Olympiad problems

1999 Ukraine Team Selection Test, 12

Tags: combinatorics

In a group of $n \ge 4$ persons, every three who know each other have a common signal. Assume that these signals are not repeated and that there are $m \ge 1$ signals in total. For any set of four persons in which there are three having a common signal, the fourth person has a common signal with at most one of them. Show that there three persons who have a common signal, such that the number of persons having no signal with anyone of them does not exceed $\left[n+3 -\frac{18m}{n}\right]$

2013 Brazil Team Selection Test, 3

Tags: modular arithmetic , number theory , Divisibility , IMO Shortlist

Let $x$ and $y$ be positive integers. If ${x^{2^n}}-1$ is divisible by $2^ny+1$ for every positive integer $n$, prove that $x=1$.

2018 PUMaC Live Round, 1.1

Tags: PuMAC , Live Round

Find the number of pairs of real numbers $(x,y)$ such that $x^4+y^4=4xy-2$.

2007 Paraguay Mathematical Olympiad, 3

Tags: ratio , analytic geometry , graphing lines , slope

Let $ABCD$ be a square, $E$ and $F$ midpoints of $AB$ and $AD$ respectively, and $P$ the intersection of $CF$ and $DE$. a) Show that $DE \perp CF$. b) Determine the ratio $CF : PC : EP$

2014 China Western Mathematical Olympiad, 6

Tags: inequalities , ceiling function , inequalities proposed

Let $n\ge 2$ is a given integer , $x_1,x_2,\ldots,x_n $ be real numbers such that $(1) x_1+x_2+\ldots+x_n=0 $, $(2) |x_i|\le 1$ $(i=1,2,\cdots,n)$. Find the maximum of Min$\{|x_1-x_2|,|x_2-x_3|,\cdots,|x_{n-1}-x_n|\}$.

2018 Online Math Open Problems, 3

Tags:

Hen Hao randomly selects two distinct squares on a standard $8\times 8$ chessboard. Given that the two squares touch (at either a vertex or a side), the probability that the two squares are the same color can be expressed in the form $\frac mn$ for relatively prime positive integers $m$ and $n$. Find $100m+n$. [i]Proposed by James Lin

2017 Junior Balkan Team Selection Tests - Romania, 4

Tags: Equilateral , geometry , angles , equal angles

Let $ABC$ be a right triangle, with the right angle at $A$. The altitude from $A$ meets $BC$ at $H$ and $M$ is the midpoint of the hypotenuse $[BC]$. On the legs, in the exterior of the triangle, equilateral triangles $BAP$ and $ACQ$ are constructed. If $N$ is the intersection point of the lines $AM$ and $PQ$, prove that the angles $\angle NHP$ and $\angle AHQ$ are equal. Miguel Ochoa Sanchez and Leonard Giugiuc

2007 IberoAmerican Olympiad For University Students, 6

Tags: group theory , abstract algebra , Ring Theory , superior algebra , superior algebra unsolved

Let $F$ be a field whose characteristic is not $2$, let $F^*=F\setminus\left\{0\right\}$ be its multiplicative group and let $T$ be the subgroup of $F^*$ constituted by its finite order elements. Prove that if $T$ is finite, then $T$ is cyclic and its order is even.

2024 Sharygin Geometry Olympiad, 8.4

Tags: geometry , Sharygin Geometry Olympiad , Sharygin 2024

A square with side $1$ is cut from the paper. Construct a segment with length $1/2024$ using at most $20$ folds. No instruments are available. It is allowed only to fold the paper and to mark the common points of folding lines.

