Found problems: 190
2023 ELMO Shortlist, N1
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 \(P(k)=n^m\).
[i]Proposed by Raymond Feng[/i]
2023 ELMO Shortlist, G5
Let \(ABC\) be an acute triangle with circumcircle \(\omega\). Let \(P\) be a variable point on the arc \(BC\) of \(\omega\) not containing \(A\). Squares \(BPDE\) and \(PCFG\) are constructed such that \(A\), \(D\), \(E\) lie on the same side of line \(BP\) and \(A\), \(F\), \(G\) lie on the same side of line \(CP\). Let \(H\) be the intersection of lines \(DE\) and \(FG\). Show that as \(P\) varies, \(H\) lies on a fixed circle.
[i]Proposed by Karthik Vedula[/i]
2024 ELMO Shortlist, N2
Call a positive integer [i]emphatic[/i] if it can be written in the form $a^2+b!$, where $a$ and $b$ are positive integers. Prove that there are infinitely many positive integers $n$ such that $n$, $n+1$, and $n+2$ are all [i]emphatic[/i].
[i]Allen Wang[/i]
2010 ELMO Problems, 2
2010 MOPpers are assigned numbers 1 through 2010. Each one is given a red slip and a blue slip of paper. Two positive integers, A and B, each less than or equal to 2010 are chosen. On the red slip of paper, each MOPper writes the remainder when the product of A and his or her number is divided by 2011. On the blue slip of paper, he or she writes the remainder when the product of B and his or her number is divided by 2011. The MOPpers may then perform either of the following two operations:
[list]
[*] Each MOPper gives his or her red slip to the MOPper whose number is written on his or her blue slip.
[*] Each MOPper gives his or her blue slip to the MOPper whose number is written on his or her red slip.[/list]
Show that it is always possible to perform some number of these operations such that each MOPper is holding a red slip with his or her number written on it.
[i]Brian Hamrick.[/i]
2023 ELMO Shortlist, C3
Find all pairs of positive integers \((a,b)\) with the following property: there exists an integer \(N\) such that for any integers \(m\ge N\) and \(n\ge N\), every \(m\times n\) grid of unit squares may be partitioned into \(a\times b\) rectangles and fewer than \(ab\) unit squares.
[i]Proposed by Holden Mui[/i]
2010 ELMO Shortlist, 5
Let $n > 1$ be a positive integer. A 2-dimensional grid, infinite in all directions, is given. Each 1 by 1 square in a given $n$ by $n$ square has a counter on it. A [i]move[/i] consists of taking $n$ adjacent counters in a row or column and sliding them each by one space along that row or column. A [i]returning sequence[/i] is a finite sequence of moves such that all counters again fill the original $n$ by $n$ square at the end of the sequence.
[list]
[*] Assume that all counters are distinguishable except two, which are indistinguishable from each other. Prove that any distinguishable arrangement of counters in the $n$ by $n$ square can be reached by a returning sequence.
[*] Assume all counters are distinguishable. Prove that there is no returning sequence that switches two counters and returns the rest to their original positions.[/list]
[i]Mitchell Lee and Benjamin Gunby.[/i]
2019 ELMO Shortlist, G4
Let triangle $ABC$ have altitudes $BE$ and $CF$ which meet at $H$. The reflection of $A$ over $BC$ is $A'$. Let $(ABC)$ meet $(AA'E)$ at $P$ and $(AA'F)$ at $Q$. Let $BC$ meet $PQ$ at $R$. Prove that $EF \parallel HR$.
[i]Proposed by Daniel Hu[/i]
2023 ELMO Shortlist, C7
A [i]discrete hexagon with center \((a,b,c)\) \emph{(where \(a\), \(b\), \(c\) are integers)[/i] and radius \(r\) [i](a nonnegative integer)[/i]} is the set of lattice points \((x,y,z)\) such that \(x+y+z=a+b+c\) and \(\max(|x-a|,|y-b|,|z-c|)\le r\).
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.
