Olympiad problems

Croatia MO (HMO) - geometry, 2020.7

Tags: geometry , perpendicular

A circle of diameter $AB$ is given. There are points $C$ and $ D$ on this circle, on different sides of the diameter such that holds $AC <BC$ or $AC<AD$. The point $P$ lies on the segment $BC$ and $\angle CAP = \angle ABC$. The perpendicular from the point $C$ to the line $AB$ intersects the direction $BD$ at the point $Q$. Lines $PQ$ and $AD$ intersect at point $R$, and the lines $PQ$ and $CD$ intersect at point $T$. If $AR=RQ$, prove that the lines $AT$ and $PQ$ are perpendicular.

PEN A Problems, 17

Tags: modular arithmetic , number theory , quadratic , relatively prime , divisibility theory

Let $m$ and $n$ be natural numbers such that \[A=\frac{(m+3)^{n}+1}{3m}\] is an integer. Prove that $A$ is odd.

1988 China Team Selection Test, 3

Tags: circumcircle , incenter , geometry

In triangle $ABC$, $\angle C = 30^{\circ}$, $O$ and $I$ are the circumcenter and incenter respectively, Points $D \in AC$ and $E \in BC$, such that $AD = BE = AB$. Prove that $OI = DE$ and $OI \bot DE$.

2022 IMC, 4

Tags: set , combinatorics , logarithm , graph coloring

Let $n > 3$ be an integer. Let $\Omega$ be the set of all triples of distinct elements of $\{1, 2, \ldots , n\}$. Let $m$ denote the minimal number of colours which suffice to colour $\Omega$ so that whenever $1\leq a<b<c<d \leq n$, the triples $\{a,b,c\}$ and $\{b,c,d\}$ have different colours. Prove that $\frac{1}{100}\log\log n \leq m \leq100\log \log n$.

2018 AMC 10, 22

Tags: greatest common divisor , number theory

Let $a, b, c,$ and $d$ be positive integers such that $\gcd(a, b)=24$, $\gcd(b, c)=36$, $\gcd(c, d)=54$, and $70<\gcd(d, a)<100$. Which of the following must be a divisor of $a$? $\textbf{(A)} \text{ 5} \qquad \textbf{(B)} \text{ 7} \qquad \textbf{(C)} \text{ 11} \qquad \textbf{(D)} \text{ 13} \qquad \textbf{(E)} \text{ 17}$

2023 Ecuador NMO (OMEC), 5

Tags: number theory

Find all positive integers $n$ such that $4^n + 4n + 1$ is a perfect square.

2020 May Olympiad, 1

Tags: combinatorics

Sofia places the dice on a table as shown in the figure, matching faces that have the same number on each die. She circles the table without touching the dice. What is the sum of the numbers of all the faces that she cannot see? $Note$. In all given the numbers on the opposite faces add up to 7.

1982 Polish MO Finals, 3

Tags: algebra , system of equations , parameter

Find all pairs of positive numbers $(x,y)$ which satisfy the system of equations $$\begin{cases} x^2 +y^2 = a^2 +b^2 \\ x^3 +y^3 = a^3 +b^3 \end{cases}$$ where $a$ and $b$ are given positive numbers.

1984 IMO Shortlist, 1

Tags: algebra , system of equations

Find all solutions of the following system of $n$ equations in $n$ variables: \[\begin{array}{c}\ x_1|x_1| - (x_1 - a)|x_1 - a| = x_2|x_2|,x_2|x_2| - (x_2 - a)|x_2 - a| = x_3|x_3|,\ \vdots \ x_n|x_n| - (x_n - a)|x_n - a| = x_1|x_1|\end{array}\] where $a$ is a given number.

1998 India Regional Mathematical Olympiad, 4

Tags: geometry , geometric transformation , reflection

Let $ABC$ be a triangle with $AB = AC$ and $\angle BAC = 30^{\circ}$, Let $A'$ be the reflection of $A$ in the line $BC$; $B'$ be the reflection of $B$ in the line $CA$; $C'$ be the reflection of $C$ in line $AB$, Show that $A'B'C'$ is an equilateral triangle.

2025 Polish MO Finals, 1

Tags: algebra , system of equations

Find all $(a, b, c, d)\in \mathbb{R}$ satisfying \[\begin{aligned} \begin{cases} a+b+c+d=0,\\ a^2+b^2+c^2+d^2=12,\\ abcd=-3.\\ \end{cases} \end{aligned}\]

2016 Ukraine Team Selection Test, 3

Tags: geometry

Let $ABC$ be a triangle with $CA \neq CB$. Let $D$, $F$, and $G$ be the midpoints of the sides $AB$, $AC$, and $BC$ respectively. A circle $\Gamma$ passing through $C$ and tangent to $AB$ at $D$ meets the segments $AF$ and $BG$ at $H$ and $I$, respectively. The points $H'$ and $I'$ are symmetric to $H$ and $I$ about $F$ and $G$, respectively. The line $H'I'$ meets $CD$ and $FG$ at $Q$ and $M$, respectively. The line $CM$ meets $\Gamma$ again at $P$. Prove that $CQ = QP$. [i]Proposed by El Salvador[/i]

2012 Kosovo National Mathematical Olympiad, 3

Tags: algebra

Solve the recurrence $R_0=1, R_n=nR_{n-1}+2^n\cdot n!$.

