Found problems: 190
2019 ELMO Shortlist, C1
Elmo and Elmo's clone are playing a game. Initially, $n\geq 3$ points are given on a circle. On a player's turn, that player must draw a triangle using three unused points as vertices, without creating any crossing edges. The first player who cannot move loses. If Elmo's clone goes first and players alternate turns, who wins? (Your answer may be in terms of $n$.)
[i]Proposed by Milan Haiman[/i]
2019 ELMO Shortlist, G2
Carl is given three distinct non-parallel lines $\ell_1, \ell_2, \ell_3$ and a circle $\omega$ in the plane. In addition to a normal straightedge, Carl has a special straightedge which, given a line $\ell$ and a point $P$, constructs a new line passing through $P$ parallel to $\ell$. (Carl does not have a compass.) Show that Carl can construct a triangle with circumcircle $\omega$ whose sides are parallel to $\ell_1,\ell_2,\ell_3$ in some order.
[i]Proposed by Vincent Huang[/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]
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]
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]
2023 ELMO Shortlist, C4
Let \(n\) be a positive integer and consider an \(n\times n\) square grid. For \(1\le k\le n\), a [i]python[/i] of length \(k\) is a snake that occupies \(k\) consecutive cells in a single row, and no other cells. Similarly, an [i]anaconda[/i] of length \(k\) is a snake that occupies \(k\) consecutive cells in a single column, and no other cells.
The grid contains at least one python or anaconda, and it satisfies the following properties: [list] [*]No cell is occupied by multiple snakes. [*]If a cell in the grid is immediately to the left or immediately to the right of a python, then that cell must be occupied by an anaconda. [*]If a cell in the grid is immediately to above or immediately below an anaconda, then that cell must be occupied by a python. [/list]
Prove that the sum of the squares of the lengths of the snakes is at least \(n^2\).
[i]Proposed by Linus Tang[/i]
2024 ELMO Shortlist, N6
Given a positive integer whose base-$10$ representation is $\overline{d_k\ldots d_0}$ for some integer $k \geq 0$, where $d_k \neq 0$, a move consists of selecting some integers $0 \leq i \leq j \leq k$, such that the digits $d_j,\ldots,d_i$ are not all $0$, erasing them from $n$, and replacing them with a divisor of $\overline{d_j\ldots d_i}$ (this divisor need not have the same number of digits as $\overline{d_j\ldots d_i}$).
Prove that for all sufficiently large even integers $n$, we may apply some sequence of moves to $n$ to transform it into $2024$.
[i]Allen Wang[/i]
2010 ELMO Shortlist, 3
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, N3
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 [i]filling machine[/i] and an [i]emptying machine[/i], which operate according to the following rules: [list] [*]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. [/list] 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 \(n\).
[i]Proposed by Raymond Feng[/i]
2012 ELMO Shortlist, 1
In acute triangle $ABC$, let $D,E,F$ denote the feet of the altitudes from $A,B,C$, respectively, and let $\omega$ be the circumcircle of $\triangle AEF$. Let $\omega_1$ and $\omega_2$ be the circles through $D$ tangent to $\omega$ at $E$ and $F$, respectively. Show that $\omega_1$ and $\omega_2$ meet at a point $P$ on $BC$ other than $D$.
[i]Ray Li.[/i]
2024 ELMO Shortlist, G6
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.
[i]Tiger Zhang[/i]
2013 ELMO Shortlist, 3
Let $a_1,a_2,...,a_9$ be nine real numbers, not necessarily distinct, with average $m$. Let $A$ denote the number of triples $1 \le i < j < k \le 9$ for which $a_i + a_j + a_k \ge 3m$. What is the minimum possible value of $A$?
[i]Proposed by Ray Li[/i]
2013 ELMO Shortlist, 5
Let $m_1,m_2,...,m_{2013} > 1$ be 2013 pairwise relatively prime positive integers and $A_1,A_2,...,A_{2013}$ be 2013 (possibly empty) sets with $A_i\subseteq \{1,2,...,m_i-1\}$ for $i=1,2,...,2013$. Prove that there is a positive integer $N$ such that
\[ N \le \left( 2\left\lvert A_1 \right\rvert + 1 \right)\left( 2\left\lvert A_2 \right\rvert + 1 \right)\cdots\left( 2\left\lvert A_{2013} \right\rvert + 1 \right) \]
and for each $i = 1, 2, ..., 2013$, there does [i]not[/i] exist $a \in A_i$ such that $m_i$ divides $N-a$.
