Olympiad problems

2000 Baltic Way, 15

Let $n$ be a positive integer not divisible by $2$ or $3$. Prove that for all integers $k$, the number $(k+1)^n-k^n-1$ is divisible by $k^2+k+1$.

2015 Kazakhstan National Olympiad, 2

Tags: diophantine equation , number theory

Solve in positive integers $x^yy^x=(x+y)^z$

2009 Canadian Mathematical Olympiad Qualification Repechage, 5

Tags: quadratic , algebra , quadratic formula , number theory

Determine all positive integers $n$ for which $n(n + 9)$ is a perfect square.

1955 Moscow Mathematical Olympiad, 308

Tags: tangent , circles , geometry , bisect arc , tangent circle

* Two circles are tangent to each other externally, and to a third one from the inside. Two common tangents to the first two circles are drawn, one outer and one inner. Prove that the inner tangent divides in halves the arc intercepted by the outer tangent on the third circle.

2007 Bosnia Herzegovina Team Selection Test, 4

Tags: polynomial , inequalities , vieta , algebra

Let $P(x)$ be a polynomial such that $P(x)=x^3-2x^2+bx+c$. Roots of $P(x)$ belong to interval $(0,1)$. Prove that $8b+9c \leq 8$. When does equality hold?

1995 Tournament Of Towns, (459) 4

Tags: combinatorics , combinatorial geometry , lattice points , geometry

Some points with integer coordinates in the plane are marked. It is known that no four of them lie on a circle. Show that there exists a circle of radius 1995 without any marked points inside. (AV Shapovelov)

1999 Mongolian Mathematical Olympiad, Problem 6

Tags: geometry

Two circles in the plane intersect at $C$ and $D$. A chord $AB$ of the first circle and a chord $EF$ of the second circle pass through a point on the common chord $CD$. Show that the points $A,B,E,F$ lie on a circle.

2019 Jozsef Wildt International Math Competition, W. 59

Tags: tetrahedron , 3d geometry , circumradius , inradius , geometry , inequalities

In the any $[ABCD]$ tetrahedron we denote with $\alpha$, $\beta$, $\gamma$ the measures, in radians, of the angles of the three pairs of opposite edges and with $r$, $R$ the lengths of the rays of the sphere inscribed and respectively circumscribed the tetrahedron. Demonstrate inequality$$\left(\frac{3r}{R}\right)^3\leq \sin \frac{\alpha +\beta +\gamma}{3}$$(A refinement of inequality $R \geq 3r$).

1981 Canada National Olympiad, 4

Tags: polynomial , function , algebra

$P(x),Q(x)$ are two polynomials such that $P(x)=Q(x)$ has no real solution, and $P(Q(x))\equiv Q(P(x))\forall x\in\mathbb{R}$. Prove that $P(P(x))=Q(Q(x))$ has no real solution.

2024 JHMT HS, 9

Tags: geometry

Compute the smallest positive integer $k$ such that the area of the region bounded by \[ k\min(x,y)+x^2+y^2=0 \] exceeds $100$.

1986 Traian Lălescu, 2.4

Tags: geometry , rectangle , 3d geometry

Prove that $ ABCD $ is a rectangle if and only if $ MA^2+MC^2=MB^2+MD^2, $ for all spatial points $ M. $

2019 CMIMC, 7

Tags: algebra , number theory

For all positive integers $n$, let \[f(n) = \sum_{k=1}^n\varphi(k)\left\lfloor\frac nk\right\rfloor^2.\] Compute $f(2019) - f(2018)$. Here $\varphi(n)$ denotes the number of positive integers less than or equal to $n$ which are relatively prime to $n$.

2022 Macedonian Team Selection Test, Problem 3

Tags: algebra , function , functional equation

We consider all functions $f: \mathbb{N} \rightarrow \mathbb{N}$ such that $f(f(n)+n)=n$ and $f(a+b-1) \leq f(a)+f(b)$ for all positive integers $a, b, n$. Prove that there are at most two values for $f(2022)$. $\textit {Proposed by Ilija Jovcheski}$

2008 Costa Rica - Final Round, 4

Tags: inequalities

Let $ x$, $ y$ and $ z$ be non negative reals, such that there are not two simultaneously equal to $ 0$. Show that $ \frac {x \plus{} y}{y \plus{} z} \plus{} \frac {y \plus{} z}{x \plus{} y} \plus{} \frac {y \plus{} z}{z \plus{} x} \plus{} \frac {z \plus{} x}{y \plus{} z} \plus{} \frac {z \plus{} x}{x \plus{} y} \plus{} \frac {x \plus{} y}{z \plus{} x}\geq\ 5 \plus{} \frac {x^{2} \plus{} y^{2} \plus{} z^{2}}{xy \plus{} yz \plus{} zx}$ and determine the equality cases.

