Olympiad problems

2014 District Olympiad, 3

Tags: incenter , geometry

The points $M, N,$ and $P$ are chosen on the sides $BC, CA$ and $AB$ of the $\Delta ABC$ such that $BM=BP$ and $CM=CN$. The perpendicular dropped from $B$ to $MP$ and the perpendicular dropped from $C$ to $MN$ intersect at $I$. Prove that the angles $\measuredangle{IPA}$ and $\measuredangle{INC}$ are congruent.

2021 Spain Mathematical Olympiad, 5

Tags: combinatorics , vandermonde identity , combinatorial identity

We have $2n$ lights in two rows, numbered from $1$ to $n$ in each row. Some (or none) of the lights are on and the others are off, we call that a "state". Two states are distinct if there is a light which is on in one of them and off in the other. We say that a state is good if there is the same number of lights turned on in the first row and in the second row. Prove that the total number of good states divided by the total number of states is: $$ \frac{3 \cdot 5 \cdot 7 \cdots (2n-1)}{2^n n!} $$

2025 Austrian MO Regional Competition, 2

Tags: geometry

Let $\triangle{ABC}$ be an isosceles triangle with $AC = BC$ and circumcircle $\omega$. The line through $B$ perpendicular to $BC$ is denoted by $\ell$. Furthermore, let $M$ be any point on $\ell$. The circle $\gamma$ with center $M$ and radius $BM$ intersects $AB$ once more at point $P$ and the circumcircle $\omega$ once more at point $Q$. Prove that the points $P,Q$ and $C$ lie on a straight line. [i](Karl Czakler)[/i]

2022 Oral Moscow Geometry Olympiad, 5

Tags: locus , geometry

Given a circle and a straight line $AB$ passing through its center (points $A$ and $B$ are fixed, $A$ is outside the circle, and $B$ is inside). Find the locus of the intersection of lines $AX$ and $BY$, where $XY$ is an arbitrary diameter of the circle. (A. Akopyan, A. Zaslavsky)

2008 Vietnam National Olympiad, 4

Tags: limit , algebra

he sequence of real number $ (x_n)$ is defined by $ x_1 \equal{} 0,$ $ x_2 \equal{} 2$ and $ x_{n\plus{}2} \equal{} 2^{\minus{}x_n} \plus{} \frac{1}{2}$ $ \forall n \equal{} 1,2,3 \ldots$ Prove that the sequence has a limit as $ n$ approaches $ \plus{}\infty.$ Determine the limit.

2023 Malaysian APMO Camp Selection Test, 4

Tags: combinatorics

Let $k$ be a fixed integer. In the town of Ivanland, there are at least $k+1$ citizens standing on a plane such that the distances between any two citizens are distinct. An election is to be held such that every citizen votes the $k$-th closest citizen to be the president. What is the maximal number of votes a citizen can have? [i]Proposed by Ivan Chan Kai Chin[/i]

2022 Moscow Mathematical Olympiad, 2

Tags: analytic geometry , ruler , compass , geometry

In a Cartesian coordinate system (with the same scale on the x and y axes)there is a graph of the exponential function $y=3^x$. Then the y-axis and all marks on the x-axis erased. Only the graph of the function and the x-axis remained without a scale and a mark of $0$. How can you restore the y-axis using a compass and ruler?

2005 Romania National Olympiad, 4

Tags:

On a circle there are written 2005 non-negative integers with sum 7022. Prove that there exist two pairs formed with two consecutive numbers on the circle such that the sum of the elements in each pair is greater or equal with 8. [i]After an idea of Marin Chirciu[/i]

2024 AMC 10, 7

Tags: integer

The product of three integers is $60$. What is the least possible positive sum of the three integers? $\textbf{(A) } 2 \qquad \textbf{(B) } 3 \qquad \textbf{(C) } 5 \qquad \textbf{(D) } 6 \qquad \textbf{(E) } 13$

2016 Saint Petersburg Mathematical Olympiad, 4

Tags: combinatorics , coloring

The cells of a square $100 \times 100$ table are colored in one of two colors, black or white. A coloring is called admissible if for any row or column, the number $b$ of black colored cells satisfies $50 \le b \le 60$. It is permitted to recolor a cell if by doing so the resulting configuration is still admissible. Prove that one can transition from any admissible coloring of the board to any other using a sequence of valid recoloring operations.

2025 Azerbaijan Junior NMO, 4

Tags: combinatorics

A $3\times3$ square is filled with numbers $1;2;3...;9$.The numbers inside four $2\times2$ squares is summed,and arranged in an increasing order. Is it possible to obtain the following sequences as a result of this operation? $\text{a)}$ $24,24,25,25$ $\text{b)}$ $20,23,26,29$

2010 All-Russian Olympiad Regional Round, 9.5

Tags: algebra

Dunno wrote down $11$ natural numbers in a circle. For every two adjacent numbers, he calculated their difference. As a result among the differences found there were four units, four twos and three threes. Prove that Dunno made a mistake somewhere an error.

