Olympiad problems

2001 IMC, 2

Let $r,s,t$ positive integers which are relatively prime and $a,b \in G$, $G$ a commutative multiplicative group with unit element $e$, and $a^r=b^s=(ab)^t=e$. (a) Prove that $a=b=e$. (b) Does the same hold for a non-commutative group $G$?

2024 Lusophon Mathematical Olympiad, 4

Tags: geometry

In the figure, the triangles $ABC$ and $CDE$ are equilateral, with side lengths $1$ and $4$, respectively. Moreover, $B$, $C$ and $D$ are collinear and $F$ and $G$ are midpoints of $BC$ and $CD$, respectively. Let $P$ be the intersection point of $AF$ and $BE$. Determine the area of the shaded triangle $BPG$. [img]https://fv5-4.failiem.lv/thumb_show.php?i=qmpfykxcek&view&v=1&PHPSESSID=1f433228a75b4117c35f707722c547c423d3d671[/img]

2004 Flanders Math Olympiad, 2

Tags:

Two bags contain some numbers, and the total number of numbers is prime. When we tranfer the number 170 from 1 bag to bag 2, the average in both bags increases by one. If the total sum of all numbers is 2004, find the number of numbers.

2021 Brazil Undergrad MO, Problem 5

Tags: linear algebra , matrix , eigenvalue

Find all triplets $(\lambda_1,\lambda_2,\lambda_3) \in \mathbb{R}^3$ such that there exists a matrix $A_{3 \times 3}$ with all entries being non-negative reals whose eigenvalues are $\lambda_1,\lambda_2,\lambda_3$.

2013 Saudi Arabia BMO TST, 1

Tags: geometry , cyclic

$ABCD$ is a cyclic quadrilateral such that $AB = BC = CA$. Diagonals $AC$ and $BD$ intersect at $E$. Given that $BE = 19$ and $ED = 6$, find the possible values of $AD$.

1909 Eotvos Mathematical Competition, 1

Tags: perfect cube , number theory , consecutive

Consider any three consecutive natural numbers. Prove that the cube of the largest cannot be the sum of the cubes of the other two.

2006 China Western Mathematical Olympiad, 4

Tags: induction , combinatorics

Given a positive integer $ n\geq 2$, let $ B_{1}$, $ B_{2}$, ..., $ B_{n}$ denote $ n$ subsets of a set $ X$ such that each $ B_{i}$ contains exactly two elements. Find the minimum value of $ \left|X\right|$ such that for any such choice of subsets $ B_{1}$, $ B_{2}$, ..., $ B_{n}$, there exists a subset $ Y$ of $ X$ such that: (1) $ \left|Y\right| \equal{} n$; (2) $ \left|Y \cap B_{i}\right|\leq 1$ for every $ i\in\left\{1,2,...,n\right\}$.

2009 Today's Calculation Of Integral, 430

Tags: calculus , integration , trigonometry , limit , calculus computations

For a natural number $ n$, let $ a_n\equal{}\int_0^{\frac{\pi}{4}} (\tan x)^{2n}dx$. Answer the following questions. (1) Find $ a_1$. (2) Express $ a_{n\plus{}1}$ in terms of $ a_n$. (3) Find $ \lim_{n\to\infty} a_n$. (4) Find $ \lim_{n\to\infty} \sum_{k\equal{}1}^n \frac{(\minus{}1)^{k\plus{}1}}{2k\minus{}1}$.

1970 IMO Longlists, 25

Tags: calculus , derivative , function , inequalities , integration , real analysis

A real function $f$ is defined for $0\le x\le 1$, with its first derivative $f'$ defined for $0\le x\le 1$ and its second derivative $f''$ defined for $0<x<1$. Prove that if $f(0)=f'(0)=f'(1)=f(1)-1 =0$, then there exists a number $0<y<1$ such that $|f''(y)|\ge 4$.

2004 China Girls Math Olympiad, 2

Tags: inequalities

Let $ a, b, c$ be positive reals. Find the smallest value of \[ \frac {a \plus{} 3c}{a \plus{} 2b \plus{} c} \plus{} \frac {4b}{a \plus{} b \plus{} 2c} \minus{} \frac {8c}{a \plus{} b \plus{} 3c}. \]

2016 China Girls Math Olympiad, 3

Tags: number theory

Let $m$ and $n$ are relatively prime integers and $m>1,n>1$. Show that:There are positive integers $a,b,c$ such that $m^a=1+n^bc$ , and $n$ and $c$ are relatively prime.

LMT Theme Rounds, 15

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A round robin tournament is held with $2016$ participants. Each player plays each other player once and exactly one game results in a tie. Let $W$ be the sum of the squares of each team's win total and let $L$ be the sum of the squares of each team's loss total. Find the maximum possible value of $W-L$. [i]Proposed by Matthew Weiss

2014 Sharygin Geometry Olympiad, 4

Tags: incenter , geometry

A square is inscribed into a triangle (one side of the triangle contains two vertices and each of two remaining sides contains one vertex. Prove that the incenter of the triangle lies inside the square.

