Olympiad problems

2024 Baltic Way, 3

Positive real numbers $a_1, a_2, \ldots, a_{2024}$ are written on the blackboard. A move consists of choosing two numbers $x$ and $y$ on the blackboard, erasing them and writing the number $\frac{x^2+6xy+y^2}{x+y}$ on the blackboard. After $2023$ moves, only one number $c$ will remain on the blackboard. Prove that \[ c<2024 (a_1+a_2+\ldots+a_{2024}).\]

1962 All-Soviet Union Olympiad, 2

Tags: geometry , locus

Given a fixed circle $C$ and a line L through the center $O$ of $C$. Take a variable point $P$ on $L$ and let $K$ be the circle with center $P$ through $O$. Let $T$ be the point where a common tangent to $C$ and $K$ meets $K$. What is the locus of $T$?

2009 China Team Selection Test, 2

Tags: ceiling function , pigeonhole principle , induction , graph theory , combinatorics

Let $ n,k$ be given positive integers satisfying $ k\le 2n \minus{} 1$. On a table tennis tournament $ 2n$ players take part, they play a total of $ k$ rounds match, each round is divided into $ n$ groups, each group two players match. The two players in different rounds can match on many occasions. Find the greatest positive integer $ m \equal{} f(n,k)$ such that no matter how the tournament processes, we always find $ m$ players each of pair of which didn't match each other.

2023 MIG, 7

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At a length of $104$ miles, the Danyang-Kushan Bridge holds the title for being the longest bridge in the world. A car travels at a constant speed of $39$ miles per hour across the Danyang-Kushan Bridge. How long does it take the car to travel across the entire bridge? $\textbf{(A) }\text{2 hours, 12 minutes} \qquad \textbf{(B) }\text{2 hours, 20 minutes} \qquad \textbf{(C) }\text{2 hours, 25 minutes}$\\ $\textbf{(D) }\text{2 hours, 30 minutes} \qquad \textbf{(E) }\text{2 hours, 40 minutes}$

2024 IFYM, Sozopol, 3

Tags: geometry

Let $ X $ be an arbitrary point on the side $ BC $ of triangle $ ABC $. The point $ M $ on the ray $ AB^\to $ beyond $ B $, the point $ N $ on the ray $ AC^\to $ beyond $ C $, and the point $ K $ inside $ ABC $ are such that $ \angle BMX = \angle CNX = \angle KBC = \angle KCB $. The line through $ A $, parallel to $ BC $, intersects the line $ KX $ at point $ P $. Prove that the points $ A $, $ P $, $ M $, $ N $ lie on a circle.

1988 Mexico National Olympiad, 7

Tags: combinatorics , sum , subset

Two disjoint subsets of the set $\{1,2, ... ,m\}$ have the same sums of elements. Prove that each of the subsets $A,B$ has less than $m / \sqrt2$ elements.

2007 Tournament Of Towns, 2

Tags: algebra

A student did not notice multiplication sign between two three-digit numbers and wrote it as a six-digit number. Result was 7 times more that it should be. Find these numbers. [i](2 points)[/i]

1970 Polish MO Finals, 1

Tags: geometry , geometric inequalities

Diameter $AB$ divides a circle into two semicircles. Points $P_1$ , $P_2$, $...$, $P_n$ are given on one of the semicircles in this order. How should a point C be chosen on the other semicircle in order to maximize the sum of the areas of triangles $CP_1P_2$, $CP_2P_3$, $...$,$CP_{n-1}P_n$?

2019 Macedonia Junior BMO TST, 3

Tags: combinatorics

Define a colouring in tha plane the following way: - we pick a positive integer $m$; - let $K_{1}$, $K_{2}$, ..., $K_{m}$ be different circles with nonzero radii such that $K_{i}\subset K_{j}$ or $K_{j}\subset K_{i}$ if $i \neq j$; - the points in the plane that lie outside an arbitrary circle (one that is amongst the circles we pick) are coloured differently than the points that lie inside the circle. There are $2019$ points in the plane such that any $3$ of them are not collinear. Determine the maximum number of colours which we can use to colour the given points.

2012 Kosovo National Mathematical Olympiad, 2

Tags: geometry , inequality

Let $a,b,c$ be the lengths of the sides of a triangle. Prove that, $\left|\frac {a}{b}+\frac {b}{c}+\frac {c}{a}-\frac {b}{a}-\frac {c}{b}-\frac {a}{c}\right|<1$

2023-24 IOQM India, 28

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On each side of an equilateral triangle with side length $n$ units, where $n$ is an integer, $1 \leq n \leq 100$, consider $n-1$ points that divide the side into $n$ equal segments. Through these points, draw lines parallel to the sides of the triangle, obtaining a net of equilateral triangles of side length one unit. On each of the vertices of these small triangles, place a coin head up. Two coins are said to be adjacent if the distance between them is 1 unit. A move consists of flipping over any three mutually adjacent coins. Find the number of values of $n$ for which it is possible to turn all coins tail up after a finite number of moves.

