Olympiad problems

VMEO III 2006 Shortlist, A5

Find all continuous functions $f : (0,+\infty) \to (0,+\infty)$ such that if $a, b, c$ are the lengths of the sides of any triangle then it is satisfied that $$\frac{f(a+b-c)+f(b+c-a)+f(c+a-b)}{3}=f\left(\sqrt{\frac{ab+bc+ca}{3}}\right)$$

2016 Nigerian Senior MO Round 2, Problem 7

Tags: irrational number , induction , algebra

Prove that $(2+\sqrt{3})^{2n}+(2-\sqrt{3})^{2n}$ is an even integer and that $(2+\sqrt{3})^{2n}-(2-\sqrt{3})^{2n}=w\sqrt{3}$ for some positive integer $w$, for all integers $n \geq 1$.

2016 District Olympiad, 3

Tags: inequalities

Let be nonnegative real numbers $ a,b,c, $ holding the inequality: $ \sum_{\text{cyc}} \frac{a}{b+c+1} \le 1. $ Prove that $ \sum_{\text{cyc}} \frac{1}{b+c+1} \ge 1. $

2023 Math Prize for Girls Problems, 3

Tags:

You have 5000 distinct finite sets. Their intersection is empty. However, the intersection of any two is nonempty. What is the smallest possible number of elements contained in their union?

2021 Israel National Olympiad, P2

Tags: number theory

Does there exist an infinite sequence of primes $p_1, p_2, p_3, \dots $ for which, \[p_{n+1}=2p_n+1\] for each $n$?

2023 Korea Summer Program Practice Test, P7

Tags: inequalities

Determine the smallest value of $M$ for which for any choice of positive integer $n$ and positive real numbers $x_1<x_2<\ldots<x_n \le 2023$ the inequality $$\sum_{1\le i < j \le n , x_j-x_i \ge 1} 2^{i-j}\le M$$ holds.

2013 Brazil Team Selection Test, 2

Tags: parallelogram , reflection , moving points , ptolemy sinus lemma , geometry

Let $ABCD$ be a cyclic quadrilateral whose diagonals $AC$ and $BD$ meet at $E$. The extensions of the sides $AD$ and $BC$ beyond $A$ and $B$ meet at $F$. Let $G$ be the point such that $ECGD$ is a parallelogram, and let $H$ be the image of $E$ under reflection in $AD$. Prove that $D,H,F,G$ are concyclic.

1970 IMO, 1

Tags: geometry , ratio , incircle , radius

$M$ is any point on the side $AB$ of the triangle $ABC$. $r,r_1,r_2$ are the radii of the circles inscribed in $ABC,AMC,BMC$. $q$ is the radius of the circle on the opposite side of $AB$ to $C$, touching the three sides of $AB$ and the extensions of $CA$ and $CB$. Similarly, $q_1$ and $q_2$. Prove that $r_1r_2q=rq_1q_2$.

2016 Costa Rica - Final Round, A1

Tags: system of equations , algebra

Find all solutions of the system $\sqrt[3]{\frac{yz^4}{x^2}}+2wx=0 $ $\sqrt[3]{\frac{xz^4}{y}}+5wy=0 $ $\sqrt[3]{\frac{xy}{x}}+7wz^{-1/3}=0$ $x^{12}+\frac{125}{4}y^5+\frac{343}{2}z^4=16$ where $x, y, z \ge 0$ and $w \in R$ [hide=PS] I attached the system, in case I have any typos[/hide]

2017 Macedonia JBMO TST, 3

Tags: inequalities

Let $x,y,z$ be positive reals such that $xyz=1$. Show that $$\frac{x^2+y^2+z}{x^2+2} + \frac{y^2+z^2+x}{y^2+2} + \frac{z^2+x^2+y}{z^2+2} \geq 3.$$ When does equality happen?

Russian TST 2017, P1

Tags: geometry , nine point center

The diagonals of a convex quadrilateral divide it into four triangles. Prove that the nine point centers of these four triangles either lie on one straight line, or are the vertices of a parallelogram.