[i]Proposed by Victor Wang[/i]
2024 ELMO Shortlist, G3
Let $ABC$ be a triangle, and let $\omega_1,\omega_2$ be centered at $O_1$, $O_2$ and tangent to line $BC$ at $B$, $C$ respectively. Let line $AB$ intersect $\omega_1$ again at $X$ and let line $AC$ intersect $\omega_2$ again at $Y$. If $Q$ is the other intersection of the circumcircles of triangles $ABC$ and $AXY$, then prove that lines $AQ$, $BC$, and $O_1O_2$ either concur or are all parallel.
[i]Advaith Avadhanam[/i]
2023 ELMO Shortlist, N5
An ordered pair \((k,n)\) of positive integers is [i]good[/i] if there exists an ordered quadruple \((a,b,c,d)\) of positive integers such that \(a^3+b^k=c^3+d^k\) and \(abcd=n\). Prove that there exist infinitely many positive integers \(n\) such that \((2022,n)\) is not good but \((2023,n)\) is good.
[i]Proposed by Luke Robitaille[/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]
2014 ELMO Shortlist, 4
Find all triples $(f,g,h)$ of injective functions from the set of real numbers to itself satisfying
\begin{align*}
f(x+f(y)) &= g(x) + h(y) \\
g(x+g(y)) &= h(x) + f(y) \\
h(x+h(y)) &= f(x) + g(y)
\end{align*}
for all real numbers $x$ and $y$. (We say a function $F$ is [i]injective[/i] if $F(a)\neq F(b)$ for any distinct real numbers $a$ and $b$.)
[i]Proposed by Evan Chen[/i]
2024 ELMO Shortlist, G4
In quadrilateral $ABCD$ with incenter $I$, points $W,X,Y,Z$ lie on sides $AB, BC,CD,DA$ with $AZ=AW$, $BW=BX$, $CX=CY$, $DY=DZ$. Define $T=\overline{AC}\cap\overline{BD}$ and $L=\overline{WY}\cap\overline{XZ}$. Let points $O_a,O_b,O_c,O_d$ be such that $\angle O_aZA=\angle O_aWA=90^\circ$ (and cyclic variants), and $G=\overline{O_aO_c}\cap\overline{O_bO_d}$. Prove that $\overline{IL}\parallel\overline{TG}$.
[i]Neal Yan[/i]
2023 ELMO Shortlist, N4
Let \(d(n)\) denote the number of positive divisors of \(n\). The sequence \(a_0\), \(a_1\), \(a_2\), \(\ldots\) is defined as follows: \(a_0=1\), and for all integers \(n\ge1\), \[a_n=d(a_{n-1})+d(d(a_{n-2}))+\cdots+ {\underbrace{d(d(\ldots d(a_0)\ldots))}_{n\text{ times}}}.\] Show that for all integers \(n\ge1\), we have \(a_n\le3n\).
[i]Proposed by Karthik Vedula[/i]
2023 ELMO Shortlist, N5
An ordered pair \((k,n)\) of positive integers is [i]good[/i] if there exists an ordered quadruple \((a,b,c,d)\) of positive integers such that \(a^3+b^k=c^3+d^k\) and \(abcd=n\). Prove that there exist infinitely many positive integers \(n\) such that \((2022,n)\) is not good but \((2023,n)\) is good.
[i]Proposed by Luke Robitaille[/i]
2024 ELMO Shortlist, G3
Let $ABC$ be a triangle, and let $\omega_1,\omega_2$ be centered at $O_1$, $O_2$ and tangent to line $BC$ at $B$, $C$ respectively. Let line $AB$ intersect $\omega_1$ again at $X$ and let line $AC$ intersect $\omega_2$ again at $Y$. If $Q$ is the other intersection of the circumcircles of triangles $ABC$ and $AXY$, then prove that lines $AQ$, $BC$, and $O_1O_2$ either concur or are all parallel.
[i]Advaith Avadhanam[/i]
2023 ELMO Shortlist, C1
Elmo has 2023 cookie jars, all initially empty. Every day, he chooses two distinct jars and places a cookie in each. Every night, Cookie Monster finds a jar with the most cookies and eats all of them. If this process continues indefinitely, what is the maximum possible number of cookies that the Cookie Monster could eat in one night?
[i]Proposed by Espen Slettnes[/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]
2015 ELMO Problems, 2
Let $m$, $n$, and $x$ be positive integers. Prove that \[ \sum_{i = 1}^n \min\left(\left\lfloor \frac{x}{i} \right\rfloor, m \right) = \sum_{i = 1}^m \min\left(\left\lfloor \frac{x}{i} \right\rfloor, n \right). \]
[i]Proposed by Yang Liu[/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]