to\mathbb{R}$ and $g:\mathbb{R}_{+}\to\mathbb{R}$ such that for all $x,y\in\mathbb{R}_{+}$,
\[ g(x)-g(y) = (x-y)f(xy). \]
Let $n$ and $k$ be positive integers and $G$ be a complete graph on $n$ vertices. Each edge of $G$ is colored one of $k$ colors such that every triangle consists of either three edges of the same color or three edges of three different colors. Furthermore, there exist two different-colored edges. Prove that \[ n \le (k-1)^2. \]
Let $n\ge 5$ be an integer. A trapezoid with base lengths of $1$ and $r$ is tiled by $n$ (not necessarily congruent) equilateral triangles. In terms of $n$, find the maximum possible value of $r$.
Let $ABC$ be a triangle. Construct rectangles $BA_1A_2C$, $CB_1B_2A$, and $AC_1C_2B$ outside $ABC$ such that $\angle BCA_1=\angle CAB_1=\angle ABC_1$. Let $A_1B_2$ and $A_2C_1$ intersect at $A'$ and define $B',C'$ similarly. Prove that line $AA'$ bisects $B'C'$.
Find all pairs $(n,d)$ of positive integers such that $d\mid n^2$ and $(n-d)^2<2d$.
Find all pairs $(a,b)$ of positive integers such that $a^2 \mid b^3+1$ and $b^2 \mid a^3+1$.
Let $f:\mathbb{R}\to\mathbb{R}$ be a function such that for all real numbers $x\ne 1$,
\[ f(x-f(x)) + f(x) = x^2 - x + \frac{1}{x-1}. \]
Find all possible values of $f(2023)$.
Let $n$ be a positive integer and consider an $n\times n$ square grid. For $1\le k\le n$, a python is a snake that occupies some consecutive cells in a single row, and no other cells. Similarly, an anaconda is a snake that occupies some consecutive cells in a single column, and no other cells. The length of a snake is the number of cells it occupies.
The grid contains at least one python or anaconda, and it satisfies the following properties:
Prove that the sum of the squares of the lengths of the snakes is at least $n^2$.
A discrete hexagon with center $(a,b,c)$ (where $a,b,c$ are integers) and radius $r$ (a nonnegative integer) is the set of lattice points $(x,y,z)$ such that $x+y+z=a+b+c$ and \[ \max\bigl(|x-a|,\ |y-b|,\ |z-c|\bigr)\le r. \]
Let $n$ be a nonnegative integer and $S$ be the set of triples $(x,y,z)$ of nonnegative integers such that $x+y+z=n$. If $S$ is partitioned into discrete hexagons, show that at least $n+1$ hexagons are needed.