For every ordered pair of integers $(i,j)$, not necessarily positive, we wish to select a point $P_{i,j}$ in the Cartesian plane whose coordinates lie inside the unit square defined by \[ i < x < i+1, \qquad j < y < j+1. \]Find all real numbers $c > 0$ for which it's possible to choose these points such that for all integers $i$ and $j$, the (possibly concave or degenerate) quadrilateral $P_{i,j} P_{i+1,j} P_{i+1,j+1} P_{i,j+1}$ has perimeter strictly less than $c$. Karthik Vedula

Let $p$ be an odd prime number. Suppose $P$ and $Q$ are polynomials with integer coefficients such that $P(0)=Q(0)=1$, there is no nonconstant polynomial dividing both $P$ and $Q$, and \[ 1 + \cfrac{x}{1 + \cfrac{2x}{1 + \cfrac{\ddots}{1 + (p-1)x}}}=\frac{P(x)}{Q(x)}. \]Show that all coefficients of $P$ except for the constant coefficient are divisible by $p$, and all coefficients of $Q$ are not divisible by $p$. Andrew Gu

Let $A = \{a_1, \dots, a_{2024}\}$ be a set of $2024$ pairwise distinct real numbers. Assume that there exist positive integers $b_1, b_2,\dotsc,b_{2024}$ such that \[ a_1b_1 + a_2b_2 + \dots + a_{2024}b_{2024} = 0. \]Prove that one can choose $a_{2025}, a_{2026}, a_{2027}, \dots$ such that $a_k \in A$ for all $k \ge 2025$ and, for every positive integer $d$, there exist infinitely many positive integers $n$ satisfying \[ \sum_{k=1}^n a_k k^d = 0. \]Daniel Zhu

Let $ABCD$ be a quadrilateral inscribed in a circle with center $O$ and $E$ be the intersection of segments $AC$ and $BD$. Let $\omega_1$ be the circumcircle of $ADE$ and $\omega_2$ be the circumcircle of $BCE$. The tangent to $\omega_1$ at $A$ and the tangent to $\omega_2$ at $C$ meet at $P$. The tangent to $\omega_1$ at $D$ and the tangent to $\omega_2$ at $B$ meet at $Q$. Show that $OP=OQ$. Merlijn Staps

For a positive integer $k$, let $s(k)$ denote the number of $1$s in the binary representation of $k$. Prove that for any positive integer $n$, \[\sum_{i=1}^{n}(-1)^{s(3i)} > 0.\]Holden Mui

Determine whether there exists a function $f: \mathbb{Z}_{> 0} \rightarrow \mathbb{Z}_{> 0}$ such that for all positive integers $m$ and $n$, \[f(m+nf(m))=f(n)^m+2024! \cdot m.\]Jaedon Whyte

An infinite sequence $a_1$, $a_2$, $a_3$, $\ldots$ of real numbers satisfies \[ a_{2n-1} + a_{2n} > a_{2n+1} + a_{2n+2} \qquad \mbox{and} \qquad a_{2n} + a_{2n+1} < a_{2n+2} + a_{2n+3} \]for every positive integer $n$. Prove that there exists a real number $C$ such that $a_{n} a_{n+1} < C$ for every positive integer $n$. Merlijn Staps

Let $ABC$ be a scalene triangle, and let $D$ be a point on side $BC$ satisfying $\angle BAD=\angle DAC$. Suppose that $X$ and $Y$ are points inside $ABC$ such that triangles $ABX$ and $ACY$ are similar and quadrilaterals $ACDX$ and $ABDY$ are cyclic. Let lines $BX$ and $CY$ meet at $S$ and lines $BY$ and $CX$ meet at $T$. Prove that lines $DS$ and $AT$ are parallel. Michael Ren

Let $n \ge 2$ be a fixed integer. The cells of an $n \times n$ table are filled with the integers from $1$ to $n^2$ with each number appearing exactly once. Let $N$ be the number of unordered quadruples of cells on this board which form an axis-aligned rectangle, with the two smaller integers being on opposite vertices of this rectangle. Find the largest possible value of $N$. Anonymous