Olympiad problems

1910 Eotvos Mathematical Competition, 2

Let $a, b, c, d$ and $u$ be integers such that each of the numbers $$ac\ \ , \ \ bc + ad \ \ , \ \ bd$$ is a multiple of $u$. Show that $bc$ and $ad$ are multiples of $u$.

2019 Junior Balkan Team Selection Tests - Moldova, 5

Tags: number theory

Find all triplets of positive integers $(a, b, c)$ that verify $\left(\frac{1}{a}+1\right)\left(\frac{1}{b}+1\right)\left(\frac{1}{c}+1\right)=2$.

2004 All-Russian Olympiad, 2

Tags: incenter , exterior angle , complex number , geometry

Let $ABCD$ be a circumscribed quadrilateral (i. e. a quadrilateral which has an incircle). The exterior angle bisectors of the angles $DAB$ and $ABC$ intersect each other at $K$; the exterior angle bisectors of the angles $ABC$ and $BCD$ intersect each other at $L$; the exterior angle bisectors of the angles $BCD$ and $CDA$ intersect each other at $M$; the exterior angle bisectors of the angles $CDA$ and $DAB$ intersect each other at $N$. Let $K_{1}$, $L_{1}$, $M_{1}$ and $N_{1}$ be the orthocenters of the triangles $ABK$, $BCL$, $CDM$ and $DAN$, respectively. Show that the quadrilateral $K_{1}L_{1}M_{1}N_{1}$ is a parallelogram.

2020 Indonesia MO, 1

Tags: circles , bisector , geometry

Since this is already 3 PM (GMT +7) in Jakarta, might as well post the problem here. Problem 1. Given an acute triangle $ABC$ and the point $D$ on segment $BC$. Circle $c_1$ passes through $A, D$ and its centre lies on $AC$. Whereas circle $c_2$ passes through $A, D$ and its centre lies on $AB$. Let $P \neq A$ be the intersection of $c_1$ with $AB$ and $Q \neq A$ be the intersection of $c_2$ with $AC$. Prove that $AD$ bisects $\angle{PDQ}$.

1994 Tuymaada Olympiad, 1

Tags: combinatorics , point

World Cup in America introduced a new point system. For a victory $3$ points are given, for a draw $1$ point and for defeat $0$ points. In the preliminary games, the teams are divided into groups of $4$ teams. In groups, teams play with each other, once, then in accordance with the points scored $a,b,c$ and $d$ ($a>b>c>d$) teams take the first, second, third and fourth place in their groups. Give all possible options for the distribution points $a,b,c$ and $d$

2013 IFYM, Sozopol, 7

Tags: algebra , equation

Let $a,b,c,$ and $d$ be real numbers and $k\geq l\geq m$ and $p\geq q\geq r$. Prove that $f(x)=a(x+1)^k (x+2)^p+b(x+1)^l (x+2)^q+c(x+1)^m (x+2)^r-d=0$ has no more than 14 positive roots.

2015 Azerbaijan National Olympiad, 4

Tags: prime divisor , number theory , divisor

Natural number $M$ has $6$ divisors, such that sum of them are equal to $3500$.Find the all values of $M$.

2018 China Team Selection Test, 3

Tags: inequalities , algebra

Prove that there exists a constant $C>0$ such that $$H(a_1)+H(a_2)+\cdots+H(a_m)\leq C\sqrt{\sum_{i=1}^{m}i a_i}$$ holds for arbitrary positive integer $m$ and any $m$ positive integer $a_1,a_2,\cdots,a_m$, where $$H(n)=\sum_{k=1}^{n}\frac{1}{k}.$$

1988 Tournament Of Towns, (172) 5

Tags: combinatorial geometry , combinatorics , cover , circles

Is it possible to cover a plane with circles in such a way that exactly $1988$ circles pass through each point? ( N . Vasiliev)

2002 Federal Math Competition of S&M, Problem 3

Tags: number theory , diophantine equation

Find all pairs $(n,k)$ of positive integers such that $\binom nk=2002$.

2023 Israel Olympic Revenge, P1

Tags: combinatorics , graph theory

Armadillo and Badger are playing a game. Armadillo chooses a nonempty tree (a simple acyclic graph) and places apples at some of its vertices (there may be several apples on a single vertex). First, Badger picks a vertex $v_0$ and eats all its apples. Next, Armadillo and Badger take turns alternatingly, with Armadillo starting. On the $i$-th turn the animal whose turn it is picks a vertex $v_i$ adjacent to $v_{i-1}$ that hasn't been picked before and eats all its apples. The game ends when there is no such vertex $v_i$. Armadillo's goal is to have eaten more apples than Badger once the game ends. Can she fulfill her wish?

2005 Tournament of Towns, 2

Tags:

The base-ten expressions of all the positive integers are written on an infinite ribbon without spacing: $1234567891011\ldots$. Then the ribbon is cut up into strips seven digits long. Prove that any seven digit integer will: (a) appear on at least one of the strips; [i](3 points)[/i] (b) appear on an infinite number of strips. [i](1 point)[/i]

2012 Graduate School Of Mathematical Sciences, The Master Course, Kyoto University, 2

Tags: calculus , integration , limit , logarithm , real analysis

For real numbers $b>a>0$, let $f : [0,\ \infty)\rightarrow \mathbb{R}$ be a continuous function. Prove that : (i) $\lim_{\epsilon\rightarrow +0} \int_{a\epsilon}^{b\epsilon} \frac{f(x)}{x}dx=f(0)\ln \frac{b}{a}.$ (ii) If $\int_1^{\infty} \frac{f(x)}{x}dx$ converges, then $\int_0^{\infty} \frac{f(bx)-f(ax)}{x}dx=f(0)\ln \frac{a}{b}.$

2001 China National Olympiad, 3

Tags: pigeonhole principle , combinatorics

Let $P$ be a regular $n$-gon $A_1A_2\ldots A_n$. Find all positive integers $n$ such that for each permutation $\sigma (1),\sigma (2),\ldots ,\sigma (n)$ there exists $1\le i,j,k\le n$ such that the triangles $A_{i}A_{j}A_{k}$ and $A_{\sigma (i)}A_{\sigma (j)}A_{\sigma (k)}$ are both acute, both right or both obtuse.

