Found problems: 31
2023 USEMO, 5
Let $n \ge 2$ be an integer. A cube of size $n \times n \times n$ is dissected into $n^3$ unit cubes. A nonzero real number is written at the center of each unit cube so that the sum of the $n^2$ numbers in each slab of size $1 \times n \times n$, $n \times 1 \times n$, or $n \times n \times 1$ equals zero. (There are a total of $3n$ such slabs, forming three groups of $n$ slabs each such that slabs in the same group are parallel and slabs in different groups are perpendicular.)
Could it happen that some plane in three-dimensional space separates the positive and the negative written numbers? (The plane should not pass through any of the numbers.)
[i]Nikolai Beluhov[/i]
2023 USEMO, 2
Each point in the plane is labeled with a real number. Show that there exist two distinct points $P$ and $Q$ whose labels differ by less than the distance from $P$ to $Q$.
[i]Holden Mui[/i]
2023 USEMO, 1
A positive integer $n$ is called [i]beautiful[/i] if, for every integer $4 \le b \le 10000$, the base-$b$ representation of $n$ contains the consecutive digits $2$, $0$, $2$, $3$ (in this order, from left to right). Determine whether the set of all beautiful integers is finite.
[i]Oleg Kryzhanovsky[/i]
2022 USEMO, 1
A [i]stick[/i] is defined as a $1 \times k$ or $k\times 1$ rectangle for any integer $k \ge 1$. We wish to partition the cells of a $2022 \times 2022$ chessboard into $m$ non-overlapping sticks, such that any two of these $m$ sticks share at most one unit of perimeter. Determine the smallest $m$ for which this is possible.
[i]Holden Mui[/i]
2019 USEMO, 4
Prove that for any prime $p,$ there exists a positive integer $n$ such that
\[1^n+2^{n-1}+3^{n-2}+\cdots+n^1\equiv 2020\pmod{p}.\]
[i]Robin Son[/i]
2020 USEMO, 6
Prove that for every odd integer $n > 1$, there exist integers $a, b > 0$ such that, if we let $Q(x) = (x + a)^
2 + b$, then the following conditions hold:
$\bullet$ we have $\gcd(a, n) = gcd(b, n) = 1$;
$\bullet$ the number $Q(0)$ is divisible by $n$; and
$\bullet$ the numbers $Q(1), Q(2), Q(3), \dots$ each have a prime factor not dividing $n$.
2022 USEMO, 5
Let $\tau(n)$ denote the number of positive integer divisors of a positive integer $n$ (for example, $\tau(2022) = 8$). Given a polynomial $P(X)$ with integer coefficients, we define a sequence $a_1, a_2,\ldots$ of nonnegative integers by setting
\[a_n =\begin{cases}\gcd(P(n), \tau (P(n)))&\text{if }P(n) > 0\\0 &\text{if }P(n) \leq0\end{cases}\]
for each positive integer $n$. We then say the sequence [i]has limit infinity[/i] if every integer occurs in this sequence only finitely many times (possibly not at all).
Does there exist a choice of $P(X)$ for which the sequence $a_1$, $a_2$, . . . has limit infinity?
[i]Jovan Vuković[/i]
2022 USEMO, 4
Let $ABCD$ be a cyclic quadrilateral whose opposite sides are not parallel. Suppose points $P, Q, R, S$ lie in the interiors of segments $AB, BC, CD, DA,$ respectively, such that $$\angle PDA = \angle PCB, \text{ } \angle QAB = \angle QDC, \text{ } \angle RBC = \angle RAD, \text{ and } \angle SCD = \angle SBA.$$ Let $AQ$ intersect $BS$ at $X$, and $DQ$ intersect $CS$ at $Y$. Prove that lines $PR$ and $XY$ are either parallel or coincide.
[i]Tilek Askerbekov[/i]
2021 USEMO, 1
Let $n$ be a fixed positive integer and consider an $n\times n$ grid of real numbers. Determine the greatest possible number of cells $c$ in the grid such that the entry in $c$ is both strictly greater than the average of $c$'s column and strictly less than the average of $c$'s row.
[i]Proposed by Holden Mui[/i]
2023 USEMO, 3
Canmoo is trying to do constructions, but doesn't have a ruler or compass. Instead, Canmoo has a device that, given four distinct points $A$, $B$, $C$, $P$ in the plane, will mark the isogonal conjugate of $P$ with respect to triangle $ABC$, if it exists. Show that if two points are marked on the plane, then Canmoo can construct their midpoint using this device, a pencil for marking additional points, and no other tools.