[i]Proposed by Linus Tang[/i]
2024 ELMO Shortlist, C1
Let $n \ge 3$ be a positive integer, and let $S$ be a set of $n$ distinct points in the plane. Call an unordered pair of distinct points ${A,B}$ [i]tasty[/i] if there exists a circle passing through $A$ and $B$ not passing through or containing any other point in $S$. Find the maximum number of tasty pairs over all possible sets $S$ of $n$ points.
[i]Tiger Zhang[/i]
2023 ELMO Shortlist, C8
Let \(n\ge3\) be a fixed integer, and let \(\alpha\) be a fixed positive real number. There are \(n\) numbers written around a circle such that there is exactly one \(1\) and the rest are \(0\)'s. An [i]operation[/i] consists of picking a number \(a\) in the circle, subtracting some positive real \(x\le a\) from it, and adding \(\alpha x\) to each of its neighbors.
Find all pairs \((n,\alpha)\) such that all the numbers in the circle can be made equal after a finite number of operations.
[i]Proposed by Anthony Wang[/i]
2024 ELMO Shortlist, A4
The number $2024$ is written on a blackboard. Each second, if there exist positive integers $a,b,k$ such that $a^k+b^k$ is written on the blackboard, you may write $a^{k'}+b^{k'}$ on the blackboard for any positive integer $k'.$ Find all positive integers that you can eventually write on the blackboard.
[i]Srinivas Arun[/i]
2024 ELMO Shortlist, N5
Let $T$ be a finite set of squarefree integers.
(a) Show that there exists an integer polynomial $P(x)$ such that the set of squarefree numbers in the range of $P(n)$ across all $n \in \mathbb{Z}$ is exactly $T$.
(b) Suppose that $T$ is allowed to be infinite. Is it still true that for all choices of $T$, such an integer polynomial $P(x)$ exists?
[i]Allen Wang[/i]
2024 ELMO Shortlist, A2
Let $n$ be a positive integer. Find the number of sequences $a_0,a_1,a_2,\dots,a_{2n}$ of integers in the range $[0,n]$ such that for all integers $0\leq k\leq n$ and all nonnegative integers $m$, there exists an integer $k\leq i\leq 2k$ such that $\lfloor k/2^m\rfloor=a_i.$
[i]Andrew Carratu[/i]
2023 ELMO Shortlist, A2
Let \(\mathbb R_{>0}\) denote the set of positive real numbers. Find all functions \(f:\mathbb R_{>0}\to\mathbb R_{>0}\) such that for all positive real numbers \(x\) and \(y\), \[f(xy+1)=f(x)f\left(\frac1x+f\left(\frac1y\right)\right).\]
[i]Proposed by Luke Robitaille[/i]
2023 ELMO Shortlist, C2
Alice is performing a magic trick. She has a standard deck of 52 cards, which she may order beforehand. She invites a volunteer to pick an integer \(0\le n\le 52\), and cuts the deck into a pile with the top \(n\) cards and a pile with the remaining \(52-n\). She then gives both piles to the volunteer, who riffles them together and hands the deck back to her face down. (Thus, in the resulting deck, the cards that were in the deck of size \(n\) appear in order, as do the cards that were in the deck of size \(52-n\).)
Alice then flips the cards over one-by-one from the top. Before flipping over each card, she may choose to guess the color of the card she is about to flip over. She stops if she guesses incorrectly. What is the maximum number of correct guesses she can guarantee?
[i]Proposed by Espen Slettnes[/i]
2024 ELMO Shortlist, C1.5
Let $m, n \ge 2$ be distinct positive integers. In an infinite grid of unit squares, each square is filled with exactly one real number so that
[list]
[*]In each $m \times m$ square, the sum of the numbers in the $m^2$ cells is equal.
[*]In each $n \times n$ square, the sum of the numbers in the $n^2$ cells is equal.
[*]There exist two cells in the grid that do not contain the same number.
[/list]
Let $S$ be the set of numbers that appear in at least one square on the grid. Find, in terms of $m$ and $n$, the least possible value of $|S|$.