2010 Flanders Math Olympiad, 2

Tags: parallelogram , geometry , ratio , trisection

A parallelogram with an angle of $60^o$ has $a$ as the longest side and a shortest side $b$. Let's take the perpendiculars down from the vertices of the obtuse angles to the longest diagonal, then it is divided into three equal parts. Determine the ratio $\frac{a}{b}$.

2024/2025 TOURNAMENT OF TOWNS, P5

Tags: geometry , combinatorics

A triangle is constructed on each side of a convex polygon in a manner that the third vertex of each triangle is the meet point of bisectors of the angles adjacent to this side. Prove that these triangles cover all the polygon. Egor Bakaev

1977 Chisinau City MO, 134

Tags: algebra

Where is the number $35 351$ in the sequence $1, 8, 22, 43,...$?

2021 All-Russian Olympiad, 4

Tags: combinatorics

Given a natural number $n>4$ and $2n+4$ cards numbered with $1, 2, \dots, 2n+4$. On the card with number $m$ a real number $a_m$ is written such that $\lfloor a_{m}\rfloor=m$. Prove that it's possible to choose $4$ cards in such a way that the sum of the numbers on the first two cards differs from the sum of the numbers on the two remaining cards by less than $$\frac{1}{n-\sqrt{\frac{n}{2}}}$$.

2018 CMIMC CS, 3

Tags: computer science

You are given the existence of an unsorted sequence $a_1,\ldots, a_5$ of five distinct real numbers. The Erdos-Szekeres theorem states that there exists a subsequence of length $3$ which is either strictly increasing or strictly decreasing. You do not have access to the $a_i$, but you do have an oracle which, when given two indexes $1\leq i < j\leq 5$, will tell you whether $a_i < a_j$ or $a_i > a_j$. What is the minimum number of calls to the oracle needed in order to identify an ordered triple of integers $(r,s,t)$ such that $a_r,a_s,a_t$ is one such sequence?

1998 Niels Henrik Abels Math Contest (Norwegian Math Olympiad) Round 2, 1

Tags:

Let $ a \geq b$ be real number such that $ a^2\plus{}b^2 \equal{} 31$ and $ ab \equal{} 3.$ Then $ a\minus{}b$ equals $ \text{(A)}\ 5 \qquad \text{(B)}\ \frac{31}{6} \qquad \text{(C)}\ 2 \sqrt{6} \qquad \text{(D)}\ \frac{5}{6} \sqrt{31} \qquad \text{(E)}\ \frac{5}{6} \sqrt{37}$

2023 Assara - South Russian Girl's MO, 6

Tags: combinatorics

Aunt Raya has $14$ wheels of cheese. She found out that out of any $6$ wheels, she could choose $4$ and put them on the scales so that the scales came into balance. Aunt Raya wants to give Daud Kazbekovich two of these $14$ wheels , and divide the rest equally (by weight) between Pavel and Kirill. Prove that she can make her wish come true.

the 16th XMO, 3

Tags: arithmetic sequence , number theory , greatest common divisor

$m$ is an integer satisfying $m \ge 2024$ , $p$ is the smallest prime factor of $m$ , for an arithmetic sequence $\{a_n\}$ of positive numbers with the common difference $m$ satisfying : for any integer $1 \le i \le \frac{p}{2} $ , there doesn’t exist an integer $x , y \le \max \{a_1 , m\}$ such that $a_i=xy$ Try to proof that there exists a positive real number $c$ such that for any $ 1\le i \le j \le n $ , $gcd(a_i , a_j ) = c \times gcd(i , j)$

1998 Canada National Olympiad, 2

Tags: algebra

Find all real numbers $x$ such that: \[ x = \sqrt{ x - \frac{1}{x} } + \sqrt{ 1 - \frac{1}{x} } \]

1982 IMO Longlists, 49

Tags: number theory

Simplify \[\sum_{k=0}^n \frac{(2n)!}{(k!)^2((n-k)!)^2}.\]

1977 USAMO, 4

Tags:

Prove that if the opposite sides of a skew (non-planar) quadrilateral are congruent, then the line joining the midpoints of the two diagonals is perpendicular to these diagonals, and conversely, if the line joining the midpoints of the two diagonals of a skew quadrilateral is perpendicular to these diagonals, then the opposite sides of the quadrilateral are congruent.

2022 Bundeswettbewerb Mathematik, 3

Tags: perpendicular bisector , geometry

In an acute triangle $ABC$ with $AC<BC$, lines $m_a$ and $m_b$ are the perpendicular bisectors of sides $BC$ and $AC$, respectively. Further, let $M_c$ be the midpoint of side $AB$. The Median $CM_c$ intersects $m_a$ in point $S_a$ and $m_b$ in point $S_b$; the lines $AS_b$ und $BS_a$ intersect in point $K$. Prove: $\angle ACM_c = \angle KCB$.