Olympiad problems

2025 Caucasus Mathematical Olympiad, 1

Tags: number theory

For given positive integers $a$ and $b$, let us consider the equation$$a + \gcd(b, x) = b + \gcd(a, x).$$ [list=a] [*]For $a = 20$ and $b = 25$, find the least positive integer $x$ satisfying this equation. [*]Prove that for any positive integers $a$ and $b$, there exist infinitely many positive integers $x$ satisfying this equation. [/list] [i](Here, $\gcd(m, n)$ denotes the greatest common divisor of positive integers $m$ and $n$.)[/i]

2024 Serbia JBMO TST, 2

Tags: inequalities

Let $a, b, c$ be positive reals such that $ab+bc+ca=\frac{3}{4}$. Show that $$(a+b+c)^6 \geq (\frac{9} {8})^3(1+(a+b)^2)(1+(b+c)^2)(1+(c+a)^2).$$ When does equality hold?

2010 Indonesia TST, 3

Tags: combinatorics

In a party, each person knew exactly $ 22$ other persons. For each two persons $ X$ and $ Y$, if $ X$ and $ Y$ knew each other, there is no other person who knew both of them, and if $ X$ and $ Y$ did not know each other, there are exactly $ 6$ persons who knew both of them. Assume that $ X$ knew $ Y$ iff $ Y$ knew $ X$. How many people did attend the party? [i]Yudi Satria, Jakarta[/i]

2001 National Olympiad First Round, 16

Tags: algebra , polynomial

The polynomial $P(x)=x^3+ax+1$ has exactly one solution on the interval $[-2,0)$ and has exactly one solution on the interval $(0,1]$ where $a$ is a real number. Which of the followings cannot be equal to $P(2)$? $ \textbf{(A)}\ \sqrt{17} \qquad\textbf{(B)}\ \sqrt[3]{30} \qquad\textbf{(C)}\ \sqrt{26}-1 \qquad\textbf{(D)}\ \sqrt {30} \qquad\textbf{(E)}\ \sqrt [3]{10} $

LMT Guts Rounds, 1

Tags:

Compute $1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}.$

2014 India IMO Training Camp, 1

Tags: incenter , analytic geometry , circumcircle , trigonometry , geometry

In a triangle $ABC$, let $I$ be its incenter; $Q$ the point at which the incircle touches the line $AC$; $E$ the midpoint of $AC$ and $K$ the orthocenter of triangle $BIC$. Prove that the line $KQ$ is perpendicular to the line $IE$.

1996 AMC 8, 21

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How many subsets containing three different numbers can be selected from the set \[\{ 89,95,99,132, 166,173 \}\] so that the sum of the three numbers is even? $\text{(A)}\ 6 \qquad \text{(B)}\ 8 \qquad \text{(C)}\ 10 \qquad \text{(D)}\ 12 \qquad \text{(E)}\ 24$

2023 Indonesia MO, 7

Tags: geometry , tangent , circumcircle

Given a triangle $ABC$ with $\angle ACB = 90^{\circ}$. Let $\omega$ be the circumcircle of triangle $ABC$. The tangents of $\omega$ at $B$ and $C$ intersect at $P$. Let $M$ be the midpoint of $PB$. Line $CM$ intersects $\omega$ at $N$ and line $PN$ intersects $AB$ at $E$. Point $D$ is on $CM$ such that $ED \parallel BM$. Show that the circumcircle of $CDE$ is tangent to $\omega$.

2020 Iranian Geometry Olympiad, 3

Tags: geometry

Assume three circles mutually outside each other with the property that every line separating two of them have intersection with the interior of the third one. Prove that the sum of pairwise distances between their centers is at most $2\sqrt{2}$ times the sum of their radii. (A line separates two circles, whenever the circles do not have intersection with the line and are on different sides of it.) [color=#45818E]Note.[/color] Weaker results with $2\sqrt{2}$ replaced by some other $c$ may be awarded points depending on the value of $c>2\sqrt{2}$ [i]Proposed by Morteza Saghafian[/i]

2018 Online Math Open Problems, 3

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Katie has a list of real numbers such that the sum of the numbers on her list is equal to the sum of the squares of the numbers on her list. Compute the largest possible value of the arithmetic mean of her numbers. [i]Proposed by Michael Ren[/i]

MIPT student olimpiad spring 2024, 2

Tags: linear algebra , matrix

Let the matrix $S$ be orthogonal and the matrix $I-S$ be invertible, where I is the identity matrix of the same size as $S$. Find $x^T(I-S)^{-1}x$ Where $x$ is a real unit vector.