[i]Kiran Reddy[/i]
2019 ELMO Shortlist, G5
Given a triangle $ABC$ for which $\angle BAC \neq 90^{\circ}$, let $B_1, C_1$ be variable points on $AB,AC$, respectively. Let $B_2,C_2$ be the points on line $BC$ such that a spiral similarity centered at $A$ maps $B_1C_1$ to $C_2B_2$. Denote the circumcircle of $AB_1C_1$ by $\omega$. Show that if $B_1B_2$ and $C_1C_2$ concur on $\omega$ at a point distinct from $B_1$ and $C_1$, then $\omega$ passes through a fixed point other than $A$.
[i]Proposed by Max Jiang[/i]
2024 ELMO Shortlist, C2
Let $n$ be a fixed positive integer. Ben is playing a computer game. The computer picks a tree $T$ such that no vertex of $T$ has degree $2$ and such that $T$ has exactly $n$ leaves, labeled $v_1,\ldots, v_n$. The computer then puts an integer weight on each edge of $T$, and shows Ben neither the tree $T$ nor the weights. Ben can ask queries by specifying two integers $1\leq i < j \leq n$, and the computer will return the sum of the weights on the path from $v_i$ to $v_j$. At any point, Ben can guess whether the tree's weights are all zero. He wins the game if he is correct, and loses if he is incorrect.
(a) Show that if Ben asks all $\binom n2$ possible queries, then he can guarantee victory.
(b) Does Ben have a strategy to guarantee victory in less than $\binom n2$ queries?
[i]Brandon Wang[/i]
2017 ELMO Problems, 5
The edges of $K_{2017}$ are each labeled with $1,2,$ or $3$ such that any triangle has sum of labels at least $5.$ Determine the minimum possible average of all $\dbinom{2017}{2}$ labels.
(Here $K_{2017}$ is defined as the complete graph on 2017 vertices, with an edge between every pair of vertices.)
[i]Proposed by Michael Ma[/i]
2023 ELMO Shortlist, A5
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 \[Mx=P_1(x)^3+P_2(x)^3+\cdots+P_n(x)^3.\]
[i]Proposed by Karthik Vedula[/i]
2024 ELMO Shortlist, N8
Let $d(n)$ be the number of divisors of a nonnegative integer $n$ (we set $d(0)=0$). Find all positive integers $d$ such that there exists a two-variable polynomial $P(x,y)$ of degree $d$ with integer coefficients such that:
[list]
[*] for any positive integer $y$, there are infinitely many positive integers $x$ such that $\gcd(x,y)=1$ and $d(|P(x,y)|) \mid x$, and
[*] for any positive integer $x$, there are infinitely many positive integers $y$ such that $\gcd(x,y)=1$ and $d(|P(x,y)|) \mid y$.
[/list]
[i]Allen Wang[/i]
2019 ELMO Shortlist, C4
Let $n \ge 3$ be a fixed integer. A game is played by $n$ players sitting in a circle. Initially, each player draws three cards from a shuffled deck of $3n$ cards numbered $1, 2, \dots, 3n$. Then, on each turn, every player simultaneously passes the smallest-numbered card in their hand one place clockwise and the largest-numbered card in their hand one place counterclockwise, while keeping the middle card.
Let $T_r$ denote the configuration after $r$ turns (so $T_0$ is the initial configuration). Show that $T_r$ is eventually periodic with period $n$, and find the smallest integer $m$ for which, regardless of the initial configuration, $T_m=T_{m+n}$.
[i]Proposed by Carl Schildkraut and Colin Tang[/i]
2017 ELMO Shortlist, 1
Let $ABC$ be a triangle with orthocenter $H,$ and let $M$ be the midpoint of $\overline{BC}.$ Suppose that $P$ and $Q$ are distinct points on the circle with diameter $\overline{AH},$ different from $A,$ such that $M$ lies on line $PQ.$ Prove that the orthocenter of $\triangle APQ$ lies on the circumcircle of $\triangle ABC.$
[i]Proposed by Michael Ren[/i]
2023 ELMO Shortlist, G2
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.
[i]Proposed by Luke Robitaille[/i]
2024 ELMO Problems, 2
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.
[i]Linus Tang[/i]