Olympiad problems

2007 Indonesia TST, 4

Tags: combinatorics

Let $ n$ and $ k$ be positive integers. Please, find an explicit formula for \[ \sum y_1y_2 \dots y_k,\] where the summation runs through all $ k\minus{}$tuples positive integers $ (y_1,y_2,\dots,y_k)$ satisfying $ y_1\plus{}y_2\plus{}\dots\plus{}y_k\equal{}n$.

2008 Kurschak Competition, 2

Tags: combinatorics

Let $n\ge 1$ and $a_1<a_2<\dots<a_n$ be integers. Let $S$ be the set of pairs $1\le i<j\le n$ for which $a_j-a_i$ is a power of $2$, and $T$ be the set of pairs $1\le i<j\le n$ with $j-i$ a power of $2$. (Here, the powers of $2$ are $1,2,4,\dots$.) Prove that $|S|\le |T|$.

1962 All Russian Mathematical Olympiad, 013

Tags: geometry , area

Given points $A' ,B' ,C' ,D',$ on the extension of the $[AB], [BC], [CD], [DA]$ sides of the convex quadrangle $ABCD$, such, that the following pairs of vectors are equal: $$[BB']=[AB], [CC']=[BC], [DD']=[CD], [AA']=[DA].$$ Prove that the quadrangle $A'B'C'D'$ area is five times more than the quadrangle $ABCD$ area.

1997 Pre-Preparation Course Examination, 1

Tags: function , algebra

Let $f: \mathbb R \to\mathbb R$ be a function such that $|f(x)| \leq 1$ for all $x \in \mathbb R$ and \[f \left( x + \frac{13}{42} \right) + f(x) = f \left( x + \frac 17 \right) + f \left( x + \frac 16 \right), \quad \forall x \in \mathbb R.\] Show that $f$ is a periodic function.

2006 Germany Team Selection Test, 1

Tags: combinatorics

A house has an even number of lamps distributed among its rooms in such a way that there are at least three lamps in every room. Each lamp shares a switch with exactly one other lamp, not necessarily from the same room. Each change in the switch shared by two lamps changes their states simultaneously. Prove that for every initial state of the lamps there exists a sequence of changes in some of the switches at the end of which each room contains lamps which are on as well as lamps which are off. [i]Proposed by Australia[/i]

1995 Baltic Way, 19

Tags: geometry

The following construction is used for training astronauts: A circle $C_2$ of radius $2R$ rolls along the inside of another, fixed circle $C_1$ of radius $nR$, where $n$ is an integer greater than $2$. The astronaut is fastened to a third circle $C_3$ of radius $R$ which rolls along the inside of circle $C_2$ in such a way that the touching point of the circles $C_2$ and $C_3$ remains at maximum distance from the touching point of the circles $C_1$ and $C_2$ at all times. How many revolutions (relative to the ground) does the astronaut perform together with the circle $C_3$ while the circle $C_2$ completes one full lap around the inside of circle $C_1$?

2022 Durer Math Competition Finals, 16

Tags: number theory , divisor

The number $60$ is written on a blackboard. In every move, Andris wipes the numbers on the board one by one, and writes all its divisors in its place (including itself). After $10$ such moves, how many times will $1$ appear on the board?

2019 CCA Math Bonanza, L5.3

Tags: function

For a positive integer $n$, let $d\left(n\right)$ be the number of positive divisors of $n$ (for example $d\left(39\right)=4$). Estimate the average value that $d\left(n\right)$ takes on as $n$ ranges from $1$ to $2019$. An estimate of $E$ earns $2^{1-\left|A-E\right|}$ points, where $A$ is the actual answer. [i]2019 CCA Math Bonanza Lightning Round #5.3[/i]

2015 NIMO Problems, 3

Tags:

How many $5$-digit numbers $N$ (in base $10$) contain no digits greater than $3$ and satisfy the equality $\gcd(N,15)=\gcd(N,20)=1$? (The leading digit of $N$ cannot be zero.) [i]Based on a proposal by Yannick Yao[/i]

1990 IMO Longlists, 52

Tags: inequalities

Let real numbers $a_1, a_2, \ldots, a_n$ satisfy $0 < a_i \leq a, \ i = 1, 2, \ldots, n$. Prove that (i) If $n = 4$, then \[\frac 1a \sum_{i=1}^4 a_i - \frac{a_1a_2 + a_2a_3 + a_3 a_4 + a_4 a_1}{a^2} \leq 2.\] (ii) If $n = 6$, then \[\frac 1a \sum_{i=1}^6 a_i - \frac{a_1a_2 + a_2a_3 + \cdots + a_5 a_6 + a_6 a_1}{a^2} \leq 3.\]

2023 Junior Balkan Team Selection Tests - Moldova, 6

Tags: algebra

Real numbers $a,b,c$ with $a\neq b$ verify $$a^2(b+c)=b^2(c+a)=2023.$$ Find the numerical value of $E=c^2(a+b)$.

