Olympiad problems

2013 Dutch BxMO/EGMO TST, 2

Consider a triple $(a, b, c)$ of pairwise distinct positive integers satisfying $a + b + c = 2013$. A step consists of replacing the triple $(x, y, z)$ by the triple $(y + z - x,z + x - y,x + y - z)$. Prove that, starting from the given triple $(a, b,c)$, after $10$ steps we obtain a triple containing at least one negative number.

2015 Romania National Olympiad, 3

Tags: geometry , median , area

Let be a point $ P $ in the interior of a triangle $ ABC. $ The lines $ AP,BP,CP $ meet $ BC,AC, $ respectively, $ AB $ at $ A_1,B_1, $ respectively, $ C_1. $ If $$ \mathcal{A}_{PBA_1} +\mathcal{A}_{PCB_1} +\mathcal{A}_{PAC_1} =\frac{1}{2}\mathcal{A}_{ABC} , $$ show that $ P $ lies on a median of $ ABC. $ $ \mathcal{A} $ [i]denotes area.[/i]

2000 Tournament Of Towns, 4

Tags: sequence , sum , algebra

(a) Does there exist an infinite sequence of real numbers such that the sum of every ten successive numbers is positive, while for every $n$ the sum of the first $10n + 1$ successive numbers is negative? (b) Does there exist an infinite sequence of integers with the same properties? (AK Tolpygo)

2025 Philippine MO, P1

Tags: combinatorics , subset , difference , set

The set $S$ is a subset of $\{1, 2, \dots, 2025\}$ such that no two elements of $S$ differ by $2$ or by $7$. What is the largest number of elements that $S$ can have?

2017-IMOC, G2

Tags: geometry , area of a triangle , geometric inequality , area

Given two acute triangles $\vartriangle ABC, \vartriangle DEF$. If $AB \ge DE, BC \ge EF$ and $CA \ge FD$, show that the area of $\vartriangle ABC$ is not less than the area of $\vartriangle DEF$

Estonia Open Junior - geometry, 2011.1.3

Tags: geometry , parallelogram , rhombus , incenter

Consider a parallelogram $ABCD$. a) Prove that if the incenter of the triangle $ABC$ is located on the diagonal $BD$, then the parallelogram $ABCD$ is a rhombus. b) Is the parallelogram $ABCD$ a rhombus whenever the circumcenter of the triangle $ABC$ is located on the diagonal $BD$?

2019 Iran Team Selection Test, 5

Tags: combinatorial geometry , combinatorics , perimeter

Let $P$ be a simple polygon completely in $C$, a circle with radius $1$, such that $P$ does not pass through the center of $C$. The perimeter of $P$ is $36$. Prove that there is a radius of $C$ that intersects $P$ at least $6$ times, or there is a circle which is concentric with $C$ and have at least $6$ common points with $P$. [i]Proposed by Seyed Reza Hosseini[/i]

2025 Czech-Polish-Slovak Junior Match., 1

Tags: number theory

Find all primes $p, q, r$ such that $$p^3+p^2+p+1=qr.$$

2024 Pan-American Girls’ Mathematical Olympiad, 3

Tags: number theory

Let $M$ be a non-empty set of positive integers and let $S_M$ be the sum of all the elements of $M$. We define the [i]tlacoyo[/i] of $M$ as the sum of the digits of $S_M$. For example, if $M=\{2,7,34\}$, then $S_M=2+7+34=43$ and the tlacoyo of the set $M$ is $4+3=7$. \\ Prove that for every positive integer $n$, there exists a set $M$ of $n$ distinct positive integers, such that all its non-empty subsets have the same tlacoyo.

2019 Israel National Olympiad, 6

Tags: number theory

A set of integers is called [b]legendary[/b] if you can reach any integer from it by using the following action multiple times: If the numbers $x,y$ are in the set, we may add the number $xy-y^2-y+x$ to the set. Prove that any legendary set contains at least 8 numbers.

2025 Kyiv City MO Round 2, Problem 1

Tags: algebra

Find the largest possible value of the expression $ y - x $, if the non-negative real numbers $ x, y $ satisfy the equation: \[ x^4 = y(y - 2025)^3. \] [i]Proposed by Mykhailo Shtandenko, Anton Trygub[/i]

2018 Stars of Mathematics, 3

Tags: geometry , trapezoid , rhombus

Let be an isosceles trapezoid such that its smaller base is equal to its legs, and a rhombus that has each of its vertexes on a different side of the trapezoid. Prove that the smaller angles of the trapezoid are equal to the smaller ones of the rhombus. [i]Vlad Robu[/i]

2021 AIME Problems, 1

Tags:

Zou and Chou are practicing their 100-meter sprints by running $6$ races against each other. Zou wins the first race, and after that, the probability that one of them wins a race is $\frac23$ if they won the previous race but only $\frac13$ if they lost the previous race. The probability that Zou will win exactly $5$ of the $6$ races is $\frac mn$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

