ELMO 2024

In convex quadrilateral $ABCD$ , let diagonals $\overline{AC}$ and $\overline{BD}$ intersect at $E$ . Let the circumcircles of $ADE$ and $BCE$ intersect $\overline{AB}$ again at $P \neq A$ and $Q \neq B$ , respectively. Let the circumcircle of $ACP$ intersect $\overline{AD}$ again at $R \neq A$ , and let the circumcircle of $BDQ$ intersect $\overline{BC}$ again at $S \neq B$ . Prove that $A$ , $B$ , $R$ , and $S$ are concyclic.

Tiger Zhang

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.

Linus Tang

For some positive integer $n,$ Elmo writes down the equation $$x_1+x_2+\dots+x_n=x_1+x_2+\dots+x_n.$$ Elmo inserts at least one $f$ to the left side of the equation and adds parentheses to create a valid functional equation. For example, if $n=3,$ Elmo could have created the equation $$f(x_1+f(f(x_2)+x_3))=x_1+x_2+x_3.$$ Cookie Monster comes up with a function $f: \mathbb{Q}\to\mathbb{Q}$ which is a solution to Elmo's functional equation. (In other words, Elmo's equation is satisfied for all choices of $x_1,\dots,x_n\in\mathbb{Q})$ . Is it possible that there is no integer $k$ (possibly depending on $f$ ) such that $f^k(x)=x$ for all $x$ ?

Srinivas Arun

In triangle $ABC$ with $AB<AC$ and $AB+AC=2BC$ , let $M$ be the midpoint of $\overline{BC}$ . Choose point $P$ on the extension of $\overline{BA}$ past $A$ and point $Q$ on segment $\overline{AC}$ such that $M$ lies on $\overline{PQ}$ . Let $X$ be on the opposite side of $\overline{AB}$ from $C$ such that $\overline{AX} \parallel \overline{BC}$ and $AX=AP=AQ$ . Let $\overline{BX}$ intersect the circumcircle of $BMQ$ again at $Y \neq B$ , and let $\overline{CX}$ intersect the circumcircle of $CMP$ again at $Z \neq C$ . Prove that $A$ , $Y$ , and $Z$ are collinear.

Tiger Zhang

For a prime $p$ , let $\mathbb{F}_p$ denote the integers modulo $p$ , and let $\mathbb{F}_p[x]$ be the set of polynomials with coefficients in $\mathbb{F}_p$ . Find all $p$ for which there exists a quartic polynomial $P(x) \in \mathbb{F}_p[x]$ such that for all integers $k$ , there exists some integer $\ell$ such that $P(\ell) \equiv k \pmod p$ . (Note that there are $p^4(p-1)$ quartic polynomials in $\mathbb{F}_p[x]$ in total.)

Aprameya Tripathy