Olympiad problems

2024 Harvard-MIT Mathematics Tournament, 1

Tags: geometry

Inside an equilateral triangle of side length $6$, three congruent equilateral triangles of side length $x$ with sides parallel to the original equilateral triangle are arranged so that each has a vertex on a side of the larger triangle, and a vertex on another one of the three equilateral triangles, as shown below. [img]https://cdn.artofproblemsolving.com/attachments/3/f/ff48c885154ce065c0d0420d1580769aa98eb1.png[/img] A smaller equilateral triangle formed between the three congruent equilateral triangles has side length $1$. Compute $x$.

2019 Kyiv Mathematical Festival, 3

Tags: Kyiv mathematical festival , combinatorics

There were $2n,$ $n\ge2,$ teams in a tournament. Each team played against every other team once without draws. A team gets 0 points for a loss and gets as many points for a win as its current number of losses. For which $n$ all the teams could end up with the same non-zero number of points?

2007 Iran MO (3rd Round), 2

Tags: inequalities , LaTeX , inequalities proposed

$ a,b,c$ are three different positive real numbers. Prove that:\[ \left|\frac{a\plus{}b}{a\minus{}b}\plus{}\frac{b\plus{}c}{b\minus{}c}\plus{}\frac{c\plus{}a}{c\minus{}a}\right|>1\]

2002 Germany Team Selection Test, 3

Tags: number theory , decimal representation , Digits , factorial , IMO Shortlist

Prove that there is no positive integer $n$ such that, for $k = 1,2,\ldots,9$, the leftmost digit (in decimal notation) of $(n+k)!$ equals $k$.

2022 Germany Team Selection Test, 2

Tags: number theory , combinatorics , construction

Let $r>1$ be a rational number. Alice plays a solitaire game on a number line. Initially there is a red bead at $0$ and a blue bead at $1$. In a move, Alice chooses one of the beads and an integer $k \in \mathbb{Z}$. If the chosen bead is at $x$, and the other bead is at $y$, then the bead at $x$ is moved to the point $x'$ satisfying $x'-y=r^k(x-y)$. Find all $r$ for which Alice can move the red bead to $1$ in at most $2021$ moves.

1970 IMO Longlists, 31

Tags: inequalities , geometry

Prove that for any triangle with sides $a, b, c$ and area $P$ the following inequality holds: \[P \leq \frac{\sqrt 3}{4} (abc)^{2/3}.\] Find all triangles for which equality holds.

2002 France Team Selection Test, 1

Tags: geometry , circumcircle , geometric transformation , reflection , geometry proposed

In an acute-angled triangle $ABC$, $A_1$ and $B_1$ are the feet of the altitudes from $A$ and $B$ respectively, and $M$ is the midpoint of $AB$. a) Prove that $MA_1$ is tangent to the circumcircle of triangle $A_1B_1C$. b) Prove that the circumcircles of triangles $A_1B_1C,BMA_1$, and $AMB_1$ have a common point.

2025 USA IMO Team Selection Test, 3

Tags: geometry , USA TST , USA TST 2025

Let $A_1A_2\dotsm A_{2025}$ be a convex 2025-gon, and let $A_i = A_{i+2025}$ for all integers $i$. Distinct points $P$ and $Q$ lie in its interior such that $\angle A_{i-1}A_iP = \angle QA_iA_{i+1}$ for all $i$. Define points $P^{j}_{i}$ and $Q^{j}_{i}$ for integers $i$ and positive integers $j$ as follows: [list] [*] For all $i$, $P^1_i = Q^1_i = A_i$. [*] For all $i$ and $j$, $P^{j+1}_{i}$ and $Q^{j+1}_i$ are the circumcenters of $PP^j_iP^j_{i+1}$ and $QQ^j_iQ^{j}_{i+1}$, respectively. [/list] Let $\mathcal{P}$ and $\mathcal{Q}$ be the polygons $P^{2025}_{1}P^{2025}_{2}\dotsm P^{2025}_{2025}$ and $Q^{2025}_{1}Q^{2025}_{2}\dotsm Q^{2025}_{2025}$, respectively. [list=a] [*] Prove that $\mathcal{P}$ and $\mathcal{Q}$ are cyclic. [*] Let $O_P$ and $O_Q$ be the circumcenters of $\mathcal{P}$ and $\mathcal{Q}$, respectively. Assuming that $O_P\neq O_Q$, show that $O_PO_Q$ is parallel to $PQ$. [/list] [i]Ruben Carpenter[/i]

