Olympiad problems

2014 ELMO Shortlist, 5

Define a [i]beautiful number[/i] to be an integer of the form $a^n$, where $a\in\{3,4,5,6\}$ and $n$ is a positive integer. Prove that each integer greater than $2$ can be expressed as the sum of pairwise distinct beautiful numbers. [i]Proposed by Matthew Babbitt[/i]

2012 Morocco TST, 2

Tags: induction , pigeonhole principle , algebra unsolved , algebra

Let $\left ( a_{n} \right )_{n \geq 1}$ be an increasing sequence of positive integers such that $a_1=1$, and for all positive integers $n$, $a_{n+1}\leq 2n$. Prove that for every positive $n$; there exists positive integers $p$ and $q$ such that $n=a_{p}-a_{q}$.

2006 Moldova National Olympiad, 8.4

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Sum of $ 100 $ natural distinct numbers is $ 9999 $. Prove that $ 2006 $ divide their product.

2021 CMIMC, 2.7

Tags: algebra , number theory

For each positive integer $n,$ let $\sigma(n)$ denote the sum of the positive integer divisors of $n.$ How many positive integers $n \leq 2021$ satisfy $$\sigma(3n) \geq \sigma(n)+\sigma(2n)?$$ [i]Proposed by Kyle Lee[/i]

2013 NIMO Summer Contest, 9

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Compute $99(99^2+3) + 3\cdot99^2$. [i]Proposed by Evan Chen[/i]

PEN H Problems, 80

Tags: Diophantine Equations

Prove that if $a, b, c, d$ are integers such that $d=( a+\sqrt[3]{2}b+\sqrt[3]{4}c)^{2}$ then $d$ is a perfect square.

1990 AMC 12/AHSME, 25

Tags: geometry , 3D geometry , sphere , AMC

Nine congruent spheres are packed inside a unit cube in such a way that one of them has its center at the center of the cube and each of the others is tangent to the center sphere and to three faces of the cube. What is the radius of each sphere? $ \textbf{(A)}\ 1-\frac{\sqrt{3}}{2} \qquad\textbf{(B)}\ \frac{2\sqrt{3}-3}{2} \qquad\textbf{(C)}\ \frac{\sqrt{2}}{6} \qquad\textbf{(D)}\ \frac{1}{4} \qquad\textbf{(E)}\ \frac{\sqrt{3}(2-\sqrt{2})}{4} $

2003 Bundeswettbewerb Mathematik, 2

Tags: function , algebra , system of equations , algebra proposed

Find all triples $\left(x,\ y,\ z\right)$ of integers satisfying the following system of equations: $x^3-4x^2-16x+60=y$; $y^3-4y^2-16y+60=z$; $z^3-4z^2-16z+60=x$.

2007 IberoAmerican Olympiad For University Students, 2

Tags: inequalities , induction , inequalities proposed

Prove that for all positive integers $n$ and for all real numbers $x$ such that $0\le x\le1$, the following inequality holds: $\left(1-x+\frac{x^2}{2}\right)^n-(1-x)^n\le\frac{x}{2}$.

1993 Tournament Of Towns, (392) 4

Tags: algebra , combinatorics

Peter wants to make an unusual die having different positive integers on each of its faces. For neighbouring faces the corresponding numbers should differ by at least two. Find the minimal sum of the six numbers. (Folklore)

1986 AMC 12/AHSME, 20

Tags: AMC

Suppose $x$ and $y$ are inversely proportional and positive. If $x$ increases by $p\%$, then $y$ decreases by $ \textbf{(A)}\ p\%\qquad\textbf{(B)}\ \frac{p}{1+p}\%\qquad\textbf{(C)}\ \frac{100}{p}\%\qquad\textbf{(D)}\ \frac{p}{100+p}\%\qquad\textbf{(E)}\ \frac{100p}{100+p}\%$

2008 China Girls Math Olympiad, 8

Tags: search , number theory unsolved , number theory

For positive integers $ n$, $ f_n \equal{} \lfloor2^n\sqrt {2008}\rfloor \plus{} \lfloor2^n\sqrt {2009}\rfloor$. Prove there are infinitely many odd numbers and infinitely many even numbers in the sequence $ f_1,f_2,\ldots$.

2000 Tournament Of Towns, 6

Tags: Tournament , combinatorics

In a chess tournament , every two participants play each other exactly once. A win is worth one point , a draw is worth half a point and a loss is worth zero points. Looking back at the end of the tournament, a game is called an upset if the total number of points obtained by the winner of that game is less than the total number of points obtained by the loser of that game. (a) Prove that the number of upsets is always strictly less than three-quarters of the total number of games in the tournament. (b) Prove that three-quarters cannot be replaced by a smaller number. (S Tokarev) PS. part (a) for Juniors, both parts for Seniors

1996 India National Olympiad, 6

Tags: linear algebra , matrix , combinatorics unsolved , combinatorics

There is a $2n \times 2n$ array (matrix) consisting of $0's$ and $1's$ and there are exactly $3n$ zeroes. Show that it is possible to remove all the zeroes by deleting some $n$ rows and some $n$ columns.