2016 District Olympiad, 4

Tags: function , analysis , darboux , real analysis

Let $ I $ be an open real interval, and let be two functions $ f,g:I\longrightarrow\mathbb{R} $ satisfying the identity: $$ x,y\in I\wedge x\neq y\implies\frac{f(x)-g(y)}{x-y} +|x-y|\ge 0. $$ [b]a)[/b] Prove that $ f,g $ are nondecreasing. [b]b)[/b] Give a concrete example for $ f\neq g. $

2010 Korea Junior Math Olympiad, 6

Tags: number theory

Let $n\in\mathbb{N}$ and $p$ is the odd prime number. Define the sequence $a_n$ such that $a_1=pn+1$ and $a_{k+1}=na_k+1$ for all $k \in \mathbb{N}$ . Prove that $a_{p-1}$ is compound number.

2019 PUMaC Geometry A, 1

Tags: geometry , 3d geometry

A right cone in $xyz$-space has its apex at $(0,0,0)$, and the endpoints of a diameter on its base are $(12,13,-9)$ and $(12,-5,15)$. The volume of the cone can be expressed as $a\pi$. What is $a$?

LMT Speed Rounds, 7

Tags: geometry , 3d geometry , prism

Isabella is making sushi. She slices a piece of salmon into the shape of a solid triangular prism. The prism is $2$ cm thick, and its triangular faces have side lengths $7$ cm, $ 24$cm, and $25$ cm. Find the volume of this piece of salmon in cm$^3$. [i]Proposed by Isabella Li[/i]

2017 Puerto Rico Team Selection Test, 4

Tags: algebra , sequence

We define the sequences $a_n =\frac{n (n + 1)}{2}$ and $b_n = a_1 + a_2 +… + a_n$. Prove that there is no integer $n$ such that $b_n = 2017$.

2012 Baltic Way, 18

Tags: quadratic , number theory

Find all triples $(a,b,c)$ of integers satisfying $a^2 + b^2 + c^2 = 20122012$.

2013 Greece Team Selection Test, 4

Tags: combinatorics , counting , circles , line

Given are $n$ different concentric circles on the plane.Inside the disk with the smallest radius (strictly inside it),we consider two distinct points $A,B$.We consider $k$ distinct lines passing through $A$ and $m$ distinct lines passing through $B$.There is no line passing through both $A$ and $B$ and all the lines passing through $k$ intersect with all the lines passing through $B$.The intersections do not lie on some of the circles.Determine the maximum and the minimum number of regions formed by the lines and the circles and are inside the circles.

2018 All-Russian Olympiad, 2

Tags: algebra

Let $n\geq 2$ and $x_{1},x_{2},\ldots,x_{n}$ positive real numbers. Prove that \[\frac{1+x_{1}^2}{1+x_{1}x_{2}}+\frac{1+x_{2}^2}{1+x_{2}x_{3}}+\cdots+\frac{1+x_{n}^2}{1+x_{n}x_{1}}\geq n.\]

2010 Contests, 3

Tags: number theory

A student adds up rational fractions incorrectly: \[\frac{a}{b}+\frac{x}{y}=\frac{a+x}{b+y}\quad (\star) \] Despite that, he sometimes obtains correct results. For a given fraction $\frac{a}{b},a,b\in\mathbb{Z},b>0$, find all fractions $\frac{x}{y},x,y\in\mathbb{Z},y>0$ such that the result obtained by $(\star)$ is correct.

2020 Taiwan TST Round 3, 1

Tags: number theory , additive number theory

Prove that there is a constant $c>0$ and infinitely many positive integers $n$ with the following property: there are infinitely many positive integers that cannot be expressed as the sum of fewer than $cn\log(n)$ pairwise coprime $n$th powers. [i]Canada[/i]

2005 Romania National Olympiad, 2

Tags: geometry , 3d geometry , pyramid , trigonometry , geometric transformation , rotation , conic

The base $A_{1}A_{2}\ldots A_{n}$ of the pyramid $VA_{1}A_{2}\ldots A_{n}$ is a regular polygon. Prove that if \[\angle VA_{1}A_{2}\equiv \angle VA_{2}A_{3}\equiv \cdots \equiv \angle VA_{n-1}A_{n}\equiv \angle VA_{n}A_{1},\] then the pyramid is regular.

2024 HMNT, 5

Tags: guts

Let $ABCD$ be a trapezoid with $AB \parallel CD, AB=20, CD=24,$ and area $880.$ Compute the area of the triangle formed by the midpoints of $AB, AC,$ and $BD.$

2009 Indonesia TST, 3

Tags: circumcircle , geometry

Let $ ABC$ be an isoceles triangle with $ AC\equal{}BC$. A point $ P$ lies inside $ ABC$ such that \[ \angle PAB \equal{} \angle PBC, \angle PAC \equal{} \angle PCB.\] Let $ M$ be the midpoint of $ AB$ and $ K$ be the intersection of $ BP$ and $ AC$. Prove that $ AP$ and $ PK$ trisect $ \angle MPC$.