Olympiad problems

1999 USAMO, 5

Tags: USA(J)MO , USAMO , algorithm , combinatorics , game , game strategy

The Y2K Game is played on a $1 \times 2000$ grid as follows. Two players in turn write either an S or an O in an empty square. The first player who produces three consecutive boxes that spell SOS wins. If all boxes are filled without producing SOS then the game is a draw. Prove that the second player has a winning strategy.

1980 USAMO, 2

Tags: AMC , USA(J)MO , USAMO , induction , algebra unsolved , algebra

Determine the maximum number of three-term arithmetic progressions which can be chosen from a sequence of $n$ real numbers \[a_1<a_2<\cdots<a_n.\]

1997 USAMO, 1

Tags: floor function , calculus , integration , AMC , USA(J)MO , USAMO , modular arithmetic

Let $p_1, p_2, p_3, \ldots$ be the prime numbers listed in increasing order, and let $x_0$ be a real number between 0 and 1. For positive integer $k$, define \[ x_k = \begin{cases} 0 & \mbox{if} \; x_{k-1} = 0, \\[.1in] {\displaystyle \left\{ \frac{p_k}{x_{k-1}} \right\}} & \mbox{if} \; x_{k-1} \neq 0, \end{cases} \] where $\{x\}$ denotes the fractional part of $x$. (The fractional part of $x$ is given by $x - \lfloor x \rfloor$ where $\lfloor x \rfloor$ is the greatest integer less than or equal to $x$.) Find, with proof, all $x_0$ satisfying $0 < x_0 < 1$ for which the sequence $x_0, x_1, x_2, \ldots$ eventually becomes 0.

1996 Taiwan National Olympiad, 6

Tags: algebra , polynomial , AMC , USA(J)MO , USAMO , number theory proposed , number theory

Let $q_{0},q_{1},...$ be a sequence of integers such that a) for any $m>n$ we have $m-n\mid q_{m}-q_{n}$, and b) $|q_{n}|\leq n^{10}, \ \forall n\geq 0$. Prove there exists a polynomial $Q$ such that $q_{n}=Q(n), \ \forall n\geq 0$.

1978 USAMO, 4

Tags: AMC , USA(J)MO , USAMO , geometry , 3D geometry , tetrahedron , sphere

(a) Prove that if the six dihedral (i.e. angles between pairs of faces) of a given tetrahedron are congruent, then the tetrahedron is regular. (b) Is a tetrahedron necessarily regular if five dihedral angles are congruent?

2019 USAJMO, 6

Tags: number theory , relatively prime , USA(J)MO , USAMO , USAJMO , Hi

Two rational numbers $\tfrac{m}{n}$ and $\tfrac{n}{m}$ are written on a blackboard, where $m$ and $n$ are relatively prime positive integers. At any point, Evan may pick two of the numbers $x$ and $y$ written on the board and write either their arithmetic mean $\tfrac{x+y}{2}$ or their harmonic mean $\tfrac{2xy}{x+y}$ on the board as well. Find all pairs $(m,n)$ such that Evan can write 1 on the board in finitely many steps. [i]Proposed by Yannick Yao[/i]

2024 USAJMO, 2

Tags: AMC , USA(J)MO , USAJMO , Hi

Let $m$ and $n$ be positive integers. Let $S$ be the set of integer points $(x,y)$ with $1\leq x\leq 2m$ and $1\leq y\leq 2n$. A configuration of $mn$ rectangles is called [i]happy[/i] if each point in $S$ is a vertex of exactly one rectangle, and all rectangles have sides parallel to the coordinate axes. Prove that the number of happy configurations is odd. [i]Proposed by Serena An and Claire Zhang[/i]

2012 USAJMO, 1

Tags: USA(J)MO , USAJMO , geometry , circumcircle , 2012 USAJMO

Given a triangle $ABC$, let $P$ and $Q$ be points on segments $\overline{AB}$ and $\overline{AC}$, respectively, such that $AP=AQ$. Let $S$ and $R$ be distinct points on segment $\overline{BC}$ such that $S$ lies between $B$ and $R$, $\angle BPS=\angle PRS$, and $\angle CQR=\angle QSR$. Prove that $P,Q,R,S$ are concyclic (in other words, these four points lie on a circle).