2009 Kosovo National Mathematical Olympiad, 3

Tags: number theory

Prove that $\sqrt 2$ is irrational.

LMT Team Rounds 2021+, B3

Tags: combinatorics

Aidan rolls a pair of fair, six sided dice. Let$ n$ be the probability that the product of the two numbers at the top is prime. Given that $n$ can be written as $a/b$ , where $a$ and $b$ are relatively prime positive integers, find $a +b$. [i]Proposed by Aidan Duncan[/i]

Kvant 2022, M2690

Tags: combinatorics , Kvant

Vasya has $n{}$ candies of several types, where $n>145$. It is known that for any group of at least 145 candies, there is a type of candy which appears exactly 10 times. Find the largest possible value of $n{}$. [i]Proposed by A. Antropov[/i]

2018 Pan-African Shortlist, A6

Tags: inequalities , inequalities proposed , algebra

Let $a, b, c$ be positive real numbers such that $a^3 + b^3 + c^3 = 5abc$. Show that \[ \left( \frac{a + b}{c} \right) \left( \frac{b + c}{a} \right) \left( \frac{c + a}{b} \right) \geq 9. \]

2019 AMC 12/AHSME, 3

Tags: AMC , AMC 12 , AMC 12 B , geometry , geometric transformation , rotation

Which one of the following rigid transformations (isometries) maps the line segment $\overline{AB}$ onto the line segment $\overline{A'B'}$ so that the image of $A(-2,1)$ is $A'(2,-1)$ and the image of $B(-1,4)$ is $B'(1,-4)?$ $\textbf{(A) } $ reflection in the $y$-axis $\textbf{(B) } $ counterclockwise rotation around the origin by $90^{\circ}$ $\textbf{(C) } $ translation by 3 units to the right and 5 units down $\textbf{(D) } $ reflection in the $x$-axis $\textbf{(E) } $ clockwise rotation about the origin by $180^{\circ}$

2020 Durer Math Competition Finals, 11

Tags: angles , geometry

The convex quadrilateral $ABCD$ has $|AB| = 8$, $|BC| = 29$, $|CD| = 24$ and $|DA| = 53$. What is the area of the quadrilateral if $\angle ABC + \angle BCD = 270^o$?

2022 Germany Team Selection Test, 1

Tags: algebra , IMO Shortlist , 2021 , A1 , ISL 2021

Let $n$ be a positive integer. Given is a subset $A$ of $\{0,1,...,5^n\}$ with $4n+2$ elements. Prove that there exist three elements $a<b<c$ from $A$ such that $c+2a>3b$. [i]Proposed by Dominik Burek and Tomasz Ciesla, Poland[/i]

2002 Canada National Olympiad, 4

Tags: geometry , circumcircle , trigonometry , angle bisector , perpendicular bisector , geometry unsolved

Let $\Gamma$ be a circle with radius $r$. Let $A$ and $B$ be distinct points on $\Gamma$ such that $AB < \sqrt{3}r$. Let the circle with centre $B$ and radius $AB$ meet $\Gamma$ again at $C$. Let $P$ be the point inside $\Gamma$ such that triangle $ABP$ is equilateral. Finally, let the line $CP$ meet $\Gamma$ again at $Q$. Prove that $PQ = r$.

2020 Latvia TST, 1.3

Tags: infinitely many solutions , number theory , fibbonacci

Prove that equation $a^2 - b^2=ab - 1$ has infinitely many solutions, if $a,b$ are positive integers

2022 Saudi Arabia IMO TST, 3

Tags: number theory , AZE IMO TST

Show that $n!=a^{n-1}+b^{n-1}+c^{n-1}$ has only finitely many solutions in positive integers. [i]Proposed by Dorlir Ahmeti, Albania[/i]

2022 Bosnia and Herzegovina IMO TST, 1

Tags: geometry , circumcircle , geometric transformation , reflection

Let $ABC$ be a triangle such that $AB=AC$ and $\angle BAC$ is obtuse. Point $O$ is the circumcenter of triangle $ABC$, and $M$ is the reflection of $A$ in $BC$. Let $D$ be an arbitrary point on line $BC$, such that $B$ is in between $D$ and $C$. Line $DM$ cuts the circumcircle of $ABC$ in $E,F$. Circumcircles of triangles $ADE$ and $ADF$ cut $BC$ in $P,Q$ respectively. Prove that $DA$ is tangent to the circumcircle of triangle $OPQ$.