2021 LMT Spring, A25 B26

Tags: number theory

Chandler the Octopus is making a concoction to create the perfect ink. He adds $1.2$ grams of melanin, $4.2$ grams of enzymes, and $6.6$ grams of polysaccharides. But Chandler accidentally added n grams of an extra ingredient to the concoction, Chemical $X$, to create glue. Given that Chemical $X$ contains none of the three aforementioned ingredients, and the percentages of melanin, enzymes, and polysaccharides in the final concoction are all integers, find the sum of all possible positive integer values of $n$. [i]Proposed by Taiki Aiba[/i]

2024 Dutch BxMO/EGMO TST, IMO TSTST, 4

Tags: rotation , board , combinatorics

Let $n$ be a positive with $n\geq 3$. Consider a board of $n \times n$ boxes. In each step taken the colors of the $5$ boxes that make up the figure bellow change color (black boxes change to white and white boxes change to black) The figure can be rotated $90°, 180°$ or $270°$. Firstly, all the boxes are white.Determine for what values of $n$ it can be achieved, through a series of steps, that all the squares on the board are black.

1975 AMC 12/AHSME, 16

Tags: ratio , geometric series

If the first term of an infinite geometric series is a positive integer, the common ratio is the reciprocal of a positive integer, and the sum of the series is 3, then the sum of the first two terms of the series is $ \textbf{(A)}\ 1/3 \qquad \textbf{(B)}\ 2/3 \qquad \textbf{(C)}\ 8/3 \qquad \textbf{(D)}\ 2 \qquad \textbf{(E)}\ 9/2$

2003 China Girls Math Olympiad, 2

Tags: combinatorics

There are 47 students in a classroom with seats arranged in 6 rows $ \times$ 8 columns, and the seat in the $ i$-th row and $ j$-th column is denoted by $ (i,j).$ Now, an adjustment is made for students’ seats in the new school term. For a student with the original seat $ (i,j),$ if his/her new seat is $ (m,n),$ we say that the student is moved by $ [a, b] \equal{} [i \minus{} m, j \minus{} n]$ and define the position value of the student as $ a\plus{}b.$ Let $ S$ denote the sum of the position values of all the students. Determine the difference between the greatest and smallest possible values of $ S.$

2010 Tournament Of Towns, 5

Tags: triangle inequality , inequalities

For each side of a given pentagon, divide its length by the total length of all other sides. Prove that the sum of all the fractions obtained is less than 2.

2018 Bulgaria EGMO TST, 3

Tags: circumcircle , geometry

Let be given a semicircle with diameter $AB$ and center $O$, and a line intersecting the semicircle at $C$ and $D$ and the line $AB$ at $M$ ($MB < MA$, $MD < MC$). The circumcircles of the triangles $AOC$ and $DOB$ meet again at $L$. Prove that $\angle MKO$ is right. [i]L. Kuptsov[/i]

2018 USAMTS Problems, 5:

Tags:

A positive integer is called [i]uphill[/i] if the digits in its decimal representation form a non-decreasing sequence from left to right. That is, a number with decimal representation $\overline{a_1a_2\cdots{}a_d}$ is uphill if $a_i\leq{}a_{i+1}$ for all $i$ (All single-digit integers are uphill.) Given a positive integer $n$, let $f(n)$ be the smallest nonnegative integer $m$ such that $n+m$ is uphill. For example, $f(520)=35$ and $f(169)=0$. Find, with proof, the value of$$f(1)-f(2)+f(3)-f(4)+\cdots{}+f(10^{2018}-1)$$

2022 BMT, Tie 2

Tags: number theory

Call a positive whole number [i]rickety [/i] if it is three times the product of its digits. There are two $2$-digit numbers that are rickety. What is their sum?

2015 Auckland Mathematical Olympiad, 4

Tags: combinatorics

In the planetary system of the star Zoolander there are $2015$ planets. On each planet an astronomer lives who observes the closest planet into his telescope (the distances between planets are all different). Prove that there is a planet who is observed by nobody.

2007 India IMO Training Camp, 3

Tags: function , ratio , algebra

Find all function(s) $f:\mathbb R\to\mathbb R$ satisfying the equation \[f(x+y)+f(x)f(y)=(1+y)f(x)+(1+x)f(y)+f(xy);\] For all $x,y\in\mathbb R.$

2023-IMOC, A6

Tags: algebra , inequalities

We define \[f(x,y,z)=|xy|\sqrt{x^2+y^2}+|yz|\sqrt{y^2+z^2}+|zx|\sqrt{z^2+x^2}.\] Find the best constants $c_1,c_2\in\mathbb{R}$ such that \[c_1(x^2+y^2+z^2)^{3/2}\leq f(x,y,z)\leq c_1(x^2+y^2+z^2)^{3/2}\] hold for all reals $x,y,z$ satisfying $x+y+z=0$. [i]Proposed by Untro368.[/i]

2012 Centers of Excellency of Suceava, 1

Tags: linear algebra , matrix

Let be a natural number $ n $ and a $ n\times n $ nilpotent real matrix $ A. $ Prove that $ 0=\det\left( A+\text{adj} A \right) . $ [i]Neculai Moraru[/i]