2013 Purple Comet Problems, 15

Tags: AMC , USA(J)MO , USAMO , algebra

Let $a$, $b$, and $c$ be positive real numbers such that $a^2+b^2+c^2=989$ and $(a+b)^2+(b+c)^2+(c+a)^2=2013$. Find $a+b+c$.

2014 India IMO Training Camp, 2

Tags: AMC , USA(J)MO , USAMO , combinatorics unsolved , combinatorics

Let $n$ be a natural number.A triangulation of a convex n-gon is a division of the polygon into $n-2$ triangles by drawing $n-3$ diagonals no two of which intersect at an interior point of the polygon.Let $f(n)$ denote the number of triangulations of a regular n-gon such that each of the triangles formed is isosceles.Determine $f(n)$ in terms of $n$.

2024 USAMO, 3

Tags: AMC , USA(J)MO , USAMO , geometry

Let $m$ be a positive integer. A triangulation of a polygon is [i]$m$-balanced[/i] if its triangles can be colored with $m$ colors in such a way that the sum of the areas of all triangles of the same color is the same for each of the $m$ colors. Find all positive integers $n$ for which there exists an $m$-balanced triangulation of a regular $n$-gon. [i]Note[/i]: A triangulation of a convex polygon $\mathcal{P}$ with $n \ge 3$ sides is any partitioning of $\mathcal{P}$ into $n-2$ triangles by $n-3$ diagonals of $\mathcal{P}$ that do not intersect in the polygon's interior. [i]Proposed by Krit Boonsiriseth[/i]

1981 USAMO, 3

Tags: AMC , USA(J)MO , USAMO , trigonometry , inequalities , geometry unsolved , geometry

If $A,B,C$ are the angles of a triangle, prove that \[-2 \le \sin{3A}+\sin{3B}+\sin{3C} \le \frac{3\sqrt{3}}{2}\] and determine when equality holds.

2008 Tuymaada Olympiad, 2

Tags: AMC , USA(J)MO , USAMO , number theory , prime numbers , number theory unsolved

Is it possible to arrange on a circle all composite positive integers not exceeding $ 10^6$, so that no two neighbouring numbers are coprime? [i]Author: L. Emelyanov[/i] [hide="Tuymaada 2008, Junior League, First Day, Problem 2."]Prove that all composite positive integers not exceeding $ 10^6$ may be arranged on a circle so that no two neighbouring numbers are coprime. [/hide]

1987 Flanders Math Olympiad, 3

Tags: function , AMC , USA(J)MO , USAMO

Find all continuous functions $f: \mathbb{R}\rightarrow\mathbb{R}$ such that \[f(x)^3 = -\frac x{12}\cdot\left(x^2+7x\cdot f(x)+16\cdot f(x)^2\right),\ \forall x \in \mathbb{R}.\]

2010 USAJMO, 4

Tags: AMC , USA(J)MO , USAJMO , conics , parabola , analytic geometry , nt

A triangle is called a parabolic triangle if its vertices lie on a parabola $y = x^2$. Prove that for every nonnegative integer $n$, there is an odd number $m$ and a parabolic triangle with vertices at three distinct points with integer coordinates with area $(2^nm)^2$.

2011 USAJMO, 4

Tags: AMC , USA(J)MO , USAJMO , induction , symmetry

A [i]word[/i] is defined as any finite string of letters. A word is a [i]palindrome[/i] if it reads the same backwards and forwards. Let a sequence of words $W_0, W_1, W_2,...$ be defined as follows: $W_0 = a, W_1 = b$, and for $n \ge 2$, $W_n$ is the word formed by writing $W_{n-2}$ followed by $W_{n-1}$. Prove that for any $n \ge 1$, the word formed by writing $W_1, W_2, W_3,..., W_n$ in succession is a palindrome.

2021 USAMO, 6

Tags: AMC , USAMO , USA(J)MO

Let $ABCDEF$ be a convex hexagon satisfying $\overline{AB} \parallel \overline{DE}$, $\overline{BC} \parallel \overline{EF}$, $\overline{CD} \parallel \overline{FA}$, and \[ AB \cdot DE = BC \cdot EF = CD \cdot FA. \] Let $X$, $Y$, and $Z$ be the midpoints of $\overline{AD}$, $\overline{BE}$, and $\overline{CF}$. Prove that the circumcenter of $\triangle ACE$, the circumcenter of $\triangle BDF$, and the orthocenter of $\triangle XYZ$ are collinear.

