Olympiad problems

2023 Bulgaria JBMO TST, 3

Tags: geometry , incircle , excircle , similarity , circumcircle , incenter

Let $ABC$ be a non-isosceles triangle with circumcircle $k$, incenter $I$ and $C$-excenter $I_C$. Let $M$ be the midpoint of $AB$ and $N$ be the midpoint of arc $\widehat{ACB}$ on $k$. Prove that $\angle IMI_C + \angle INI_C = 180^{\circ}$.

2016 Switzerland - Final Round, 3

Tags: algebra , number theory

Find all primes $p, q$ and natural numbers $n$ such that: $p(p+1)+q(q+1)=n(n+1)$

2005 Turkey Team Selection Test, 1

Tags: search , function , number theory

Show that for any integer $n\geq2$ and all integers $a_{1},a_{2},...,a_{n}$ the product $\prod_{i<j}{(a_{j}-a_{i})}$ is divisible by $\prod_{i<j}{(j-i)}$ .

2007 IMO Shortlist, 3

Tags: combinatorics , modular arithmetic , counting , triplets

Find all positive integers $ n$ for which the numbers in the set $ S \equal{} \{1,2, \ldots,n \}$ can be colored red and blue, with the following condition being satisfied: The set $ S \times S \times S$ contains exactly $ 2007$ ordered triples $ \left(x, y, z\right)$ such that: [b](i)[/b] the numbers $ x$, $ y$, $ z$ are of the same color, and [b](ii)[/b] the number $ x \plus{} y \plus{} z$ is divisible by $ n$. [i]Author: Gerhard Wöginger, Netherlands[/i]

2017 Estonia Team Selection Test, 8

Tags: inequalities , three variable inequality , ineqstd

Let $a$, $b$, $c$ be positive real numbers such that $\min(ab,bc,ca) \ge 1$. Prove that $$\sqrt[3]{(a^2+1)(b^2+1)(c^2+1)} \le \left(\frac{a+b+c}{3}\right)^2 + 1.$$ [i]Proposed by Tigran Margaryan, Armenia[/i]

2008 F = Ma, 8

Tags:

Riders in a carnival ride stand with their backs against the wall of a circular room of diameter $\text{8.0 m}$. The room is spinning horizontally about an axis through its center at a rate of $\text{45 rev/min}$ when the floor drops so that it no longer provides any support for the riders. What is the minimum coefficient of static friction between the wall and the rider required so that the rider does not slide down the wall? (a) $\text{0.0012}$ (b) $\text{0.056}$ (c) $\text{0.11}$ (d) $\text{0.53}$ (e) $\text{8.9}$

2009 Harvard-MIT Mathematics Tournament, 7

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Simplify the product \[ \prod_{m=1}^{100}\prod_{n=1}^{100}\frac{x^{n+m}+x^{n+m+2}+x^{2n+1}+x^{2m+1}}{x^{2n}+2x^{n+m}+x^{2m}} \] Express your answer in terms of $x$.

2019 USMCA, 24

Tags:

Let $ABC$ be a triangle with $\angle A = 60^\circ$, $AB = 12$, $AC = 14$. Point $D$ is on $BC$ such that $\angle BAD = \angle CAD$. Extend $AD$ to meet the circumcircle at $M$. The circumcircle of $BDM$ intersects $AB$ at $K \neq B$, and line $KM$ intersects the circumcircle of $CDM$ at $L \neq M$. Find $\frac{KM}{LM}$.

2007 Korea Junior Math Olympiad, 4

Tags: geometry , perpendicular bisector , concyclic , collinear

Let $P$ be a point inside $\triangle ABC$. Let the perpendicular bisectors of $PA,PB,PC$ be $\ell_1,\ell_2,\ell_3$. Let $D =\ell_1 \cap \ell_2$ , $E=\ell_2 \cap \ell_3$, $F=\ell_3 \cap \ell_1$. If $A,B,C,D,E,F$ lie on a circle, prove that $C, P,D$ are collinear.

2002 China Team Selection Test, 2

Tags: 3d geometry , sphere , vector , induction , geometry

There are $ n$ points ($ n \geq 4$) on a sphere with radius $ R$, and not all of them lie on the same semi-sphere. Prove that among all the angles formed by any two of the $ n$ points and the sphere centre $ O$ ($ O$ is the vertex of the angle), there is at least one that is not less than $ \displaystyle 2 \arcsin{\frac{\sqrt{6}}{3}}$.

1968 Yugoslav Team Selection Test, Problem 6

Tags: geometry , 3d geometry , tetrahedron

Prove that the incenter coincides with the circumcenter of a tetrahedron if and only if each pair of opposite edges are of equal length.