2024 Bundeswettbewerb Mathematik, 2

Tags: digit , perfect square , number theory

Can a number of the form $44\dots 41$, with an odd number of decimal digits $4$ followed by a digit $1$, be a perfect square?

2009 Switzerland - Final Round, 7

Tags: geometry , common tangent , circles , concurrency

Points $A, M_1, M_2$ and $C$ are on a line in this order. Let $k_1$ the circle with center $M_1$ passing through $A$ and $k_2$ the circle with center $M_2$ passing through $C$. The two circles intersect at points $E$ and $F$. A common tangent of $k_1$ and $k_2$, touches $k_1$ at $B$ and $k_2$ at $D$. Show that the lines $AB, CD$ and $EF$ intersect at one point.

2013 Balkan MO Shortlist, A3

Tags: algebra , polynomial , factoring polynomials , integer polynomial

Prove that the polynomial $P (x) = (x^2- 8x + 25) (x^2 - 16x + 100) ... (x^2 - 8nx + 25n^2)- 1$, $n \in N^*$, cannot be written as the product of two polynomials with integer coefficients of degree greater or equal to $1$.

2025 Malaysian IMO Team Selection Test, 8

Tags: geometry

Let $ABC$ be an equilateral triangle, and $P$ is a point on its incircle. Let $\omega_a$ be the circle tangent to $AB$ passing through $P$ and $A$. Similarly, let $\omega_b$ be the circle tangent to $BC$ passing through $P$ and $B$, and $\omega_c$ be the circle tangent to $CA$ passing through $P$ and $C$. Prove that the circles $\omega_a$, $\omega_b$, $\omega_c$ has a common tangent line. [i]Proposed by Ivan Chan Kai Chin[/i]

2003 AMC 12-AHSME, 24

Tags: ceiling function , analytic geometry , graphing lines , slope , function , algebra , system of equations

Positive integers $ a$, $ b$, and $ c$ are chosen so that $ a<b<c$, and the system of equations \[ 2x\plus{}y\equal{}2003\text{ and }y\equal{}|x\minus{}a|\plus{}|x\minus{}b|\plus{}|x\minus{}c| \]has exactly one solution. What is the minimum value of $ c$? $ \textbf{(A)}\ 668 \qquad \textbf{(B)}\ 669 \qquad \textbf{(C)}\ 1002 \qquad \textbf{(D)}\ 2003 \qquad \textbf{(E)}\ 2004$

2007 Peru IMO TST, 4

Tags: inequalities

Let $a,b$ and $c$ be sides of a triangle. Prove that: $\frac{\sqrt{b+c-a}}{\sqrt{b}+\sqrt{c}-\sqrt{a}}+\frac{\sqrt{c+a-b}}{\sqrt{c}+\sqrt{a}-\sqrt{b}}+\frac{\sqrt{a+b-c}}{\sqrt{a}+\sqrt{b}-\sqrt{c}}\leq 3$

1970 Canada National Olympiad, 8

Tags: parameterization , ellipse , geometry , conic

Consider all line segments of length 4 with one end-point on the line $y=x$ and the other end-point on the line $y=2x$. Find the equation of the locus of the midpoints of these line segments.

PEN O Problems, 55

Tags:

The set $M$ consists of integers, the smallest of which is $1$ and the greatest $100$. Each member of $M$, except $1$, is the sum of two (possibly identical) numbers in $M$. Of all such sets, find one with the smallest possible number of elements.

2015 Postal Coaching, Problem 5

Tags: geometry , incenter

Let $ABCD$ be a convex quadrilateral. In the triangle $ABC$ let $I$ and $J$ be the incenter and the excenter opposite to vertex $A$, respectively. In the triangle $ACD$ let $K$ and $L$ be the incenter and the excenter opposite to vertex $A$, respectively. Show that the lines $IL$ and $JK$, and the bisector of the angle $BCD$ are concurrent.

1992 IMO Longlists, 35

Tags: algebra , polynomial , functional equation , iteration

Let $ f(x)$ be a polynomial with rational coefficients and $ \alpha$ be a real number such that \[ \alpha^3 \minus{} \alpha \equal{} [f(\alpha)]^3 \minus{} f(\alpha) \equal{} 33^{1992}.\] Prove that for each $ n \geq 1,$ \[ \left [ f^{n}(\alpha) \right]^3 \minus{} f^{n}(\alpha) \equal{} 33^{1992},\] where $ f^{n}(x) \equal{} f(f(\cdots f(x))),$ and $ n$ is a positive integer.

1967 IMO Shortlist, 4

Tags: geometry , geometric inequality , construction , median , triangle

Suppose medians $m_a$ and $m_b$ of a triangle are orthogonal. Prove that: a.) Using medians of that triangle it is possible to construct a rectangular triangle. b.) The following inequality: \[5(a^2+b^2-c^2) \geq 8ab,\] is valid, where $a,b$ and $c$ are side length of the given triangle.