Olympiad problems

2012 Canadian Mathematical Olympiad Qualification Repechage, 3

We say that $(a,b,c)$ form a [i]fantastic triplet[/i] if $a,b,c$ are positive integers, $a,b,c$ form a geometric sequence, and $a,b+1,c$ form an arithmetic sequence. For example, $(2,4,8)$ and $(8,12,18)$ are fantastic triplets. Prove that there exist infinitely many fantastic triplets.

2017 Bosnia And Herzegovina - Regional Olympiad, 4

Tags: arithmetic mean , number theory , set , combinatorics

Let $S$ be a set of $n$ distinct real numbers, and $A_S$ set of arithemtic means of two distinct numbers from $S$. For given $n \geq 2$ find minimal number of elements in $A_S$

2021 Simon Marais Mathematical Competition, B1

Tags: probability , linear algebra , matrix

Let $n \ge 2$ be an integer, and let $O$ be the $n \times n$ matrix whose entries are all equal to $0$. Two distinct entries of the matrix are chosen uniformly at random, and those two entries are changed from $0$ to $1$. Call the resulting matrix $A$. Determine the probability that $A^2 = O$, as a function of $n$.

2006 All-Russian Olympiad Regional Round, 11.2

Tags: algebra , polynomial , trinomial

Product of square trinomials $x^2 - a_1x + b_1$, $x^2 - a_2x + b_2$, $...$, $x^2-a_nx + b_n$ is equal to the polynomial $P(x) = x^{2n} +c_1x^{2n-1} +c_2x^{2n-2} +...+ c_{2n-1}x + c_{2n}$, where the coefficients are $c_1$, $c_2$, $...$ , $c_{2n}$ are positive. Show that for some $k$ ($1\le k \le n$) the coefficients $a_k$ and $b_k$ are positive.

2014 Junior Balkan MO, 3

Tags: algebra , inequality

For positive real numbers $a,b,c$ with $abc=1$ prove that $\left(a+\frac{1}{b}\right)^{2}+\left(b+\frac{1}{c}\right)^{2}+\left(c+\frac{1}{a}\right)^{2}\geq 3(a+b+c+1)$

2008 Postal Coaching, 5

Tags: fraction , combinatorics , irreducible

Let $n \in N$. Find the maximum number of irreducible fractions a/b (i.e., $gcd(a, b) = 1$) which lie in the interval $(0,1/n)$.

2022/2023 Tournament of Towns, P6

Tags: geometry , tetrahedron , 3d geometry

The midpoints of all heights of a certain tetrahedron lie on its inscribed sphere. Is this tetrahedron necessarily regular then?

2022 AMC 10, 5

Tags: fraction

What is the value of $\frac{(1+\frac{1}{3})(1+\frac{1}{5})(1+\frac{1}{7})}{\sqrt{(1-\frac{1}{3^2})(1-\frac{1}{5^2})(1-\frac{1}{7^2})}}?$ $\textbf{(A) }\sqrt{3} \qquad \textbf{(B) }2 \qquad \textbf{(C) }\sqrt{15} \qquad \textbf{(D) }4 \qquad \textbf{(E) }\sqrt{105}$

2009 JBMO Shortlist, 4

Tags: number theory , sum of powers

Determine all prime numbers $p_1, p_2,..., p_{12}, p_{13}, p_1 \le p_2 \le ... \le p_{12} \le p_{13}$, such that $p_1^2+ p_2^2+ ... + p_{12}^2 = p_{13}^2$ and one of them is equal to $2p_1 + p_9$.

Indonesia MO Shortlist - geometry, g2.6

Tags: circumcircle , geometry

Let $ABC$ be a triangle. Suppose $D$ is on $BC$ such that $AD$ bisects $\angle BAC$. Suppose $M$ is on $AB$ such that $\angle MDA = \angle ABC$, and $N$ is on $AC$ such that $\angle NDA = \angle ACB$. If $AD$ and $MN$ intersect on $P$, prove that $AD^3 = AB \cdot AC \cdot AP$.

2006 China Western Mathematical Olympiad, 1

Tags: inequalities

Let $n$ be a positive integer with $n \geq 2$, and $0<a_{1}, a_{2},...,a_{n}< 1$. Find the maximum value of the sum $\sum_{i=1}^{n}(a_{i}(1-a_{i+1}))^{\frac{1}{6}}$ where $a_{n+1}=a_{1}$

2011 China Team Selection Test, 3

Tags: inequalities

Let $n$ be a positive integer. Find the largest real number $\lambda$ such that for all positive real numbers $x_1,x_2,\cdots,x_{2n}$ satisfying the inequality \[\frac{1}{2n}\sum_{i=1}^{2n}(x_i+2)^n\geq \prod_{i=1}^{2n} x_i,\] the following inequality also holds \[\frac{1}{2n}\sum_{i=1}^{2n}(x_i+1)^n\geq \lambda\prod_{i=1}^{2n} x_i.\]

2022 MMATHS, 2

Tags: geometry , reflection , area

Triangle $ABC$ has $AB = 3$, $BC = 4$, and $CA = 5$. Points $D$, $E$, $F$, $G$, $H$, and $I$ are the reflections of $A$ over $B$, $B$ over $A$, $B$ over $C$, $C$ over $B$, $C$ over $A$, and $A$ over $C$, respectively. Find the area of hexagon $EFIDGH$.

