Olympiad problems

1996 Estonia National Olympiad, 3

Tags: combinatorics

There are $1,000,000$ piles of $1996$ coins in each of them, and in one pile there are only fake coins, and in all the others - only real ones. What is the smallest weighing number that can be used to determine a heap containing counterfeit coins if the scales used have one bowl and allow weighing as much weight as desired with an accuracy of one gram, and it is also known that each counterfeit coin weighs $9$ grams, and each real coin weighs $10$ grams?

1964 Swedish Mathematical Competition, 3

Tags: radical , integer polynomial , polynomial , algebra

Find a polynomial with integer coefficients which has $\sqrt2 + \sqrt3$ and $\sqrt2 + \sqrt[3]{3}$ as roots.

2017 Harvard-MIT Mathematics Tournament, 2

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How many ways are there to insert $+$'s between the digits of $111111111111111$ (fifteen $1$'s) so that the result will be a multiple of $30$?

2018 Hong Kong TST, 4

Tags: number theory

Find infinitely many positive integers $m$ such that for each $m$, the number $\dfrac{2^{m-1}-1}{8191m}$ is an integer.

2004 Poland - Second Round, 3

Tags: induction , combinatorics

There are $ n\geq 5$ people in a party. Assume that among any three of them some two know each other. Show that one can select at least $ \frac{n}{2}$ people and arrange them at a round table so that each person sits between two of his/her acquaintances.

2025 USAJMO, 5

Tags: geometry

Let $H$ be the orthocenter of acute triangle $ABC$, let $F$ be the foot of the altitude from $C$ to $AB$, and let $P$ be the reflection of $H$ across $BC$. Suppose that the circumcircle of triangle $AFP$ intersects line $BC$ at two distinct points $X$ and $Y$. Prove that $C$ is the midpoint of $XY$.

May Olympiad L2 - geometry, 2003.2

Tags: perpendicular , rectangle , geometry

Let $ABCD$ be a rectangle of sides $AB = 4$ and $BC = 3$. The perpendicular on the diagonal $BD$ drawn from $A$ cuts $BD$ at point $H$. We call $M$ the midpoint of $BH$ and $N$ the midpoint of $CD$. Calculate the measure of the segment $MN$.

1989 IMO Longlists, 26

Tags: algebra

Let $ b_1, b_2, \ldots, b_{1989}$ be positive real numbers such that the equations \[ x_{r\minus{}1} \minus{} 2x_r \plus{} x_{r\plus{}1} \plus{} b_rx_r \equal{} 0 \quad (1 \leq r \leq 1989)\] have a solution with $ x_0 \equal{} x_{1989} \equal{} 0$ but not all of $ x_1, \ldots, x_{1989}$ are equal to zero. Prove that \[ \sum^{1989}_{k\equal{}1} b_k \geq \frac{2}{995}.\]

IV Soros Olympiad 1997 - 98 (Russia), 11.5

Tags: logarithm , number theory

Find all integers $n$ for which $\log_{2n-2} (n^2 + 2)$ is a rational number.

2021 Bundeswettbewerb Mathematik, 2

Tags: graph cycles , graph theory , combinatorics

A school has 2021 students, each of which knows at least 45 of the other students (where "knowing" is mutual). Show that there are four students who can be seated at a round table such that each of them knows both of her neighbours.

2005 Taiwan TST Round 1, 1

Tags: algebra , function , triangle inequality , inequalities , polynomial , quadratic , complex number

Let $f(x)=Ax^2+Bx+C$, $g(x)=ax^2+bx+c$ be two quadratic polynomial functions with real coefficients that satisfy the relation \[|f(x)| \ge |g(x)|\] for all real $x$. Prove that $|b^2-4ac| \le |B^2-4AC|.$ My solution was nearly complete...

1965 IMO, 3

Tags: geometry , 3d geometry , tetrahedron , prism

Given the tetrahedron $ABCD$ whose edges $AB$ and $CD$ have lengths $a$ and $b$ respectively. The distance between the skew lines $AB$ and $CD$ is $d$, and the angle between them is $\omega$. Tetrahedron $ABCD$ is divided into two solids by plane $\epsilon$, parallel to lines $AB$ and $CD$. The ratio of the distances of $\epsilon$ from $AB$ and $CD$ is equal to $k$. Compute the ratio of the volumes of the two solids obtained.

2016 Saint Petersburg Mathematical Olympiad, 1

Tags: divisor , product , number theory

Sasha multiplied all the divisors of the natural number $n$. Fedya increased each divider by $1$, and then multiplied the results. If the product found Fedya is divided by the product found by Sasha , what can $n$ be equal to ?

PEN A Problems, 94

Tags: algorithm , divisibility theory

Find all $n \in \mathbb{N}$ such that $3^{n}-n$ is divisible by $17$.

