Olympiad problems

2025 Philippine MO, P3

Let $d$ be a positive integer. Define the sequence $a_1, a_2, a_3, \dots$ such that \[\begin{cases} a_1 = 1 \\ a_{n+1} = n\left\lfloor\frac{a_n}{n}\right\rfloor + d, \quad n \ge 1.\end{cases}\] Prove that there exists a positive integer $M$ such that $a_M, a_{M+1}, a_{M+2}, \dots$ is an arithmetic sequence.

2014 PUMaC Team, 3

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How many integer $x$ are there such that $\frac{x^2-6}{x-6}$ is a positive integer?

2021 VIASM Math Olympiad Test, Problem 1

Tags: geometry , rectangle , combinatorics

Given a $8$x$8$ square board a) Prove that: for any ways to color the board, we are always be able to find a rectangle consists of $8$ squares such that these squares are not colored. b) Prove that: we can color $7$ squares on the board such that for any rectangles formed by $\geq 9$ squares, there are at least $1$ colored square.

2004 Croatia National Olympiad, Problem 2

Tags: geometry

Prove that the medians from the vertices $A$ and $B$ of a triangle $ABC$ are orthogonal if and only if $BC^2+AC^2=5AB^2$.

1935 Moscow Mathematical Olympiad, 006

Tags: geometry , 3d geometry , pyramid , volume

The base of a right pyramid is a quadrilateral whose sides are each of length $a$. The planar angles at the vertex of the pyramid are equal to the angles between the lateral edges and the base. Find the volume of the pyramid.

2012 Stanford Mathematics Tournament, 8

Tags: quadratic

For real numbers $(x, y, z)$ satisfying the following equations, ﬁnd all possible values of $x+y+z$ $x^2y+y^2z+z^2x=-1$ $xy^2+yz^2+zx^2=5$ $xyz=-2$

2018 Mathematical Talent Reward Programme, SAQ: P 6

Tags: coloring

Let $d(n)$ be the number of divisors of $n,$ where $n$ is a natural number. Prove that the natural numbers can be colured by 2 colours in such way, that for any infinite increasing sequence $\left\{a_{1}, a_{2}, \cdots\right\}$ if $\left\{d\left(a_{1}\right), d\left(a_{2}\right), \cdots\right\}$ is an nonconstant geometric series then $\left\{a_{1}, a_{2}, \cdots\right\}$ does not bear same colour.

2022 Czech-Polish-Slovak Junior Match, 5

Tags: number theory , divisible

An integer $n\ge1$ is [i]good [/i] if the following property is satisfied: If a positive integer is divisible by each of the nine numbers $n + 1, n + 2, ..., n + 9$, this is also divisible by $n + 10$. How many good integers are $n\ge 1$?

2023 Assam Mathematics Olympiad, 4

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Real numbers $a, b, c$ satisfy $(2b - a)^2 + (2b - c)^2 = 2(2b^2 - ac)$. Prove that $a + c = 2b$.

2011 Baltic Way, 15

Tags: geometry

Let $ABCD$ be a convex quadrilateral such that $\angle ADB=\angle BDC$. Suppose that a point $E$ on the side $AD$ satisfies the equality \[AE\cdot ED + BE^2=CD\cdot AE.\] Show that $\angle EBA=\angle DCB$.

1962 Miklós Schweitzer, 1

Tags: advanced fields , algebra , polynomial , search

Let $ f$ and $ g$ be polynomials with rational coefficients, and let $ F$ and $ G$ denote the sets of values of $ f$ and $ g$ at rational numbers. Prove that $ F \equal{} G$ holds if and only if $ f(x) \equal{} g(ax \plus{} b)$ for some suitable rational numbers $ a\not \equal{} 0$ and $ b$. [i]E. Fried[/i]

2009 Germany Team Selection Test, 1

Tags: geometry , rectangle , combinatorics , combinatorial geometry , extremal combinatorics

In the plane we consider rectangles whose sides are parallel to the coordinate axes and have positive length. Such a rectangle will be called a [i]box[/i]. Two boxes [i]intersect[/i] if they have a common point in their interior or on their boundary. Find the largest $ n$ for which there exist $ n$ boxes $ B_1$, $ \ldots$, $ B_n$ such that $ B_i$ and $ B_j$ intersect if and only if $ i\not\equiv j\pm 1\pmod n$. [i]Proposed by Gerhard Woeginger, Netherlands[/i]

2008 International Zhautykov Olympiad, 1

Tags: rhombus , geometry

Points $ K,L,M,N$ are repectively the midpoints of sides $ AB,BC,CD,DA$ in a convex quadrliateral $ ABCD$.Line $ KM$ meets dioganals $ AC$ and $ BD$ at points $ P$ and $ Q$,respectively.Line $ LN$ meets dioganals $ AC$ and $ BD$ at points $ R$ and $ S$,respectively. Prove that if $ AP\cdot PC\equal{}BQ\cdot QD$,then $ AR\cdot RC\equal{}BS\cdot SD$.

