ELMO 2023

Let $m$ be a positive integer. Find, in terms of $m$, all polynomials $P(x)$ with integer coefficients such that for every integer $n$, there exists an integer $k$ such that $P(k)=n^m$.

Proposed by Raymond Feng

Let $a$, $b$, and $n$ be positive integers. A lemonade stand owns $n$ cups, all of which are initially empty. The lemonade stand has a filling machine and an emptying machine, which operate according to the following rules:

• If at any moment, $a$ completely empty cups are available, the filling machine spends the next $a$ minutes filling those $a$ cups simultaneously and doing nothing else.
• If at any moment, $b$ completely full cups are available, the emptying machine spends the next $b$ minutes emptying those $b$ cups simultaneously and doing nothing else. Suppose that after a sufficiently long time has passed, both the filling machine and emptying machine work without pausing. Find, in terms of $a$ and $b$, the least possible value of $n$.

Proposed by Raymond Feng

Convex quadrilaterals $ABCD$, $A_1B_1C_1D_1$, and $A_2B_2C_2D_2$ are similar with vertices in order. Points $A$, $A_1$, $B_2$, $B$ are collinear in order, points $B$, $B_1$, $C_2$, $C$ are collinear in order, points $C$, $C_1$, $D_2$, $D$ are collinear in order, and points $D$, $D_1$, $A_2$, $A$ are collinear in order. Diagonals $AC$ and $BD$ intersect at $P$, diagonals $A_1C_1$ and $B_1D_1$ intersect at $P_1$, and diagonals $A_2C_2$ and $B_2D_2$ intersect at $P_2$. Prove that points $P$, $P_1$, and $P_2$ are collinear.

Proposed by Holden Mui

Let $ABC$ be an acute scalene triangle with orthocenter $H$. Line $BH$ intersects $\overline{AC}$ at $E$ and line $CH$ intersects $\overline{AB}$ at $F$. Let $X$ be the foot of the perpendicular from $H$ to the line through $A$ parallel to $\overline{EF}$. Point $B_1$ lies on line $XF$ such that $\overline{BB_1}$ is parallel to $\overline{AC}$, and point $C_1$ lies on line $XE$ such that $\overline{CC_1}$ is parallel to $\overline{AB}$. Prove that points $B$, $C$, $B_1$, $C_1$ are concyclic.

Proposed by Luke Robitaille

Find the least positive integer $M$ for which there exist a positive integer $n$ and polynomials $P_1(x)$, $P_2(x)$, $\ldots$, $P_n(x)$ with integer coefficients satisfying $$Mx=P_1(x)^3+P_2(x)^3+\cdots+P_n(x)^3.$$

Proposed by Karthik Vedula

For a set $S$ of positive integers and a positive integer $n$, consider the game of $(n,S)$-nim, which is as follows. A pile starts with $n$ watermelons. Two players, Deric and Erek, alternate turns eating watermelons from the pile, with Deric going first. On any turn, the number of watermelons eaten must be an element of $S$. The last player to move wins. Let $f(S)$ denote the set of positive integers $n$ for which Deric has a winning strategy in $(n,S)$-nim.

Let $T$ be a set of positive integers. Must the sequence $$T, f(T), f(f(T)), \;\ldots$$ be eventually constant?

Proposed by Brandon Wang and Edward Wan