2011 USA TSTST, 3

Tags: analytic geometry , logarithms , USA(J)MO , USAMO , number theory , relatively prime , USA TSTST

Prove that there exists a real constant $c$ such that for any pair $(x,y)$ of real numbers, there exist relatively prime integers $m$ and $n$ satisfying the relation \[ \sqrt{(x-m)^2 + (y-n)^2} < c\log (x^2 + y^2 + 2). \]

2025 USAMO, 4

Tags: AMC , USA(J)MO , USAMO , USAJMO , geometry , anakin_skywalker , padme_amidala

Let $H$ be the orthocenter of acute triangle $ABC$, let $F$ be the foot of the altitude from $C$ to $AB$, and let $P$ be the reflection of $H$ across $BC$. Suppose that the circumcircle of triangle $AFP$ intersects line $BC$ at two distinct points $X$ and $Y$. Prove that $C$ is the midpoint of $XY$.

2023 USAMO, 5

Tags: AMC , USA(J)MO , USAMO , arithmetic sequence , modular arithmetic , primes

Let $n\geq3$ be an integer. We say that an arrangement of the numbers $1$, $2$, $\dots$, $n^2$ in a $n \times n$ table is [i]row-valid[/i] if the numbers in each row can be permuted to form an arithmetic progression, and [i]column-valid[/i] if the numbers in each column can be permuted to form an arithmetic progression. For what values of $n$ is it possible to transform any row-valid arrangement into a column-valid arrangement by permuting the numbers in each row?

2014 Contests, 2

Tags: quadratics , USA(J)MO , USAMO , induction , modular arithmetic , number theory proposed , Quadratic Residues

Let $a_1,a_2,a_3,\ldots$ be a sequence of integers, with the property that every consecutive group of $a_i$'s averages to a perfect square. More precisely, for every positive integers $n$ and $k$, the quantity \[\frac{a_n+a_{n+1}+\cdots+a_{n+k-1}}{k}\] is always the square of an integer. Prove that the sequence must be constant (all $a_i$ are equal to the same perfect square). [i]Evan O'Dorney and Victor Wang[/i]

2007 AIME Problems, 12

Tags: calculus , integration , logarithms , ratio , AMC , USA(J)MO , USAMO

The increasing geometric sequence $x_{0},x_{1},x_{2},\ldots$ consists entirely of integral powers of $3.$ Given that \[\sum_{n=0}^{7}\log_{3}(x_{n}) = 308\qquad\text{and}\qquad 56 \leq \log_{3}\left ( \sum_{n=0}^{7}x_{n}\right ) \leq 57,\] find $\log_{3}(x_{14}).$

2013 NIMO Problems, 8

Tags: AMC , USA(J)MO , USAMO , geometry , perimeter , floor function

Let $AXYZB$ be a convex pentagon inscribed in a semicircle with diameter $AB$. Suppose that $AZ-AX=6$, $BX-BZ=9$, $AY=12$, and $BY=5$. Find the greatest integer not exceeding the perimeter of quadrilateral $OXYZ$, where $O$ is the midpoint of $AB$. [i]Proposed by Evan Chen[/i]

2011 USAMO, 3

Tags: AMC , USA(J)MO , USAMO , geometry , geometric transformation , reflection , 3D geometry

In hexagon $ABCDEF$, which is nonconvex but not self-intersecting, no pair of opposite sides are parallel. The internal angles satisfy $\angle A=3\angle D$, $\angle C=3\angle F$, and $\angle E=3\angle B$. Furthermore $AB=DE$, $BC=EF$, and $CD=FA$. Prove that diagonals $\overline{AD}$, $\overline{BE}$, and $\overline{CF}$ are concurrent.

1974 USAMO, 1

Tags: algebra , polynomial , USA(J)MO , USAMO , AIME

Let $ a,b,$ and $ c$ denote three distinct integers, and let $ P$ denote a polynomial having integer coefficients. Show that it is impossible that $ P(a) \equal{} b, P(b) \equal{} c,$ and $ P(c) \equal{} a$.