Olympiad problems

1953 Miklós Schweitzer, 10

Tags: probability

[b]10.[/b] Consider a point performing a random walk on a planar triangular lattice and suppose that it moves away with equal probability from any lattice point along any one of the six lattice lines issuing from it. Prove that if the walk is continued indefinitely, then the point will return to its starting point with probability 1. [b](P. 5)[/b]

1989 China National Olympiad, 6

Tags: function , inequalities , algebra

Find all functions $f:(1,+\infty) \rightarrow (1,+\infty)$ that satisfy the following condition: for arbitrary $x,y>1$ and $u,v>0$, inequality $f(x^uy^v)\le f(x)^{\dfrac{1}{4u}}f(y)^{\dfrac{1}{4v}}$ holds.

2021 Israel TST, 2

Tags: combinatorics

Given 10 light switches, each can be in two states: on and off. For each pair of switches there is a light bulb which is on if and only if when both switches are on (45 bulbs in total). The bulbs and the switches are unmarked so it is unclear which switches correspond to which bulb. In the beginning all switches are off. How many flips are needed to find out regarding all bulbs which switches are connected to it? On each step you can flip precisely one switch

2014 Belarus Team Selection Test, 1

Tags: geometry , circles , concyclic , collinear

Circles $\Gamma_1$ and $\Gamma_2$ meet at points $X$ and $Y$. A circle $S_1$ touches internally $\Gamma_1$ at $A$ and $\Gamma_2$ externally at $B$. A circle $S_2$ touches $\Gamma_2$ internally at $C$ and $\Gamma_1$ externally at $D$. Prove that the points $A, B, C, D$ are either collinear or concyclic. (A. Voidelevich)

PEN A Problems, 102

Tags: divisibility theory

Determine all three-digit numbers $N$ having the property that $N$ is divisible by $11,$ and $\frac{N}{11}$ is equal to the sum of the squares of the digits of $N.$

2003 AMC 10, 9

Tags:

Simplify \[ \sqrt[3]{x\sqrt[3]{x\sqrt[3]{x\sqrt{x}}}} \]$ \textbf{(A)}\ \sqrt{x} \qquad \textbf{(B)}\ \sqrt[3]{x^2} \qquad \textbf{(C)}\ \sqrt[27]{x^2} \qquad \textbf{(D)}\ \sqrt[54]{x} \qquad \textbf{(E)}\ \sqrt[81]{x^{80}}$

2013 Silk Road, 3

Tags: function , inequalities , algebra

Find all non-decreasing functions $ f\,:\,\mathbb{N}\to\mathbb{N} $, such that $f(f(m)f(n)+m)=f(mf(n))+f(m)$

2007 India IMO Training Camp, 2

Tags: function , inequalities

Let $a,b,c$ be non-negative real numbers such that $a+b\leq c+1, b+c\leq a+1$ and $c+a\leq b+1.$ Show that \[a^2+b^2+c^2\leq 2abc+1.\]

May Olympiad L2 - geometry, 2023.4

Tags: geometry , rectangle , paper , folding

Matías has a rectangular sheet of paper $ABCD$, with $AB<AD$.Initially, he folds the sheet along a straight line $AE$, where $E$ is a point on the side $DC$ , so that vertex $D$ is located on side $BC$, as shown in the figure. Then folds the sheet again along a straight line $AF$, where $F$ is a point on side $BC$, so that vertex $B$ lies on the line $AE$; and finally folds the sheet along the line $EF$. Matías observed that the vertices $B$ and $C$ were located on the same point of segment $AE$ after making the folds. Calculate the measure of the angle $\angle DAE$. [img]https://cdn.artofproblemsolving.com/attachments/0/9/b9ab717e1806c6503a9310ee923f20109da31a.png[/img]

2011 Kosovo National Mathematical Olympiad, 1

Tags: function , trigonometry , algebra

It is given the function $f:\mathbb{R} \to \mathbb{R}$ such that it holds $f(\sin x)=\sin (2011x)$. Find the value of $f(\cos x)$.