2013 Greece JBMO TST, 1

Tags: algebra , inequalities

If x,y<0 prove that $\left(x+\frac{2}{y} \right) \left(\frac{y}{x}+2 \right)\geq 8$. When do we have equality?

2014 VJIMC, Problem 3

Tags: inequalities

Let $n\ge2$ be an integer and let $x>0$ be a real number. Prove that $$\left(1-\sqrt{\tanh x}\right)^n+\sqrt{\tanh(nx)}<1.$$

2009 Harvard-MIT Mathematics Tournament, 1

Tags:

If $a$ and $b$ are positive integers such that $a^2-b^4= 2009$, find $a+b$.

VMEO III 2006, 10.3

Tags: algebra , functional

Find all functions $f : R \to R$ that satisfy $f(x^2 + f(y) - y) = (f(x))^2$ for all $x,y \in R$.

Kvant 2021, M2657

Tags: number theory

Given are positive integers $n>20$ and $k>1$, such that $k^2$ divides $n$. Prove that there exist positive integers $a, b, c$, such that $n=ab+bc+ca$.

2017 European Mathematical Cup, 2

Tags: combinatorics

A regular hexagon in the plane is called sweet if its area is equal to $1$. Is it possible to place $2000000$ sweet hexagons in the plane such that the union of their interiors is a convex polygon of area at least $1900000$? Remark: A subset $S$ of the plane is called convex if for every pair of points in $S$, every point on the straight line segment that joins the pair of points also belongs to $S$. The hexagons may overlap.

1996 All-Russian Olympiad, 1

Tags: geometry

Points $E$ and $F$ are given on side $BC$ of convex quadrilateral $ABCD$ (with $E$ closer than $F$ to $B$). It is known that $\angle BAE = \angle CDF$ and $\angle EAF = \angle FDE$. Prove that $\angle FAC = \angle EDB$. [i]M. Smurov[/i]

2015 Purple Comet Problems, 9

Tags:

The ﬁgure below has only two sizes for its internal angles. The larger angles are three times the size of the smaller angles. Find the degree measure of one of the larger angles. For figure here: http://www.purplecomet.org/welcome/practice P.S The figure has 9 sides.

2010 ELMO Problems, 1

Tags: function , induction , algebra

Determine all strictly increasing functions $f: \mathbb{N}\to\mathbb{N}$ satisfying $nf(f(n))=f(n)^2$ for all positive integers $n$. [i]Carl Lian and Brian Hamrick.[/i]

2008 JBMO Shortlist, 9

Tags: geometry

Let $O$ be a point inside the parallelogram $ABCD$ such that $\angle AOB + \angle COD = \angle BOC + \angle AOD$. Prove that there exists a circle $k$ tangent to the circumscribed circles of the triangles $\vartriangle AOB, \vartriangle BOC, \vartriangle COD$ and $\vartriangle DOA$.

2019 Silk Road, 2

Tags: algebra , inequalities

Let $ a_1, $ $ a_2, $ $ \ldots, $ $ a_ {99} $ be positive real numbers such that $ ia_j + ja_i \ge i + j $ for all $ 1 \le i <j \le 99. $ Prove , that $ (a_1 + 1) (a_2 + 2) \ldots (a_ {99} +99) \ge 100!$ .

1985 IMO Shortlist, 16

Tags: geometry , construction , triangle , circles

If possible, construct an equilateral triangle whose three vertices are on three given circles.

2018 Puerto Rico Team Selection Test, 3

Tags: equal angles , geometry

Let $M$ be the point of intersection of diagonals $AC$ and $BD$ of the convex quadrilateral $ABCD$. Let $K$ be the point of intersection of the extension of side $AB$ (beyond$A$) with the bisector of the angle $ACD$. Let $L$ be the intersection of $KC$ and $BD$. If $MA \cdot CD = MB \cdot LD$, prove that the angle $BKC$ is equal to the angle $CDB$.

2007 Denmark MO - Mohr Contest, 1

Tags: decagon , geometry , area

Triangle $ABC$ lies in a regular decagon as shown in the figure. What is the ratio of the area of the triangle to the area of the entire decagon? Write the answer as a fraction of integers. [img]https://1.bp.blogspot.com/-Ld_-4u-VQ5o/Xzb-KxPX0wI/AAAAAAAAMWg/-qPtaI_04CQ3vvVc1wDTj3SoonocpAzBQCLcBGAsYHQ/s0/2007%2BMohr%2Bp1.png[/img]