2012 Tuymaada Olympiad, 3

Tags: incenter , geometric transformation , symmetry , reflection , trapezoid , circumcircle , geometry

Point $P$ is taken in the interior of the triangle $ABC$, so that \[\angle PAB = \angle PCB = \dfrac {1} {4} (\angle A + \angle C).\] Let $L$ be the foot of the angle bisector of $\angle B$. The line $PL$ meets the circumcircle of $\triangle APC$ at point $Q$. Prove that $QB$ is the angle bisector of $\angle AQC$. [i]Proposed by S. Berlov[/i]

2009 AMC 12/AHSME, 2

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Paula the painter had just enough paint for $ 30$ identically sized rooms. Unfortunately, on the way to work, three cans of paint fell of her truck, so she had only enough paint for $ 25$ rooms. How many cans of paint did she use for the $ 25$ rooms? $ \textbf{(A)}\ 10 \qquad \textbf{(B)}\ 12 \qquad \textbf{(C)}\ 15 \qquad \textbf{(D)}\ 18 \qquad \textbf{(E)}\ 25$

2002 Bosnia Herzegovina Team Selection Test, 1

Tags: algebra

Let $x,y,z$ be real numbers that satisfy \[x+y+z= 3 \ \ \text{ and } \ \ xy+yz+zx= a\]where $a$ is a real parameter. Find the value of $a$ for which the difference between the maximum and minimum possible values of $x$ equals $8$.

2000 239 Open Mathematical Olympiad, 2

Tags: tournament , combinatorics

100 volleyball teams played a one-round tournament. No two matches held at the same time. It turned out that before each match the teams playing against each other had the same number of wins. Find all possible number of wins for the winner of this tournament.

2010 VJIMC, Problem 4

Tags: rectangle , geometry , function

Let $f:[0,1]\to\mathbb R$ be a function satisfying $$|f(x)-f(y)|\le|x-y|$$for every $x,y\in[0,1]$. Show that for every $\varepsilon>0$ there exists a countable family of rectangles $(R_i)$ of dimensions $a_i\times b_i$, $a_i\le b_i$ in the plane such that $$\{(x,f(x)):x\in[0,1]\}\subset\bigcup_iR_i\text{ and }\sum_ia_i<\varepsilon.$$(The edges of the rectangles are not necessarily parallel to the coordinate axes.)

2020 ITAMO, 2

Tags: number theory

Determine all the pairs $(a,b)$ of positive integers, such that all the following three conditions are satisfied: 1- $b>a$ and $b-a$ is a prime number 2- The last digit of the number $a+b$ is $3$ 3- The number $ab$ is a square of an integer.

2005 ISI B.Math Entrance Exam, 7

Tags: locus , geometry , area of a triangle

Let $M$ be a point in the triangle $ABC$ such that \[\text{area}(ABM)=2 \cdot \text{area}(ACM)\] Show that the locus of all such points is a straight line.

2018 Macedonia JBMO TST, 5

Tags: combinatorics

A regular $2018$-gon is inscribed in a circle. The numbers $1, 2, ..., 2018$ are arranged on the vertices of the $2018$-gon, with each vertex having one number on it, such that the sum of any $2$ neighboring numbers ($2$ numbers are neighboring if the vertices they are on lie on a side of the polygon) equals the sum of the $2$ numbers that are on the antipodes of those $2$ vertices (with respect to the given circle). Determine the number of different arrangements of the numbers. (Two arrangements are identical if you can get from one of them to the other by rotating around the center of the circle).

2010 Korea National Olympiad, 3

Tags: combinatorics

There are $ 2000 $ people, and some of them have called each other. Two people can call each other at most $1$ time. For any two groups of three people $ A$ and $ B $ which $ A \cap B = \emptyset $, there exist one person from each of $A$ and $B$ that haven't called each other. Prove that the number of two people called each other is less than $ 201000 $.

2005 ISI B.Math Entrance Exam, 8

Tags: linear algebra

In how many ways can one fill an $n*n$ matrix with $+1$ and $-1$ so that the product of the entries in each row and each column equals $-1$?

2019 USAJMO, 1

Tags:

There are $a+b$ bowls arranged in a row, numbered $1$ through $a+b$, where $a$ and $b$ are given positive integers. Initially, each of the first $a$ bowls contains an apple, and each of the last $b$ bowls contains a pear. A legal move consists of moving an apple from bowl $i$ to bowl $i+1$ and a pear from bowl $j$ to bowl $j-1$, provided that the difference $i-j$ is even. We permit multiple fruits in the same bowl at the same time. The goal is to end up with the first $b$ bowls each containing a pear and the last $a$ bowls each containing an apple. Show that this is possible if and only if the product $ab$ is even.