2021 Saint Petersburg Mathematical Olympiad, 7

Tags: geometry , rectangle , ratio

A square is cut into red and blue rectangles. The sum of areas of red triangles is equal to the sum of areas of the blue ones. For each blue rectangle, we write the ratio of the length of its vertical side to the length of its horizontal one and for each red rectangle, the ratio of the length of its horizontal side to the length of its vertical side. Find the smallest possible value of the sum of all the written numbers.

2023 Chile National Olympiad, 6

Tags: geometry , angle bisector

Let $\vartriangle ABC$ be a triangle such that $\angle ABC = 30^o$, $\angle ACB = 15^o$. Let $M$ be midpoint of segment $BC$ and point $N$ lies on segment $MC$, such that the length of $NC$ is equal to length of $AB$. Proce that $AN$ is the bisector of the angle $\angle MAC$. [img]https://cdn.artofproblemsolving.com/attachments/2/7/4c554b53f03288ee69931fdd2c6fbd3e27ab13.png[/img]

2011 AMC 10, 20

Tags: probability , geometry , perimeter , AMC

Two points on the circumference of a circle of radius r are selected independently and at random. From each point a chord of length r is drawn in a clockwise direction. What is the probability that the two chords intersect? $ \textbf{(A)}\ \frac{1}{6}\qquad\textbf{(B)}\ \frac{1}{5}\qquad\textbf{(C)}\ \frac{1}{4}\qquad\textbf{(D)}\ \frac{1}{3}\qquad\textbf{(E)}\ \frac{1}{2} $

2013 Peru IMO TST, 5

Tags: number theory , prime numbers , binomial coefficients , IMO Shortlist

Determine all integers $m \geq 2$ such that every $n$ with $\frac{m}{3} \leq n \leq \frac{m}{2}$ divides the binomial coefficient $\binom{n}{m-2n}$.

2011 NIMO Problems, 2

Tags:

The sum of three consecutive integers is $15$. Determine their product.

1999 Portugal MO, 6

Tags: geometry , angles , equal angles

In the triangle $[ABC], D$ is the midpoint of $[AB]$ and $E$ is the trisection point of $[BC]$ closer to $C$. If $\angle ADC= \angle BAE$ , find the measue of $\angle BAC$ .

2012 AMC 12/AHSME, 13

Tags: conics , parabola , probability , symmetry , LaTeX , AMC

Two parabolas have equations $y=x^2+ax+b$ and $y=x^2+cx+d$, where $a$, $b$, $c$, and $d$ are integers (not necessarily different), each chosen independently by rolling a fair six-sided die. What is the probability that the parabolas have at least one point in common? $\textbf{(A)}\ \frac{1}{2} \qquad\textbf{(B)}\ \frac{25}{36} \qquad\textbf{(C)}\ \frac{5}{6} \qquad\textbf{(D)}\ \frac{31}{36} \qquad\textbf{(E)}\ 1 $

1998 Romania National Olympiad, 4

Tags: superior algebra , superior algebra solved

Let $K\subseteq \mathbb C$ be a field with the operations from $\mathbb C$ s.t. i) K has exactly two endomorphisms, namely f and g ii) if f(x)=g(x) then $x\in\mathbb Q$. Prove that there exists an integer $d\neq 1$ free from squares so that $K=\mathbb Q(\sqrt d)$.

