Olympiad problems

1998 Austrian-Polish Competition, 9

Given a triangle $ABC$, points $K,L,M$ are the midpoints of the sides $BC,CA,AB$, and points $X,Y,Z$ are the midpoints of the arcs $BC,CA,AB$ of the circumcircle not containing $A,B,C$ respectively. If $R$ denotes the circumradius and $r$ the inradius of the triangle, show that $r+KX+LY+MZ=2R$.

2024 USEMO, 4

Tags: algebra , polynomial

Find all sequences $a_1$, $a_2$, $\dots$ of nonnegative integers such that for all positive integers $n$, the polynomial \[1+x^{a_1}+x^{a_2}+\dots+x^{a_n}\] has at least one integer root. (Here $x^0=1$.) [i]Kornpholkrit Weraarchakul[/i]

Novosibirsk Oral Geo Oly VIII, 2019.1

Tags: distance , equilateral , geometry

Kikoriki live on the shores of a pond in the form of an equilateral triangle with a side of $600$ m, Krash and Wally live on the same shore, $300$ m from each other. In summer, Dokko to Krash walk $900$ m, and Wally to Rosa - also $900$ m. Prove that in winter, when the pond freezes and it will be possible to walk directly on the ice, Dokko will walk as many meters to Krash as Wally to Rosa. [url=https://en.wikipedia.org/wiki/Kikoriki]about Kikoriki/GoGoRiki / Smeshariki [/url]

2008 Sharygin Geometry Olympiad, 5

Tags: 3d geometry , pyramid , ratio , geometry

(I.Bogdanov) A section of a regular tetragonal pyramid is a regular pentagon. Find the ratio of its side to the side of the base of the pyramid.

2015 European Mathematical Cup, 3

Tags: number theory , divisor

Let $d(n)$ denote the number of positive divisors of $n$. For positive integer $n$ we define $f(n)$ as $$f(n) = d\left(k_1\right) + d\left(k_2\right)+ \cdots + d\left(k_m\right),$$ where $1 = k_1 < k_2 < \cdots < k_m = n$ are all divisors of the number $n$. We call an integer $n > 1$ [i]almost perfect[/i] if $f(n) = n$. Find all almost perfect numbers. [i]Paulius Ašvydis[/i]

1979 IMO Longlists, 36

Tags: 3d geometry , tetrahedron , geometry

A regular tetrahedron $A_1B_1C_1D_1$ is inscribed in a regular tetrahedron $ABCD$, where $A_1$ lies in the plane $BCD$, $B_1$ in the plane $ACD$, etc. Prove that $A_1B_1 \ge\frac{ AB}{3}$.

1965 Putnam, B5

Tags:

Consider collections of unordered pairs of $V$ different objects $a$, $b$, $c$, $\ldots$, $k$. Three pairs such as $ab$, $bc$, $ab$ are said to form a triangle. Prove that, if $4E\leq V^2$, it is possible to choose $E$ pairs so that no triangle is formed.

2007 Italy TST, 3

Tags: function , polynomial , functional equation , algebra

Find all $f: R \longrightarrow R$ such that \[f(xy+f(x))=xf(y)+f(x)\] for every pair of real numbers $x,y$.

2010 Slovenia National Olympiad, 1

Tags: algebra

Let $a,b$ be real numbers such that $|a| \neq |b|$ and $\frac{a+b}{a-b}+\frac{a-b}{a+b}=6.$ Find the value of the expression $\frac{a^3+b^3}{a^3-b^3}+\frac{a^3-b^3}{a^3+b^3}.$

2016 European Mathematical Cup, 2

Tags: geometry

Two circles $C_{1}$ and $C_{2}$ intersect at points $A$ and $B$. Let $P$, $Q$ be points on circles $C_{1}$, $C_{2}$ respectively, such that $|AP| = |AQ|$. The segment $PQ$ intersects circles $C_{1}$ and $C_{2}$ in points $M$, $N$ respectively. Let $C$ be the center of the arc $BP$ of $C_{1}$ which does not contain point $A$ and let $D$ be the center of arc $BQ$ of $C_{2}$ which does not contain point $A$ Let $E$ be the intersection of $CM$ and $DN$. Prove that $AE$ is perpendicular to $CD$. Proposed by Steve Dinh

2000 Abels Math Contest (Norwegian MO), 2a

Tags: algebra , sum

Let $x, y$ and $z$ be real numbers such that $x + y + z = 0$. Show that $x^3 + y^3 + z^3 = 3xyz$.

2023 Assam Mathematics Olympiad, 2

Tags:

An umbrella seller has umbrellas of $7$ different colours. He has a total of $2023$ umbrellas in stock but because of the plastic packaging, the colours are not visible. What is the minimum number of umbrellas that one must buy in order to ensure that at least $23$ umbrellas are of the same colour ?