(Recall that the [i]isogonal conjugate[/i] of $P$ with respect to triangle $ABC$ is the point $Q$ such that lines $AP$ and $AQ$ are reflections around the bisector of $\angle BAC$, lines $BP$ and $BQ$ are reflections around the bisector of $\angle CBA$, lines $CP$ and $CQ$ are reflections around the bisector of $\angle ACB$. Additional points marked by the pencil can be assumed to be in general position, meaning they don't lie on any line through two existing points or any circle through three existing points.)
[i]Maxim Li[/i]
2019 USEMO, 1
Let $ABCD$ be a cyclic quadrilateral. A circle centered at $O$ passes through $B$ and $D$ and meets lines $BA$ and $BC$ again at points $E$ and $F$ (distinct from $A,B,C$). Let $H$ denote the orthocenter of triangle $DEF.$ Prove that if lines $AC,$ $DO,$ $EF$ are concurrent, then triangle $ABC$ and $EHF$ are similar.
[i]Robin Son[/i]
2023 USEMO, 4
Let $ABC$ be an acute triangle with orthocenter $H$. Points $A_1$, $B_1$, $C_1$ are chosen in the interiors of sides $BC$, $CA$, $AB$, respectively, such that $\triangle A_1B_1C_1$ has orthocenter $H$. Define $A_2 = \overline{AH} \cap \overline{B_1C_1}$, $B_2 = \overline{BH} \cap \overline{C_1A_1}$, and $C_2 = \overline{CH} \cap \overline{A_1B_1}$.
Prove that triangle $A_2B_2C_2$ has orthocenter $H$.
[i]Ankan Bhattacharya[/i]
2022 USEMO, 6
Find all positive integers $k$ for which there exists a nonlinear function $f:\mathbb{Z} \rightarrow\mathbb{Z}$ such that the equation $$f(a)+f(b)+f(c)=\frac{f(a-b)+f(b-c)+f(c-a)}{k}$$ holds for any integers $a,b,c$ satisfying $a+b+c=0$ (not necessarily distinct).
[i]Evan Chen[/i]
2023 USEMO, 6
Let $n \ge 2$ be a fixed integer. [list=a] [*]Determine the largest positive integer $m$ (in terms of $n$) such that there exist complex numbers $r_1$, $\dots$, $r_n$, not all zero, for which \[ \prod_{k=1}^n (r_k+1) = \prod_{k=1}^n (r_k^2+1) = \dots = \prod_{k=1}^n (r_k^m+1) = 1. \] [*]For this value of $m$, find all possible values of \[ \prod\limits_{k=1}^n (r_k^{m+1}+1). \] [/list]
[i]Kaixin Wang[/i]
2019 USEMO, 6
Let $ABC$ be an acute scalene triangle with circumcenter $O$ and altitudes $\overline{AD}$, $\overline{BE}$, $\overline{CF}$. Let $X$, $Y$, $Z$ be the midpoints of $\overline{AD}$, $\overline{BE}$, $\overline{CF}$. Lines $AD$ and $YZ$ intersect at $P$, lines $BE$ and $ZX$ intersect at $Q$, and lines $CF$ and $XY$ intersect at $R$.
Suppose that lines $YZ$ and $BC$ intersect at $A'$, and lines $QR$ and $EF$ intersect at $D'$. Prove that the perpendiculars from $A$, $B$, $C$, $O$, to the lines $QR$, $RP$, $PQ$, $A'D'$, respectively, are concurrent.
[i]Ankan Bhattacharya[/i]
2021 USEMO, 5
Given a polynomial $p(x)$ with real coefficients, we denote by $S(p)$ the sum of the squares of its coefficients. For example $S(20x+ 21)=20^2+21^2=841$.
Prove that if $f(x)$, $g(x)$, and $h(x)$ are polynomials with real coefficients satisfying the indentity $f(x) \cdot g(x)=h(x)^ 2$, then $$S(f) \cdot S(g) \ge S(h)^2$$
[i]Proposed by Bhavya Tiwari[/i]
2020 USEMO, 2
Calvin and Hobbes play a game. First, Hobbes picks a family $F$ of subsets of $\{1, 2, . . . , 2020\}$, known to both players. Then, Calvin and Hobbes take turns choosing a number from $\{1, 2, . . . , 2020\}$ which is not already chosen, with Calvin going first, until all numbers are taken (i.e., each player has $1010$ numbers). Calvin wins if he has chosen all the elements of some member of $F$, otherwise Hobbes wins. What is the largest possible size of a family $F$ that Hobbes could pick while still having a winning strategy?