2024 Philippine Math Olympiad, P5

Tags: combinatorics , string , palindrome

Find the largest positive integer $k$ so that any binary string of length $2024$ contains a palindromic substring of length at least $k$.

2018 MIG, 20

Tags: 2018 Individual

Point $O$ is selected in equilateral $\triangle ABC$ such that the sum of the distances from $O$ to each side of $ABC$ is $15$. Compute the area of $ABC$. [center][img]https://cdn.artofproblemsolving.com/attachments/4/0/dd573985a7c98f23fd05d11e95c4b908eaa895.png[/img][/center] $\textbf{(A) } 15\sqrt3\qquad\textbf{(B) } 30\sqrt3\qquad\textbf{(C) } 50\sqrt3\qquad\textbf{(D) } 75\sqrt3\qquad\textbf{(E) } 225\sqrt3$

2009 China National Olympiad, 1

Tags: inequalities , symmetry , function , algebra , polynomial , absolute value , inequalities proposed

Given an integer $ n > 3.$ Let $ a_{1},a_{2},\cdots,a_{n}$ be real numbers satisfying $ min |a_{i} \minus{} a_{j}| \equal{} 1, 1\le i\le j\le n.$ Find the minimum value of $ \sum_{k \equal{} 1}^n|a_{k}|^3.$

2015 Spain Mathematical Olympiad, 2

Tags: number theory unsolved , number theory

Let $p$ and $n$ be a natural numbers such that $p$ is a prime and $1+np$ is a perfect square. Prove that the number $n+1$ is sum of $p$ perfect squares.

2002 Stanford Mathematics Tournament, 1

Tags: algebra , polynomial

Completely factor the polynomial $x^4-x^3-5x^2+3x+6$

2011 Mathcenter Contest + Longlist, 7 sl9

Tags: functional equation , algebra , functional

Find the function $\displaystyle{f : \mathbb{R}-\left\{ 0\,\right\} \rightarrow \mathbb{R} }$ such that $$f(x)+f(1-\frac{1}{x}) = \frac{1}{x},\,\,\, \forall x \in \mathbb{R}- \{ 0, 1\,\}$$ [i](-InnoXenT-)[/i]

1983 Tournament Of Towns, (043) A5

Tags: Coloring , combinatorics , combinatorial geometry , regular polygon

$k$ vertices of a regular $n$-gon $P$ are coloured. A colouring is called almost uniform if for every positive integer $m$ the following condition is satisfied: If $M_1$ is a set of m consecutive vertices of $P$ and $M_2$ is another such set then the number of coloured vertices of $M_1$ differs from the number of coloured vertices of $M_2$ at most by $1$. Prove that for all positive integers $k$ and $n$ ($k \le n$) an almost uniform colouring exists and that it is unique within a rotation. (M Kontsevich, Moscow)

2021 May Olympiad, 4

Tags: combinatorics

At each vertex of a $13$-sided polygon we write one of the numbers $1,2,3,…, 12,13$, without repeating. Then, on each side of the polygon we write the difference of the numbers of the vertices of its ends (the largest minus the smallest). For example, if two consecutive vertices of the polygon have the numbers $2$ and $11$, the number $9$ is written on the side they determine. a) Is it possible to number the vertices of the polygon so that only the numbers $3, 4$ and $5$ are written on the sides? b) Is it possible to number the vertices of the polygon so that only the numbers $3, 4$ and $6$ are written on the sides?

1983 Polish MO Finals, 5

Tags: Vectors , inequalities , algebra , combinatorial geometry

On the plane are given unit vectors $\overrightarrow{a_1},\overrightarrow{a_2},\overrightarrow{a_3}$. Show that one can choose numbers $c_1,c_2,c_3 \in \{-1,1\}$ such that the length of the vector $c_1\overrightarrow{a_1}+c_2\overrightarrow{a_2}+c_3\overrightarrow{a_3}$ is at least $2$.