2013 China Western Mathematical Olympiad, 2

Tags: induction , inequalities

Let the integer $n \ge 2$, and the real numbers $x_1,x_2,\cdots,x_n\in \left[0,1\right] $.Prove that\[\sum_{1\le k<j\le n} kx_kx_j\le \frac{n-1}{3}\sum_{k=1}^n kx_k.\]

2010 AIME Problems, 12

Tags: geometry , perimeter , ratio

Two noncongruent integer-sided isosceles triangles have the same perimeter and the same area. The ratio of the lengths of the bases of the two triangles is $ 8: 7$. Find the minimum possible value of their common perimeter.

2020 Swedish Mathematical Competition, 6

Tags: combinatorial geometry , combinatorics

A finite set of [i]axis parallel [/i]cubes in space has the property of each point of the room is located in a maximum of M different cubes. Show that you can divide the amount of cubes in $8 (M - 1) + 1$ subsets (or less) with the property that the cubes in each subset lacks common points. (An axis parallel cube is a cube whose edges are parallel to the coordinate axes.)

2017 Singapore Senior Math Olympiad, 4

Tags: inequalities , functional inequality , functional , algebra

Find all functions $f : Z^+ \to Z^+$ such that $f(k + 1) >f(f(k))$ for $k > 1$, where $Z^+$ is the set of positive integers.

2009 USA Team Selection Test, 6

Tags: gauss , combinatorics

Let $ N > M > 1$ be fixed integers. There are $ N$ people playing in a chess tournament; each pair of players plays each other once, with no draws. It turns out that for each sequence of $ M \plus{} 1$ distinct players $ P_0, P_1, \ldots P_M$ such that $ P_{i \minus{} 1}$ beat $ P_i$ for each $ i \equal{} 1, \ldots, M$, player $ P_0$ also beat $ P_M$. Prove that the players can be numbered $ 1,2, \ldots, N$ in such a way that, whenever $ a \geq b \plus{} M \minus{} 1$, player $ a$ beat player $ b$. [i]Gabriel Carroll.[/i]

2004 Tournament Of Towns, 2

Tags: combinatorics , consecutive , prime , sum

Find all possible values of $n \ge 1$ for which there exist $n$ consecutive positive integers whose sum is a prime number.

1996 Estonia Team Selection Test, 3

Tags: combinatorics

Each face of a cube is divided into $n^2$ equal squares. The vertices of the squares are called [i]nodes[/i], so each face has $(n+1)^2$ nodes. $(a)$ If $n=2$,does there exist a closed polygonal line whose links are sids of the squares and which passes through each node exactly once? $(b)$ Prove that, for each $n$, such a polygonal line divides the surface area of the cube into two equal parts

2012 Oral Moscow Geometry Olympiad, 6

Tags: geometry , circumcircle , tangent circle

Tangents drawn to the circumscribed circle of an acute-angled triangle $ABC$ at points $A$ and $C$, intersect at point $Z$. Let $AA_1, CC_1$ be altitudes. Line $A_1C_1$ intersects $ZA, ZC$ at points $X$ and $Y$, respectively. Prove that the circumscribed circles of the triangles $ABC$ and $XYZ$ are tangent.

2016 Serbia National Math Olympiad, 5

Tags: probabilistic method , combinatorics

There are $2n-1$ twoelement subsets of set $1,2,...,n$. Prove that one can choose $n$ out of these such that their union contains no more than $\frac{2}{3}n+1$ elements.

2008 Mongolia Team Selection Test, 3

Tags: inequalities , algebra , polynomial , three variable inequality

Find the maximum number $ C$ such that for any nonnegative $ x,y,z$ the inequality $ x^3 \plus{} y^3 \plus{} z^3 \plus{} C(xy^2 \plus{} yz^2 \plus{} zx^2) \ge (C \plus{} 1)(x^2 y \plus{} y^2 z \plus{} z^2 x)$ holds.

2021 Cyprus JBMO TST, 4

Tags: combinatorics , pigeonhole principle , ramsey theory

We colour every square of a $4\times 19$ chess board with one of the colours red, green and blue. Prove that however this colouring is done, we can always find two horizontal rows and two vertical columns such that the $4$ squares on the intersections of these lines all have the same colour.

May Olympiad L2 - geometry, 2021.3

Tags: geometry

Let $ABC$ be a triangle and $D$ is a point inside of the triangle, such that $\angle DBC=60^{\circ}$ and $\angle DCB=\angle DAB=30^{\circ}$. Let $M$ and $N$ be the midpoints of $AC$ and $BC$, respectively. Prove that $\angle DMN=90^{\circ}$.

1967 Vietnam National Olympiad, 1

Tags: analysis , algebra , graph

Draw the graph of the function $y = \frac{| x^3 - x^2 - 2x | }{3} - | x + 1 |$.