2020 ELMO Problems, P3

Tags: perpendicular bisector , orthocenter

Janabel has a device that, when given two distinct points $U$ and $V$ in the plane, draws the perpendicular bisector of $UV$. Show that if three lines forming a triangle are drawn, Janabel can mark the orthocenter of the triangle using this device, a pencil, and no other tools. [i]Proposed by Fedir Yudin.[/i]

1994 Putnam, 6

Tags: Putnam , function , college contests

Let $f_1,f_2,\cdots ,f_{10}$ be bijections on $\mathbb{Z}$ such that for each integer $n$, there is some composition $f_{\ell_1}\circ f_{\ell_2}\circ \cdots \circ f_{\ell_m}$ (allowing repetitions) which maps $0$ to $n$. Consider the set of $1024$ functions \[ \mathcal{F}=\{f_1^{\epsilon_1}\circ f_2^{\epsilon_2}\circ \cdots \circ f_{10}^{\epsilon_{10}}\} \] where $\epsilon _i=0$ or $1$ for $1\le i\le 10.\; (f_i^{0}$ is the identity function and $f_i^1=f_i)$. Show that if $A$ is a finite set of integers then at most $512$ of the functions in $\mathcal{F}$ map $A$ into itself.

2014 Contests, 2

Tags: function , ceiling function , algorithm , combinatorics unsolved , combinatorics

Let $k\ge 1$ be a positive integer. We consider $4k$ chips, $2k$ of which are red and $2k$ of which are blue. A sequence of those $4k$ chips can be transformed into another sequence by a so-called move, consisting of interchanging a number (possibly one) of consecutive red chips with an equal number of consecutive blue chips. For example, we can move from $r\underline{bb}br\underline{rr}b$ to $r\underline{rr}br\underline{bb}b$ where $r$ denotes a red chip and $b$ denotes a blue chip. Determine the smallest number $n$ (as a function of $k$) such that starting from any initial sequence of the $4k$ chips, we need at most $n$ moves to reach the state in which the first $2k$ chips are red.

1955 AMC 12/AHSME, 42

Tags:

If $ a$, $ b$, and $ c$ are positive integers, the radicals $ \sqrt{a\plus{}\frac{b}{c}}$ and $ a\sqrt{\frac{b}{c}}$ are equal when and only when: $ \textbf{(A)}\ a\equal{}b\equal{}c\equal{}1 \qquad \textbf{(B)}\ a\equal{}b\text{ and }c\equal{}a\equal{}1 \qquad \textbf{(C)}\ c\equal{}\frac{b(a^2\minus{}1)}{2} \\ \textbf{(D)}\ a\equal{}b \text{ and }c\text{ is any value} \qquad \textbf{(E)}\ a\equal{}b \text{ and }c\equal{}a\minus{}1$

1988 IberoAmerican, 3

Tags: geometry , perimeter , conics , ellipse , geometry unsolved

Prove that among all possible triangles whose vertices are $3,5$ and $7$ apart from a given point $P$, the ones with the largest perimeter have $P$ as incentre.

1998 AMC 12/AHSME, 18

Tags: geometry , 3D geometry , sphere , AMC

A right circular cone of volume $ A$, a right circular cylinder of volume $ M$, and a sphere of volume $ C$ all have the same radius, and the common height of the cone and the cylinder is equal to the diameter of the sphere. Then $ \textbf{(A)}\ A \minus{} M \plus{} C \equal{} 0 \qquad \textbf{(B)}\ A \plus{} M \equal{} C \qquad \textbf{(C)}\ 2A \equal{} M \plus{} C$ $ \textbf{(D)}\ A^2 \minus{} M^2 \plus{} C^2 \equal{} 0 \qquad \textbf{(E)}\ 2A \plus{} 2M \equal{} 3C$

2021 Kyiv City MO Round 1, 11.5

Tags: number theory , prime numbers

For positive integers $m, n$ define the function $f_n(m) = 1^{2n} + 2^{2n} + 3^{2n} + \ldots +m^{2n}$. Prove that there are only finitely many pairs of positive integers $(a, b)$ such that $f_n(a) + f_n(b)$ is a prime number. [i]Proposed by Nazar Serdyuk[/i]

2023 Balkan MO Shortlist, C2

Tags: combinatorics

For an integer $n>2$, the tuple $(1, 2, \ldots, n)$ is written on a blackboard. On each turn, one can choose two numbers from the tuple such that their sum is a perfect square and swap them to obtain a new tuple. Find all integers $n > 2$ for which all permutations of $\{1, 2,\ldots, n\}$ can appear on the blackboard in this way.