2023 Iranian Geometry Olympiad, 3

Tags: tangent circle , geometry

Let $\omega$ be the circumcircle of the triangle $ABC$ with $\angle B = 3\angle C$. The internal angle bisector of $\angle A$, intersects $\omega$ and $BC$ at $M$ and $D$, respectively. Point $E$ lies on the extension of the line $MC$ from $M$ such that $ME$ is equal to the radius of $\omega$. Prove that circumcircles of triangles $ACE$ and $BDM$ are tangent. [i]Proposed by Mehran Talaei - Iran[/i]

2024 LMT Fall, 6

Tags: team

A kite with $AB = BC$ and $AD = CD$ has diagonals which satisfy $AC = 80$ and $BD = 71$. Let $AC$ and $BD$ intersect at a point $O$. Find the area of the quadrilateral formed by the circumcenters of $ABO$, $BCO$, $CDO$, and $ADO$.

1966 Putnam, B4

Tags:

Let $0<a_1<a_2< \dots < a_{mn+1}$ be $mn+1$ integers. Prove that you can select either $m+1$ of them no one of which divides any other, or $n+1$ of them each dividing the following one.

2023 AIME, 12

Tags: geometry

Let $\triangle ABC$ be an equilateral triangle with side length $55$. Points $D$, $E$, and $F$ lie on sides $\overline{BC}$, $\overline{CA}$, and $\overline{AB}$, respectively, with $BD=7$, $CE=30$, and $AF=40$. A unique point $P$ inside $\triangle ABC$ has the property that \[\measuredangle AEP=\measuredangle BFP=\measuredangle CDP.\] Find $\tan^{2}\left(\measuredangle AEP\right)$.

2011 IMO, 6

Tags: geometry , circumcircle , reflection

Let $ABC$ be an acute triangle with circumcircle $\Gamma$. Let $\ell$ be a tangent line to $\Gamma$, and let $\ell_a, \ell_b$ and $\ell_c$ be the lines obtained by reflecting $\ell$ in the lines $BC$, $CA$ and $AB$, respectively. Show that the circumcircle of the triangle determined by the lines $\ell_a, \ell_b$ and $\ell_c$ is tangent to the circle $\Gamma$. [i]Proposed by Japan[/i]

2022 VN Math Olympiad For High School Students, Problem 5

Tags: geometry

Given a convex quadrilateral $MNPQ$. Assume that there exists 2 points $U, V$ inside $MNPQ$ satifying:$$\angle MUN = \angle MUV = \angle NUV = \angle QVU = \angle PVU = \angle PVQ$$Consider another 2 points $X, Y$ in the plane. Prove that the sum$$XM + XN + XY + YP + YQ$$get its minimum value iff $X\equiv U, Y\equiv V$.

2023 Kazakhstan National Olympiad, 5

Tags: number theory

Given are positive integers $a, b, m, k$ with $k \geq 2$. Prove that there exist infinitely many $n$, such that $\gcd (\varphi_m(n), \lfloor \sqrt[k] {an+b} \rfloor)=1$, where $\varphi_m(n)$ is the $m$-th iteration of $\varphi(n)$.

1966 IMO Longlists, 58

Tags: combinatorics , system of equations , algebra

In a mathematical contest, three problems, $A,B,C$ were posed. Among the participants ther were 25 students who solved at least one problem each. Of all the contestants who did not solve problem $A$, the number who solved $B$ was twice the number who solved $C$. The number of students who solved only problem $A$ was one more than the number of students who solved $A$ and at least one other problem. Of all students who solved just one problem, half did not solve problem $A$. How many students solved only problem $B$?

2001 Argentina National Olympiad, 1

Tags: combinatorics , number theory

Sergio thinks of a positive integer $S$, less than or equal to $100$. Iván must guess the number that Sergio thought of, using the following procedure: in each step, he chooses two positive integers $A$ and $B$ less than $100$, and asks Sergio what is the greatest common factor between $A+ S$ and $B$. Give a sequence of seven steps that ensures Iván guesses the number $S$ that Sergio thought of. Clarification:In each step, Sergio correctly answers Iván's question.

2022 IFYM, Sozopol, 4

Tags: number theory

a) Prove that for each positive integer $n$ the number or ordered pairs of integers $(x,y)$ for which $x^2-xy+y^2=n$ is finite and is multiple of 6. b) Find all ordered pairs of integers $(x,y)$ for which $x^2-xy+y^2=727$.

2020 LMT Fall, B7

Tags: number theory

Zachary tries to simplify the fraction $\frac{2020}{5050}$ by dividing the numerator and denominator by the same integer to get the fraction $\frac{m}{n}$ , where $m$ and $n$ are both positive integers. Find the sum of the (not necessarily distinct) prime factors of the sum of all the possible values of $m +n$

Kyiv City MO 1984-93 - geometry, 1991.10.2

Tags: orthocenter , altitude , geometry

In an acute-angled triangle $ABC$ on the sides $AB$, $BC$, $AC$, the points $C_1$, $A_1$, and $B_1$ are marked such that the segments $AA_1$, $BB_1$, $CC_1$ intersect at some point $O$ and the angles $AA_1C$, $BB_1A$, $CC_1B$ are equal. Prove that $AA_1$, $BB_1$, and $CC_1$ are the altitudes of the triangle.