1936 Moscow Mathematical Olympiad, 030

Tags: combinatorics , number theory , product

How many ways are there to represent $10^6$ as the product of three factors? Factorizations which only differ in the order of the factors are considered to be distinct.

2024 AMC 12/AHSME, 3

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For how many integer values of $x$ is $|2x|\leq 7\pi?$ $\textbf{(A) }16 \qquad\textbf{(B) }17\qquad\textbf{(C) }19\qquad\textbf{(D) }20\qquad\textbf{(E) }21$

2021 HMNT, 6

Tags: combinatorics

Mario has a deck of seven pairs of matching number cards and two pairs of matching Jokers, for a total of $18$ cards. He shuffles the deck, then draws the cards from the top one by one until he holds a pair of matching Jokers. The expected number of complete pairs that Mario holds at the end (including the Jokers) is $\frac{m}{n}$ , where $m$, $n$ are positive integers and gcd $(m,n) = 1$. Find $100m + n$.

2016 Purple Comet Problems, 9

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Find the sum of all perfect squares that divide 2016.

1968 IMO, 1

Tags: trigonometry , algebra , triangle , triangle inequality

Find all triangles whose side lengths are consecutive integers, and one of whose angles is twice another.

2017 Harvard-MIT Mathematics Tournament, 2

Tags: perimeter , geometry

Let $ABC$ be a triangle with $AB=13$, $BC=14$, and $CA=15$. Let $\ell$ be a line passing through two sides of triangle $ABC$. Line $\ell$ cuts triangle $ABC$ into two figures, a triangle and a quadrilateral, that have equal perimeter. What is the maximum possible area of the triangle?

2018 PUMaC Combinatorics A, 2

Tags: combinatorics , counting , distinguishability

In an election between $\text{A}$ and $\text{B}$, during the counting of the votes, neither candidate was more than $2$ votes ahead, and the vote ended in a tie, $6$ votes to $6$ votes. Two votes for the same candidate are indistinguishable. In how many orders could the votes have been counted? One possibility is $\text{AABBABBABABA}$.

2024 Middle European Mathematical Olympiad, 5

Tags: circumcircle , geometry

Let $ABC$ be a triangle with $\angle BAC=60^\circ$. Let $D$ be a point on the line $AC$ such that $AB = AD$ and $A$ lies between $C$ and $D$. Suppose that there are two points $E \ne F$ on the circumcircle of the triangle $DBC$ such that $AE = AF = BC$. Prove that the line $EF$ passes through the circumcenter of $ABC$.

2023 USAMTS Problems, 5

Tags: lattice points , number theory

Let $m$ and $n$ be positive integers. Let $S$ be the set of all points $(x, y)$ with integer coordinates such that $1 \leq x,y \leq m+n-1$ and $m+1 \leq x +y \leq 2m+n-1.$ Let $L$ be the set of the $3m+3n-3$ lines parallel to one of $x = 0, y = 0,$ or $x + y = 0$ and passing through at least one point in $S$. For which pairs $(m, n)$ does there exist a subset $T$ of $S$ such that every line in $L$ intersects an odd number of elements of $T$?

2006 Moldova National Olympiad, 11.8

Tags: induction , combinatorics

Given an alfabet of $n$ letters. A sequence of letters such that between any 2 identical letters there are no 2 identical letters is called a [i]word[/i]. a) Find the maximal possible length of a [i]word[/i]. b) Find the number of the [i]words[/i] of maximal length.

2000 Manhattan Mathematical Olympiad, 1

Tags:

There are 6 people at a party. Prove that one can [b]either[/b] find a group of $3$ people in which each person is friend with the other two, [b]or[/b] one can find a group of $3$ people in which no two people are friends.

2017 Sharygin Geometry Olympiad, P24

Tags: geometry , tetrahedron , similarity , 3d geometry

Two tetrahedrons are given. Each two faces of the same tetrahedron are not similar, but each face of the first tetrahedron is similar to some face of the second one. Does this yield that these tetrahedrons are similar?

2015 Estonia Team Selection Test, 11

Tags: circles , geometry

Let $M$ be the midpoint of the side $AB$ of a triangle $ABC$. A circle through point $C$ that has a point of tangency to the line $AB$ at point $A$ and a circle through point $C$ that has a point of tangency to the line $AB$ at point $B$ intersect the second time at point $N$. Prove that $|CM|^2 + |CN|^2 - |MN|^2 = |CA|^2 + |CB|^2 - |AB|^2$.

2017 Greece Team Selection Test, 2

Tags: binomial coefficient , number theory , combinatorics

Prove that the number $A=\frac{(4n)!}{(2n)!n!}$ is an integer and divisible by $2^{n+1}$, where $n$ is a positive integer.