2020 South East Mathematical Olympiad, 4

Tags: inequalities , algebra , n-variable inequality

Let $0\leq a_1\leq a_2\leq \cdots\leq a_{n-1}\leq a_n $ and $a_1+a_2+\cdots+a_n=1.$ Prove that: For any non-negative numbers $x_1,x_2,\cdots,x_n ; y_1, y_2,\cdots, y_n$ , have $$\left(\sum_{i=1}^n a_ix_i - \prod_{i=1}^n x_i^{a_i}\right) \left(\sum_{i=1}^n a_iy_i - \prod_{i=1}^n y_i^{a_i}\right) \leq a_n^2\left(n\sqrt{\sum_{i=1}^n x_i\sum_{i=1}^n y_i} - \sum_{i=1}^n\sqrt{x_i} \sum_{i=1}^n\sqrt{y_i}\right)^2.$$

2005 Tournament of Towns, 4

Tags: square , angle bisector , geometry

$M$ and $N$ are the midpoints of sides $BC$ and $AD$, respectively, of a square $ABCD$. $K$ is an arbitrary point on the extension of the diagonal $AC$ beyond $A$. The segment $KM$ intersects the side $AB$ at some point $L$. Prove that $\angle KNA = \angle LNA$. [i](5 points)[/i]

2022 Azerbaijan IMO TST, 6

Tags: number theory

Determine all integers $n\geqslant 2$ with the following property: every $n$ pairwise distinct integers whose sum is not divisible by $n$ can be arranged in some order $a_1,a_2,\ldots, a_n$ so that $n$ divides $1\cdot a_1+2\cdot a_2+\cdots+n\cdot a_n.$ [i]Arsenii Nikolaiev, Anton Trygub, Oleksii Masalitin, and Fedir Yudin[/i]

2025 CMIMC Team, 9

Tags: team

Given a triangle, $AB=78, BC=50, AC=112,$ construct squares $ABXY, BCPQ, ACMN$ outside the triangle. Let $L_1, L_2, L_3$ be the midpoints of $\overline{MP}, \overline{QX}, \overline{NY},$ respectively. Find the area of $L_1L_2L_3.$

1980 Bulgaria National Olympiad, Problem 1

Tags: algebra , digit , number theory

Show that there exists a unique sequence of decimal digits $p_0=5,p_1,p_2,\ldots$ such that, for any $k$, the square of any positive integer ending with $\overline{p_kp_{k-1}\cdots p_0}$ ends with the same digits.

2001 Czech-Polish-Slovak Match, 1

Tags: inequalities , n-variable inequality , holder , convexity

Let $n\ge2$ be a natural number, and $a_i$ be positive numbers, where $i=1,2,\cdots,n.$ Show that \[\left(a_1^3+1\right)\left(a_2^3+1\right)\cdots\left(a_n^3+1\right) \geq \left(a_1^2a_2+1\right)\left(a_2^2a_3+1\right)\cdots\left(a_n^2a_1+1\right)\]

2002 AIME Problems, 11

Tags: ratio , geometric series

Two distinct, real, infinite geometric series each have a sum of $1$ and have the same second term. The third term of one of the series is $1/8,$ and the second term of both series can be written in the form $\frac{\sqrt{m}-n}{p},$ where $m,$ $n,$ and $p$ are positive integers and $m$ is not divisible by the square of any prime. Find $100m+10n+p.$

2023 Iran Team Selection Test, 1

Tags: al , algebra

Suppose that $n\ge3$ is a natural number. Find the maximum value $k$ such that there are real numbers $a_1,a_2,...,a_n \in [0,1)$ (not necessarily distinct) that for every natural number like $j \le k$ , sum of some $a_i$-s is $j$. [i]Proposed by Navid Safaei [/i]

1940 Moscow Mathematical Olympiad, 068

Tags: triangle construction , geometric construction , circumcenter , geometry

The center of the circle circumscribing $\vartriangle ABC$ is mirrored through each side of the triangle and three points are obtained: $O_1, O_2, O_3$. Reconstruct $\vartriangle ABC$ from $O_1, O_2, O_3$ if everything else is erased.

2016 India Regional Mathematical Olympiad, 2

Tags: counting , combinatorics

At an international event there are $100$ countries participating, each with its own flag. There are $10$ distinct flagpoles at the stadium, labelled 1,#2,...,#10 in a row. In how many ways can all the $100$ flags be hoisted on these $10$ flagpoles, such that for each $i$ from $1$ to $10$, the flagpole #i has at least $i$ flags? (Note that the vertical order of the flagpoles on each flag is important)

2020 Mexico National Olympiad, 1

Tags: combinatorics , number theory , greatest common divisor , vector

A set of five different positive integers is called [i]virtual[/i] if the greatest common divisor of any three of its elements is greater than $1$, but the greatest common divisor of any four of its elements is equal to $1$. Prove that, in any virtual set, the product of its elements has at least $2020$ distinct positive divisors. [i]Proposed by Víctor Almendra[/i]