2018 Serbia JBMO TST, 2

Tags: inequalities , algebra

Show that for $a,b,c > 0$ the following inequality holds: $\frac{\sqrt{ab}}{a+b+2c}+\frac{\sqrt{bc}}{b+c+2a}+\frac{\sqrt{ca}}{c+a+2b} \le \frac {3}{4}$.

2002 All-Russian Olympiad, 2

Tags: induction , pigeonhole principle , combinatorics

We are given one red and $k>1$ blue cells, and a pack of $2n$ cards, enumerated by the numbers from $1$ to $2n$. Initially, the pack is situated on the red cell and arranged in an arbitrary order. In each move, we are allowed to take the top card from one of the cells and place it either onto the top of another cell on which the number on the top card is greater by $1$, or onto an empty cell. Given $k$, what is the maximal $n$ for which it is always possible to move all the cards onto a blue cell?

1995 Cono Sur Olympiad, 2

Tags:

There are ten points marked on a circumference, numbered from $1$ to $10$ and join all points with segments. I color the segments, with red someones and others with blue. Without changing the colors of the segments, renumber all the points from the $1$ to the $10$. Will be possible to color the segments and to renumber the points so that those numbers that were jointed with red are jointed now with blue and the numbers that were jointed with blue they are jointed now with red?

2011 Hanoi Open Mathematics Competitions, 5

Tags: number theory

Let M = 7!.8!.9!.10!.11!.12!. How many factors of M are perfect squares ?

2021 Caucasus Mathematical Olympiad, 8

Tags: functional equation , algebra , function

An infinite table whose rows and columns are numbered with positive integers, is given. For a sequence of functions $f_1(x), f_2(x), \ldots $ let us place the number $f_i(j)$ into the cell $(i,j)$ of the table (for all $i, j\in \mathbb{N}$). A sequence $f_1(x), f_2(x), \ldots $ is said to be {\it nice}, if all the numbers in the table are positive integers, and each positive integer appears exactly once. Determine if there exists a nice sequence of functions $f_1(x), f_2(x), \ldots $, such that each $f_i(x)$ is a polynomial of degree 101 with integer coefficients and its leading coefficient equals to 1.

1972 IMO Longlists, 35

Tags: function , inequalities

$(a)$ Prove that for $a, b, c, d \in\mathbb{R}, m \in [1,+\infty)$ with $am + b =-cm + d = m$, \[(i)\sqrt{a^2 + b^2}+\sqrt{c^2 + d^2}+\sqrt{(a-c)^2 + (b-d)^2}\ge \frac{4m^2}{1+m^2},\text{ and}\] \[(ii) 2 \le \frac{4m^2}{1+m^2} < 4.\] $(b)$ Express $a, b, c, d$ as functions of $m$ so that there is equality in $(i).$

2022 BMT, 8

Tags: geometry , combinatorics

Seven equally-spaced points are drawn on a circle of radius $1$. Three distinct points are chosen uniformly at random. What is the probability that the center of the circle lies in the triangle formed by the three points?

1997 Baltic Way, 17

Tags: number theory , ratio , geometry , rectangle

A rectangle can be divided into $n$ equal squares. The same rectangle can also be divided into $n+76$ equal squares. Find $n$.

2016 Estonia Team Selection Test, 12

Tags: geometry , parallel , circles

The circles $k_1$ and $k_2$ intersect at points $M$ and $N$. The line $\ell$ intersects with the circle $k_1$ at points $A$ and $C$ and with circle $k_2$ at points $B$ and $D$, so that points $A, B, C$ and $D$ are on the line $\ell$ in that order. Let $X$ be a point on line $MN$ such that the point $M$ is between points $X$ and $N$. Lines $AX$ and $BM$ intersect at point $P$ and lines $DX$ and $CM$ intersect at point $Q$. Prove that $PQ \parallel \ell $.

1987 IMO Longlists, 75

Tags: inequalities

Let $a_k$ be positive numbers such that $a_1 \geq 1$ and $a_{k+1} -a_k \geq 1 \ (k = 1, 2, . . . )$. Prove that for every $n \in \mathbb N,$ \[\sum_{k=1}^{1987}\frac{1}{a_{k+1} \sqrt[1987]{a_k}} <1987\]

2008 India Regional Mathematical Olympiad, 5

Tags: number theory

Three nonzero real numbers $ a,b,c$ are said to be in harmonic progression if $ \frac {1}{a} \plus{} \frac {1}{c} \equal{} \frac {2}{b}$. Find all three term harmonic progressions $ a,b,c$ of strictly increasing positive integers in which $ a \equal{} 20$ and $ b$ divides $ c$. [17 points out of 100 for the 6 problems]

Cono Sur Shortlist - geometry, 1993.7

Tags: parallelogram , geometry , centroid

Let $ABCD$ be a convex quadrilateral, where $M$ is the midpoint of $DC$, $N$ is the midpoint of $BC$, and $O$ is the intersection of diagonals $AC$ and $BD$. Prove that $O$ is the centroid of the triangle $AMN$ if and only if $ABCD$ is a parallelogram.