2010 Today's Calculation Of Integral, 587

Tags: calculus , integration , calculus computations

Evaluate $ \int_0^1 \frac{(x^2\plus{}3x)e^x\minus{}(x^2\minus{}3x)e^{\minus{}x}\plus{}2}{\sqrt{1\plus{}x(e^x\plus{}e^{\minus{}x})}}\ dx$.

2009 Puerto Rico Team Selection Test, 3

Tags: altitude , geometry , geometric inequality

Show that if $ h_A, h_B,$ and $ h_C$ are the altitudes of $ \triangle ABC$, and $ r$ is the radius of the incircle, then $$ h_A + h_B + h_C \ge 9r$$

2018 HMIC, 4

Tags: college , combinatorics

Find all functions $f: \mathbb{R}^+\to\mathbb{R}^+$ such that \[f(x+f(y+xy))=(y+1)f(x+1)-1\]for all $x,y\in\mathbb{R}^+$. ($\mathbb{R}^+$ denotes the set of positive real numbers.)

2010 LMT, 2

Tags:

Let points $A,B,$ and $C$ lie on a line such that $AB=1, BC=1,$ and $AC=2.$ Let $C_1$ be the circle centered at $A$ passing through $B,$ and let $C_2$ be the circle centered at $A$ passing through $C.$ Find the area of the region outside $C_1,$ but inside $C_2.$

2008 Purple Comet Problems, 9

Tags: geometry , 3d geometry

Find the sum of all the integers $N > 1$ with the properties that the each prime factor of $N $ is either $2, 3,$ or $5,$ and $N$ is not divisible by any perfect cube greater than $1.$

2002 China Team Selection Test, 2

Tags: inequalities , trigonometry , geometry

$ A_1$, $ B_1$ and $ C_1$ are the projections of the vertices $ A$, $ B$ and $ C$ of triangle $ ABC$ on the respective sides. If $ AB \equal{} c$, $ AC \equal{} b$, $ BC \equal{} a$ and $ AC_1 \equal{} 2t AB$, $ BA_1 \equal{} 2rBC$, $ CB_1 \equal{} 2 \mu AC$. Prove that: \[ \frac {a^2}{b^2} \cdot \left( \frac {t}{1 \minus{} 2t} \right)^2 \plus{} \frac {b^2}{c^2} \cdot \left( \frac {r}{1 \minus{} 2r} \right)^2 \plus{} \frac {c^2}{a^2} \cdot \left( \frac {\mu}{1 \minus{} 2\mu} \right)^2 \plus{} 16tr \mu \geq 1 \]

1925 Eotvos Mathematical Competition, 3

Tags: geometry , inradius , geometric inequalities , right triangle

Let $r$ be the radius of the inscribed circle of a right triangle $ABC$. Show that $r$ is less than half of either leg and less than one fourth of the hypotenuse.

2025 Polish MO Finals, 3

Tags: combinatorics

Positive integer $k$ and $k$ colors are given. We will say that a set of $2k$ points on a plane is $colorful$, if it contains exactly 2 points of each color and if lines connecting every two points of the same color are pairwise distinct. Find, in terms of $k$ the least integer $n\geq 2$ such that: in every set of $nk$ points of a plane, no three of which are collinear, consisting of $n$ points of every color there exists a $colorful$ subset.

1986 Brazil National Olympiad, 3

Tags: combinatorial geometry , angle , geometry

The Poincare plane is a half-plane bounded by a line $R$. The lines are taken to be (1) the half-lines perpendicular to $R$, and (2) the semicircles with center on $R$. Show that given any line $L$ and any point $P$ not on $L$, there are infinitely many lines through $P$ which do not intersect $L$. Show that if $ ABC$ is a triangle, then the sum of its angles lies in the interval $(0, \pi)$.

Sri Lankan Mathematics Challenge Competition 2022, P1

Tags: number theory

[b]Problem 1[/b] : Find the smallest positive integer $n$, such that $\sqrt[5]{5n}$, $\sqrt[6]{6n}$ , $\sqrt[7]{7n}$ are integers.

2015 ASDAN Math Tournament, 9

Tags: geometry test

Regular tetrahedron $ABCD$ has center $O$ and side length $1$. Points $A'$, $B'$, $C'$, and $D'$ are defined by reflecting $A$, $B$, $C$, and $D$ about $O$. Compute the volume of the polyhedron with vertices $ABCDA'B'C'D'$.

2016 Grand Duchy of Lithuania, 3

Tags: equal segments , isosceles , geometry

Let $ABC$ be an isosceles triangle with $AB = AC$. Let $D, E$ and $F$ be points on line segments $BC, CA$ and $AB$, respectively, such that $BF = BE$ and such that $ED$ is the angle bisector of $\angle BEC$. Prove that $BD = EF$ if and only if $AF = EC$.