CIME II 2018, 12

Tags:

Let $\Omega$ be a circle with radius $18$ and let $\mathcal{S}$ be the region inside $\Omega$ that the centroid of $\triangle XYZ$ sweeps through as $X$ varies along all possible points lying outside of $\Omega$, $Y$ varies along all possible points lying on $\Omega$ and $XZ$ is tangent to the circle. Compute the greatest integer less than or equal to the area of $\mathcal{S}$. [I]Proposed by [b]FedeX333X[/b][/I]

1997 Brazil Team Selection Test, Problem 5

Tags: geometry , geometric inequality , inequalities , geometric inequalities , triangle , area

Let $ABC$ be an acute-angled triangle with incenter $I$. Consider the point $A_1$ on $AI$ different from $A$, such that the midpoint of $AA_1$ lies on the circumscribed circle of $ABC$. Points $B_1$ and $C_1$ are defined similarly. (a) Prove that $S_{A_1B_1C_1}=(4R+r)p$, where $p$ is the semi-perimeter, $R$ is the circumradius and $r$ is the inradius of $ABC$. (b) Prove that $S_{A_1B_1C_1}\ge9S_{ABC}$.

2009 Tournament Of Towns, 3

Tags: 3d geometry , tetrahedron , sphere , geometry

Every edge of a tetrahedron is tangent to a given sphere. Prove that the three line segments joining the points of tangency of the three pairs of opposite edges of the tetrahedron are concurrent. [i](7 points)[/i]

2015 Postal Coaching, 1

Tags: polynomial , algebra

Find all real polynomials $P(x)$ that satisfy $$P(x^3-2)=P(x)^3-2$$

1980 AMC 12/AHSME, 29

Tags: algebra , system of equations

How many ordered triples $(x,y,z)$ of integers satisfy the system of equations below? \[ \begin{array}{l} x^2-3xy+2yz-z^2=31 \\ -x^2+6yz+2z^2=44 \\ x^2+xy+8z^2=100\\ \end{array} \] $\text{(A)} \ 0 \qquad \text{(B)} \ 1 \qquad \text{(C)} \ 2 \qquad \text{(D)} \ \text{a finite number greater than 2} \qquad \text{(E)} \ \text{infinately many}$

JOM 2023, 4

Tags: algebra

Given $n$ positive real numbers $x_1,x_2,x_3,...,x_n$ such that $$\left (1+\frac{1}{x_1}\right )\left(1+\frac{1}{x_2}\right)...\left(1+\frac{1}{x_n}\right)=(n+1)^n$$ Determine the minimum value of $x_1+x_2+x_3+...+x_n$. [i]Proposed by Loh Kwong Weng[/i]

2021 Baltic Way, 19

Tags: polynomial , number theory

Find all polynomials $p$ with integer coefficients such that the number $p(a) - p(b)$ is divisible by $a + b$ for all integers $a, b$, provided that $a + b \neq 0$.

2024 Mozambican National MO Selection Test, P3

Tags: positive integer , nt , number theory , diophantine equation , factorization , prime number

Find all triples of positive integers $(a,b,c)$ such that: $a^2bc-2ab^2c-2abc^2+b^3c+bc^3+2b^2c^2=11$

2007 ITest, 39

Tags: number theory , relatively prime

Let $a$ and $b$ be relatively prime positive integers such that $a/b$ is the sum of the real solutions to the equation \[\sqrt[3]{3x-4}+\sqrt[3]{5x-6}=\sqrt[3]{x-2}+\sqrt[3]{7x-8}.\] Find $a+b$.

2015 South East Mathematical Olympiad, 8

Tags: algebra , number theory

Find all prime number $p$ such that there exists an integer-coefficient polynomial $f(x)=x^{p-1}+a_{p-2}x^{p-2}+…+a_1x+a_0$ that has $p-1$ consecutive positive integer roots and $p^2\mid f(i)f(-i)$, where $i$ is the imaginary unit.

Ukraine Correspondence MO - geometry, 2016.11

Tags: geometry , trapezoid , square , angle

Inside the square $ABCD$ mark the point $P$, for which $\angle BAP = 30^o$ and $\angle BCP = 15^o$. The point $Q$ was chosen so that $APCQ$ is an isosceles trapezoid ($PC\parallel AQ$). Find the angles of the triangle $CAM$, where $M$ is the midpoint of $PQ$.

2001 All-Russian Olympiad Regional Round, 8.4

Tags: rectangle , tiles , combinatorics

An angle of size $n \times m$, where $m, n \ge 2$, is called a figure, resulting from a rectangle of size $n \times m$ cells by removing the rectangle size $(n - 1) \times (m - 1)$ cells. Two players take turns making moves consisting in painting in a corner an arbitrary non-zero number of cells forming a rectangle or square.

VMEO III 2006 Shortlist, N11

Tags: power , number theory

Prove that the composition of the sets of one of the following two forms is finite: (a) $2^{2^n}+1$ (b) $6^{2^n}+1$

2014 BMT Spring, 2

Tags: probability , combinatorics

If I roll three fair $4$-sided dice, what is the probability that the sum of the resulting numbers is relatively prime to the product of the resulting numbers?