1997 IMC, 1

Tags: limit , calculus , integration , logarithm , real analysis

Let $\{\epsilon_n\}^\infty_{n=1}$ be a sequence of positive reals with $\lim\limits_{n\rightarrow+\infty}\epsilon_n = 0$. Find \[ \lim\limits_{n\rightarrow\infty}\dfrac{1}{n}\sum\limits^{n}_{k=1}\ln\left(\dfrac{k}{n}+\epsilon_n\right) \]

2006 Taiwan National Olympiad, 1

Tags: circumcircle , angle bisector , geometry

$P,Q$ are two fixed points on a circle centered at $O$, and $M$ is an interior point of the circle that differs from $O$. $M,P,Q,O$ are concyclic. Prove that the bisector of $\angle PMQ$ is perpendicular to line $OM$.

2009 Jozsef Wildt International Math Competition, W. 5

Tags: number theory , prime number

Let $p_1$, $p_2$ be two odd prime numbers and $\alpha $, $n$ be positive integers with $\alpha >1$, $n>1$. Prove that if the equation $\left (\frac{p_2 -1}{2} \right )^{p_1} + \left (\frac{p_2 +1}{2} \right )^{p_1} = \alpha^n$ does not have integer solutions for both $p_1 =p_2$ and $p_1 \neq p_2$.

2021 Sharygin Geometry Olympiad, 8.2

Tags: geometry , reflection , concurrency

Three parallel lines $\ell_a, \ell_b, \ell_c$ pass through the vertices of triangle $ABC$. A line $a$ is the reflection of altitude $AH_a$ about $\ell_a$. Lines $b, c$ are defined similarly. Prove that $a, b, c$ are concurrent.

2010 Contests, 2

Tags: number theory , remainder

Let $n$ be an integer, $n \ge 2$. Find the remainder of the division of the number $n(n + 1)(n + 2)$ by $n - 1$.

2001 AIME Problems, 8

Tags:

Call a positive integer $N$ a $\textit{7-10 double}$ if the digits of the base-7 representation of $N$ form a base-10 number that is twice $N.$ For example, 51 is a 7-10 double because its base-7 representation is 102. What is the largest 7-10 double?

2008 Bulgaria Team Selection Test, 3

Tags: function , algebra

Let $\mathbb{R}^{+}$ be the set of positive real numbers. Find all real numbers $a$ for which there exists a function $f :\mathbb{R}^{+} \to \mathbb{R}^{+}$ such that $3(f(x))^{2}=2f(f(x))+ax^{4}$, for all $x \in \mathbb{R}^{+}$.

2007 AMC 8, 4

Tags:

A haunted house has six windows. In how many ways can Georgie the Ghost enter the house by one window and leave by a different window? $\textbf{(A)}\ 12 \qquad \textbf{(B)}\ 15 \qquad \textbf{(C)}\ 18 \qquad \textbf{(D)}\ 30 \qquad \textbf{(E)}\ 36$

2021/2022 Tournament of Towns, P5

Tags: geometry

A quadrilateral ABCD is inscribed into a circle ω with center O. The circumcircle of the triangle AOC intersects the lines AB, BC, CD and DA the second time at the points M, N, K and L respectively. Prove that the lines MN, KL and the tangents to ω at the points A и C all touch the same circle.

2010 Romania National Olympiad, 4

Tags: angle bisector , geometry

In the isosceles triangle $ABC$, with $AB=AC$, the angle bisector of $\angle B$ meets the side $AC$ at $B'$. Suppose that $BB'+B'A=BC$. Find the angles of the triangle $ABC$. [i]Dan Nedeianu[/i]

2009 Albania Team Selection Test, 4

Tags: relatively prime , number theory

Find all the natural numbers $m,n$ such that $1+5 \cdot 2^m=n^2$.

2005 China Team Selection Test, 1

Tags: rearrangement inequality , combinatorics , inequalities

Let $a_{1}$, $a_{2}$, …, $a_{6}$; $b_{1}$, $b_{2}$, …, $b_{6}$ and $c_{1}$, $c_{2}$, …, $c_{6}$ are all permutations of $1$, $2$, …, $6$, respectively. Find the minimum value of $\sum_{i=1}^{6}a_{i}b_{i}c_{i}$.

2019 IMO, 6

Tags: mixtilinear incircle , geometry

Let $I$ be the incentre of acute triangle $ABC$ with $AB\neq AC$. The incircle $\omega$ of $ABC$ is tangent to sides $BC, CA$, and $AB$ at $D, E,$ and $F$, respectively. The line through $D$ perpendicular to $EF$ meets $\omega$ at $R$. Line $AR$ meets $\omega$ again at $P$. The circumcircles of triangle $PCE$ and $PBF$ meet again at $Q$. Prove that lines $DI$ and $PQ$ meet on the line through $A$ perpendicular to $AI$. [i]Proposed by Anant Mudgal, India[/i]

2008 District Olympiad, 2

Tags: geometry , square , perpendicular

Consider the square $ABCD$ and $E \in (AB)$. The diagonal $AC$ intersects the segment $[DE]$ at point $P$. The perpendicular taken from point $P$ on $DE$ intersects the side $BC$ at point $F$. Prove that $EF = AE + FC$.