2021 USEMO, 4
Let $ABC$ be a triangle with circumcircle $\omega$, and let $X$ be the reflection of $A$ in $B$. Line $CX$ meets $\omega$ again at $D$. Lines $BD$ and $AC$ meet at $E$, and lines $AD$ and $BC$ meet at $F$. Let $M$ and $N$ denote the midpoints of $AB$ and $ AC$.
Can line $EF$ share a point with the circumcircle of triangle $AMN?$
[i]Proposed by Sayandeep Shee[/i]
2021 USEMO, 6
A bagel is a loop of $2a+2b+4$ unit squares which can be obtained by cutting a concentric $a\times b$ hole out of an $(a +2)\times (b+2)$ rectangle, for some positive integers a and b. (The side of length a of the hole is parallel to the side of length $a+2$ of the rectangle.)
Consider an infinite grid of unit square cells. For each even integer $n \ge 8$, a bakery of order $n$ is a finite set of cells $ S$ such that, for every $n$-cell bagel $B$ in the grid, there exists a congruent copy of $B$ all of whose cells are in $S$. (The copy can be translated and rotated.) We denote by $f(n)$ the smallest possible number of cells in a bakery of order $ n$.
Find a real number $\alpha$ such that, for all sufficiently large even integers $n \ge 8$, we have $$\frac{1}{100}<\frac{f (n)}{n^ {\alpha}}<100$$
[i]Proposed by Nikolai Beluhov[/i]
2019 USEMO, 3
Consider an infinite grid $\mathcal G$ of unit square cells. A [i]chessboard polygon[/i] is a simple polygon (i.e. not self-intersecting) whose sides lie along the gridlines of $\mathcal G$.
Nikolai chooses a chessboard polygon $F$ and challenges you to paint some cells of $\mathcal G$ green, such that any chessboard polygon congruent to $F$ has at least $1$ green cell but at most $2020$ green cells. Can Nikolai choose $F$ to make your job impossible?
[i]Nikolai Beluhov[/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]
2021 USEMO, 3
Let $A_1C_2B_1A_2C_1B_2$ be an equilateral hexagon. Let $O_1$ and $H_1$ denote the circumcenter and orthocenter of $\triangle A_1B_1C_1$, and let $O_2$ and $H_2$ denote the circumcenter and orthocenter of $\triangle A_2B_2C_2$. Suppose that $O_1 \ne O_2$ and $H_1 \ne H_2$. Prove that the lines $O_1O_2$ and $H_1H_2$ are either parallel or coincide.
[i]Ankan Bhattacharya[/i]
2021 USEMO, 2
Find all integers $n\ge1$ such that $2^n-1$ has exactly $n$ positive integer divisors.
[i]Proposed by Ankan Bhattacharya [/i]
2019 USEMO, 2
Let $\mathbb{Z}[x]$ denote the set of single-variable polynomials in $x$ with integer coefficients. Find all functions $\theta : \mathbb{Z}[x] \to \mathbb{Z}[x]$ (i.e. functions taking polynomials to polynomials)
such that
[list]
[*] for any polynomials $p, q \in \mathbb{Z}[x]$, $\theta(p + q) = \theta(p) + \theta(q)$;
[*] for any polynomial $p \in \mathbb{Z}[x]$, $p$ has an integer root if and only if $\theta(p)$ does.
[/list]
[i]Carl Schildkraut[/i]
2022 USEMO, 2
A function $\psi \colon {\mathbb Z} \to {\mathbb Z}$ is said to be [i]zero-requiem[/i] if for any positive integer $n$ and any integers $a_1$, $\ldots$, $a_n$ (not necessarily distinct), the sums $a_1 + a_2 + \dots + a_n$ and $\psi(a_1) + \psi(a_2) + \dots + \psi(a_n)$ are not both zero.
Let $f$ and $g$ be two zero-requiem functions for which $f \circ g$ and $g \circ f$ are both the identity function (that is, $f$ and $g$ are mutually inverse bijections). Given that $f+g$ is [i]not[/i] a zero-requiem function, prove that $f \circ f$ and $g \circ g$ are both zero-requiem.
[i]Sutanay Bhattacharya[/i]