2004 Germany Team Selection Test, 1

Tags: vector , geometry , geometric transformation , similar triangles , geometry proposed

Let $ABC$ be an acute triangle, and let $M$ and $N$ be two points on the line $AC$ such that the vectors $MN$ and $AC$ are identical. Let $X$ be the orthogonal projection of $M$ on $BC$, and let $Y$ be the orthogonal projection of $N$ on $AB$. Finally, let $H$ be the orthocenter of triangle $ABC$. Show that the points $B$, $X$, $H$, $Y$ lie on one circle.

2021 Switzerland - Final Round, 6

Tags: function , algebra , nice problem

Let $\mathbb{N}$ be the set of positive integers. Let $f: \mathbb{N} \rightarrow \mathbb{N}$ be a function such that for every positive integer $n \in \mathbb{N}$ $$ f(n) -n<2021 \quad \text{and} \quad f^{f(n)}(n) =n$$ Prove that $f(n)=n$ for infinitely many $n \in \mathbb{N}$

2011 Middle European Mathematical Olympiad, 1

Tags: quadratics , number theory proposed , number theory

Initially, only the integer $44$ is written on a board. An integer a on the board can be re- placed with four pairwise different integers $a_1, a_2, a_3, a_4$ such that the arithmetic mean $\frac 14 (a_1 + a_2 + a_3 + a_4)$ of the four new integers is equal to the number $a$. In a step we simultaneously replace all the integers on the board in the above way. After $30$ steps we end up with $n = 4^{30}$ integers $b_1, b2,\ldots, b_n$ on the board. Prove that \[\frac{b_1^2 + b_2^2+b_3^2+\cdots+b_n^2}{n}\geq 2011.\]

2016 Saudi Arabia BMO TST, 4

Tags: combinatorics , Coloring

How many ways are there to color the vertices of a square with $n$ colors $1,2, .. ., n$. (The colorings must be different so that we can’t get one from the other by a rotation.)

2008 Sharygin Geometry Olympiad, 8

Tags: geometry unsolved , geometry

(A.Akopyan, V.Dolnikov) Given a set of points inn the plane. It is known that among any three of its points there are two such that the distance between them doesn't exceed 1. Prove that this set can be divided into three parts such that the diameter of each part does not exceed 1.

2000 India Regional Mathematical Olympiad, 3

Tags: inequalities , integration , calculus , induction

Suppose $\{ x_n \}_{n\geq 1}$ is a sequence of positive real numbers such that $x_1 \geq x_2 \geq x_3 \ldots \geq x_n \ldots$, and for all $n$ \[ \frac{x_1}{1} + \frac{x_4}{2} + \frac{x_9}{3} + \ldots + \frac{x_{n^2}}{n} \leq 1 . \] Show that for all $k$ \[ \frac{x_1}{1} + \frac{x_2}{2} +\ldots + \frac{x_k}{k} \leq 3. \]

2016 Benelux, 1

Tags: number theory

Find the greatest positive integer $N$ with the following property: there exist integers $x_1, . . . , x_N$ such that $x^2_i - x_ix_j$ is not divisible by $1111$ for any $i\ne j.$

1993 French Mathematical Olympiad, Problem 2

Tags: number theory , Sequences

Let $n$ be a given positive integer. (a) Do there exist $2n+1$ consecutive positive integers $a_0,a_1,\ldots,a_{2n}$ in the ascending order such that $a_0+a_1+\ldots+a_n=a_{n+1}+\ldots+a_{2n}$? (b) Do there exist consecutive positive integers $a_0,a+1,\ldots,a_{2n}$ in ascending order such that $a_0^2+a_1^2+\ldots+a_n^2=a_{n+1}^2+\ldots+a_{2n}^2$? (c) Do there exist consecutive positive integers $a_0,a_1,\ldots,a_{2n}$ in ascending order such that $a_0^3+a_1^3+\ldots+a_n^3=a_{n+1}^3+\ldots+a_{2n}^3$? [hide=Official Hint]You may study the function $f(x)=(x-n)^3+\ldots+x^3-(x+1)^3-\ldots-(x+n)^3$ and prove that the equation $f(x)=0$ has a unique solution $x_n$ with $3n(n+1)<x_n<3n(n+1)+1$. You may use the identity $1^3+2^3+\ldots+n^3=\frac{n^2(n+1)^2}2$.[/hide]