1999 Greece JBMO TST, 1

Tags: combinatorics , coloring , number theory

A circle is divided in $100$ equal parts and the points of this division are colored green or yellow, such that when between two points of division $A,B$ there are exactly $4$ division points and the point $A$ is green, then the point $B$ shall be yellow. Which points are more, the green or the yellow ones?

2007 Mongolian Mathematical Olympiad, Problem 1

Tags: combinatorics

Find the number of subsets of the set $\{1,2,3,...,5n\}$ such that the sum of the elements in each subset are divisible by $5$.

2021 Moldova EGMO TST, 9

Tags: geometry

Let $ABCD$ be a square and $E$ a on point diagonal $(AC)$, different from its midpoint. $H$ and $K$ are the orthoceneters of triangles $ABE$ and $ADE$. Prove that $AH$ and $CK$ are parallel.

2021 Belarusian National Olympiad, 11.1

Tags: algebra , functional equation

Find all functions $f: \mathbb{R} \to \mathbb{R}$, such that for all real $x,y$ the following equation holds:$$f(x-0.25)+f(y-0.25)=f(x+\lfloor y+0.25 \rfloor - 0.25)$$

2015 Peru Cono Sur TST, P7

Tags: combinatorics , geometry

In the plan $6$ points were located such that the distance between two damages of them is greater than or equal to $1$. Prove that it is possible to choose two of those points such that their distance is greater than or equal to $2 \cos{18}$ Observation: It might help you to know that $\cos{18} = 0.95105\ldots$ and $\cos{24} = 0.91354\ldots$

1983 IMO Longlists, 26

Tags: number theory

Let $a, b, c$ be positive integers satisfying $\gcd (a, b) = \gcd (b, c) = \gcd (c, a) = 1$. Show that $2abc-ab-bc-ca$ cannot be represented as $bcx+cay +abz$ with nonnegative integers $x, y, z.$

2016 Brazil Team Selection Test, 3

Tags: combinatorics , game

Let $n$ be a positive integer. Two players $A$ and $B$ play a game in which they take turns choosing positive integers $k \le n$. The rules of the game are: (i) A player cannot choose a number that has been chosen by either player on any previous turn. (ii) A player cannot choose a number consecutive to any of those the player has already chosen on any previous turn. (iii) The game is a draw if all numbers have been chosen; otherwise the player who cannot choose a number anymore loses the game. The player $A$ takes the first turn. Determine the outcome of the game, assuming that both players use optimal strategies. [i]Proposed by Finland[/i]

2017 ASDAN Math Tournament, 2

Tags:

Let $5$ and $13$ be lengths of two sides of a right triangle. Compute the sum of all possible lengths of the third side.

2006 China Second Round Olympiad, 3

Tags: inequalities

Suppose $A = {x|5x-a \le 0}$, $B = {x|6x-b > 0}$, $a,b \in \mathbb{N}$, and $A \cap B \cap \mathbb{N} = {2,3,4}$. The number of such pairs $(a,b)$ is ${ \textbf{(A)}\ 20\qquad\textbf{(B)}\ 25\qquad\textbf{(C)}\ 30\qquad\textbf{(D)}} 42\qquad $

2013 Korea Junior Math Olympiad, 3

Tags: algebra , sequence , recurrence relation , sum , positive integer , diophantine equation

$\{a_n\}$ is a positive integer sequence such that $a_{i+2} = a_{i+1} +a_i$ (for all $i \ge 1$). For positive integer $n$, define as $$b_n=\frac{1}{a_{2n+1}}\Sigma_{i=1}^{4n-2}a_i$$ Prove that $b_n$ is positive integer.

1984 Brazil National Olympiad, 3

Tags: geometry , rectangle , dodecahedron , regular polygon , 3d geometry

Given a regular dodecahedron of side $a$. Take two pairs of opposite faces: $E, E' $ and $F, F'$. For the pair $E, E'$ take the line joining the centers of the faces and take points $A$ and $C$ on the line each a distance $m$ outside one of the faces. Similarly, take $B$ and $D$ on the line joining the centers of $F, F'$ each a distance $m$ outside one of the faces. Show that $ABCD$ is a rectangle and find the ratio of its side lengths.