Found problems: 6
2024 USEMO, 2
Let $k$ be a fixed positive integer. For each integer $1 \leq i \leq 4$, let $x_i$ and $y_i$ be positive integers such that their least common multiple is $k$. Suppose that the four points $(x_1, y_1)$, $(x_2, y_2)$, $(x_3, y_3)$, $(x_4, y_4)$ are the vertices of a non-degenerate rectangle in the Cartesian plane. Prove that $x_1x_2x_3x_4$ is a perfect square.
[i]Andrei Chirita[/i]
2024 USEMO, 4
Find all sequences $a_1$, $a_2$, $\dots$ of nonnegative integers such that for all positive integers $n$, the polynomial \[1+x^{a_1}+x^{a_2}+\dots+x^{a_n}\] has at least one integer root. (Here $x^0=1$.)
[i]Kornpholkrit Weraarchakul[/i]
2024 USEMO, 3
Let $ABC$ be a triangle with incenter $I$. Two distinct points $P$ and $Q$ are chosen on the circumcircle of $ABC$ such that
\[ \angle API = \angle AQI = 45^\circ. \]
Lines $PQ$ and $BC$ meet at $S$. Let $H$ denote the foot of the altitude from $A$ to $BC$. Prove that $\angle AHI = \angle ISH$.
[i]Matsvei Zorka[/i]
2024 USEMO, 6
Let $n$ be an odd positive integer and consider an $n \times n$ chessboard of $n^2$ unit squares. In some of the cells of the chessboard, we place a knight. A knight in a cell $c$ is said to [i]attack [/i] a cell $c'$ if the distance between the centers of $c$ and $c'$ is exactly $\sqrt{5}$ (in particular, a knight does not attack the cell which it occupies).
Suppose each cell of the board is attacked by an even number of knights (possibly zero). Show that the configuration of knights is symmetric with respect to all four axes of symmetry of the board (i.e. the configuration of knights is both horizontally and vertically symmetric, and also unchanged by reflection along either diagonal of the chessboard).
[i]NIkolai Beluhov[/i]
2024 USEMO, 5
Let $ABC$ be a scalene triangle whose incircle is tangent to $BC$, $CA$, $AB$ at $D$, $E$, $F$ respectively. Lines $BE$ and $CF$ meet at $G$. Prove that there exists a point $X$ on the circumcircle of triangle $EFG$ such that the circumcircles of triangles $BCX$ and $EFG$ are tangent, and \[\angle BGC = \angle BXC + \angle EDF.\]
[i]Kornpholkrit Weraarchakul[/i]
2024 USEMO, 1
There are $1001$ stacks of coins $S_1, S_2, \dots, S_{1001}$. Initially, stack $S_k$ has $k$ coins for each $k = 1,2,\dots,1001$. In an operation, one selects an ordered pair $(i,j)$ of indices $i$ and $j$ satisfying $1 \le i < j \le 1001$ subject to two conditions:
[list]
[*]The stacks $S_i$ and $S_j$ must each have at least $1$ coin.
[*]The ordered pair $(i,j)$ must [i]not[/i] have been selected before.
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Then, if $S_i$ and $S_j$ have $a$ coins and $b$ coins respectively, one removes $\gcd(a,b)$ coins from each stack.
What is the maximum number of times this operation could be performed?
[i]Galin Totev[/i]