1986 AMC 12/AHSME, 7

Tags: floor function , ceiling function

The sum of the greatest integer less than or equal to $x$ and the least integer greater than or equal to $x$ is $5$. The solution set for $x$ is $ \textbf{(A)}\ \Big\{\frac{5}{2}\Big\}\qquad\textbf{(B)}\ \big\{x\ |\ 2 \le x \le 3\big\}\qquad\textbf{(C)}\ \big\{x\ |\ 2\le x < 3\big\}\qquad \\ \textbf{(D)}\ \Big\{x\ |\ 2 < x \le 3\Big\}\qquad\textbf{(E)}\ \Big\{x\ |\ 2 < x < 3\Big\} $

1987 IMO Shortlist, 11

Tags: partition , combinatorics , counting , extremal combinatorics

Find the number of partitions of the set $\{1, 2, \cdots, n\}$ into three subsets $A_1,A_2,A_3$, some of which may be empty, such that the following conditions are satisfied: $(i)$ After the elements of every subset have been put in ascending order, every two consecutive elements of any subset have different parity. $(ii)$ If $A_1,A_2,A_3$ are all nonempty, then in exactly one of them the minimal number is even . [i]Proposed by Poland.[/i]

Kvant 2023, M2763

Tags: number theory , sum of digits , modular arithmetic

Let $k\geqslant 2$ be a natural number. Prove that the natural numbers with an even sum of digits give all the possible residues when divided by $k{}$. [i]Proposed by P. Kozlov and I. Bogdanov[/i]

the 9th XMO, 4

Tags: combinatorics , graph theory

One hundred million cities lie on Planet MO. Initially, there are no air routes between any two cities. Now an airline company comes. It plans to establish $5050$ two-way routes, each route connects two different cities, and no two routes connect the same two cities. The "degree" of a city is defined to be the number of routes departing from that city. The "benefit" of a route is the product of the "degrees" of the two cities it connects. Find the maximum possible value of the sum of the benefits of these $5050$ routes.

2002 AMC 12/AHSME, 14

Tags:

Find $i+2i^2+3i^3+\ldots+2002i^{2002}$. $\textbf{(A) }-999+1002i\qquad\textbf{(B) }-1002+999i\sqrt2\qquad\textbf{(C) }-1001+1000i$ $\textbf{(D) }-1002+1001i\qquad\textbf{(E) }i$

2015 IMC, 1

Tags: matrices

For any integer $n\ge 2$ and two $n\times n$ matrices with real entries $A,\; B$ that satisfy the equation $$A^{-1}+B^{-1}=(A+B)^{-1}\;$$ prove that $\det (A)=\det(B)$. Does the same conclusion follow for matrices with complex entries? (Proposed by Zbigniew Skoczylas, Wroclaw University of Technology)

2022 Kosovo National Mathematical Olympiad, 3

Tags: parallelogram , geometry

Let $ABCD$ be a parallelogram and $l$ the line parallel to $AC$ which passes through $D$. Let $E$ and $F$ points on $l$ such that $DE=DF=DB$. Show that $EA,FC$ and $BD$ are concurrent.

2007 Moldova National Olympiad, 11.7

Tags: geometry , 3d geometry , tetrahedron , inequalities

Given a tetrahedron $VABC$ with edges $VA$, $VB$ and $VC$ perpendicular any two of them. The sum of the lengths of the tetrahedron's edges is $3p$. Find the maximal volume of $VABC$.

2010 National Olympiad First Round, 35

Tags:

Which one below is not less than $x^3+y^5$ for all reals $x,y$ such that $0<x<1$ and $0<y<1$? $ \textbf{(A)}\ x^2y \qquad\textbf{(B)}\ x^2y^2 \qquad\textbf{(C)}\ x^2y^3 \qquad\textbf{(D)}\ x^3y \qquad\textbf{(E)}\ xy^4 $

1970 Putnam, A6

Tags: probability , expected value , interval

Three numbers are chosen independently at random, one from each of the three intervals $[0, L_i ]$ ($i=1,2,3$). If the distribution of each random number is uniform with respect to the length of the interval it is chosen from, determine the expected value of the smallest number chosen.

2000 AMC 8, 16

Tags: geometry , perimeter

In order for Mateen to walk a kilometer ($1000$m) in his rectangular backyard, he must walk the length $25$ times or walk its perimeter $10$ times. What is the area of Mateen's backyard in square meters? $\text{(A)}\ 40 \qquad \text{(B)}\ 200 \qquad \text{(C)}\ 400 \qquad \text{(D)}\ 500 \qquad \text{(E)}\ 1000$

1999 Romania National Olympiad, 2

Tags: sequence

Let $k$ be a positive integer, let $z_1,z_2, \ldots, z_k \in \mathbb{C}$ be distinct and let $u_1,u_2,\ldots,u_k \in \mathbb{C}$ be such that the set $\big\{a_n=u_1z_1^n+u_2z_2^n+\ldots+u_kz_k^n : n \in \mathbb{Z}_{>0} \big\}$ is finite. Prove that there exists a positive integer $p$ such that $a_n=a_{n+p},$ for any positive integer $n.$

2009 China Western Mathematical Olympiad, 2

Tags: circumcircle , geometry

Given an acute triangle $ABC$, $D$ is a point on $BC$. A circle with diameter $BD$ intersects line $AB,AD$ at $X,P$ respectively (different from $B,D$).The circle with diameter $CD$ intersects $AC,AD$ at $Y,Q$ respectively (different from $C,D$). Draw two lines through $A$ perpendicular to $PX,QY$, the feet are $M,N$ respectively.Prove that $\triangle AMN$ is similar to $\triangle ABC$ if and only if $AD$ passes through the circumcenter of $\triangle ABC$.

2010 National Olympiad First Round, 24

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How many $7$-digit positive integers are there such that the number remains same when its digits are reversed and is multiple of $11$? $ \textbf{(A)}\ 900 \qquad\textbf{(B)}\ 854 \qquad\textbf{(C)}\ 818 \qquad\textbf{(D)}\ 726 \qquad\textbf{(E)}\ \text{None} $