Olympiad problems

2001 Moldova National Olympiad, Problem 5

Tags: number theory

Show that there are nine distinct nonzero integers such that their sum is a perfect square and the sum of any eight of them is a perfect cube.

Ukrainian TYM Qualifying - geometry, 2015.20

Tags: ratio , inradii , geometric inequality , geometry

What is the smallest value of the ratio of the lengths of the largest side of the triangle to the radius of its inscribed circle?

2005 Thailand Mathematical Olympiad, 6

Tags: algebra , inequalities

Let $a, b, c$ be distinct real numbers. Prove that $$\left(\frac{2a - b}{a -b} \right)^2+\left(\frac{2b - c}{b - c} \right)^2+\left(\frac{2c - a}{c - a} \right)^2 \ge 5$$

2009 Estonia Team Selection Test, 1

Tags: inequalities , algebra

For arbitrary pairwise distinct positive real numbers $a, b, c$, prove the inequality $$\frac{(a^2- b^2)^3 + (b^2-c^2)^3+(c^2-a^2)^3}{(a- b)^3 + (b-c)^3+(c-a)^3}> 8abc$$

2014 Baltic Way, 9

Tags: combinatorics

What is the least posssible number of cells that can be marked on an $n \times n$ board such that for each $m >\frac{ n}{2}$ both diagonals of any $m \times m$ sub-board contain a marked cell?

2019 Purple Comet Problems, 3

Tags: algebra

The mean of $\frac12 , \frac34$ , and $\frac56$ differs from the mean of $\frac78$ and $\frac{9}{10}$ by $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

2008 Vietnam Team Selection Test, 1

Tags: geometric transformation , reflection , ratio , angle bisector , perpendicular bisector , geometry

On the plane, given an angle $ xOy$. $ M$ be a mobile point on ray $ Ox$ and $ N$ a mobile point on ray $ Oy$. Let $ d$ be the external angle bisector of angle $ xOy$ and $ I$ be the intersection of $ d$ with the perpendicular bisector of $ MN$. Let $ P$, $ Q$ be two points lie on $ d$ such that $ IP \equal{} IQ \equal{} IM \equal{} IN$, and let $ K$ the intersection of $ MQ$ and $ NP$. $ 1.$ Prove that $ K$ always lie on a fixed line. $ 2.$ Let $ d_1$ line perpendicular to $ IM$ at $ M$ and $ d_2$ line perpendicular to $ IN$ at $ N$. Assume that there exist the intersections $ E$, $ F$ of $ d_1$, $ d_2$ from $ d$. Prove that $ EN$, $ FM$ and $ OK$ are concurrent.

2015 Iran MO (3rd round), 5

Tags: number theory , prime number

$p>30$ is a prime number. Prove that one of the following numbers is in form of $x^2+y^2$. $$ p+1 , 2p+1 , 3p+1 , .... , (p-3)p+1$$

2025 Bulgarian Winter Tournament, 10.4

Tags: function , number theory , functional equation

The function $f: \mathbb{Z}_{>0} \times \mathbb{Z}_{>0} \to \mathbb{Z}_{>0}$ is such that $f(a,b) + f(b,c) = f(ac, b^2) + 1$ for any positive integers $a,b,c$. Assume there exists a positive integer $n$ such that $f(n, m) \leq f(n, m + 1)$ for all positive integers $m$. Determine all possible values of $f(2025, 2025)$.

2019 ELMO Shortlist, G3

Tags: geometry

Let $\triangle ABC$ be an acute triangle with incenter $I$ and circumcenter $O$. The incircle touches sides $BC,CA,$ and $AB$ at $D,E,$ and $F$ respectively, and $A'$ is the reflection of $A$ over $O$. The circumcircles of $ABC$ and $A'EF$ meet at $G$, and the circumcircles of $AMG$ and $A'EF$ meet at a point $H\neq G$, where $M$ is the midpoint of $EF$. Prove that if $GH$ and $EF$ meet at $T$, then $DT\perp EF$. [i]Proposed by Ankit Bisain[/i]

2007 Bulgarian Autumn Math Competition, Problem 8.1

Tags: algebra , system of equations , parameter , parameterization

Determine all real $a$, such that the solutions to the system of equations $\begin{cases} \frac{3x-5}{3}+\frac{3x+5}{4}\geq \frac{x}{7}-\frac{1}{15}\\ (2x-a)^3+(2x+a)(1-4x^2)+16x^2a-6x^2a+a^3\leq 2a^2+a \end{cases}$ form an interval with length $\frac{32}{225}$.

2016 Switzerland - Final Round, 9

Tags: combinatorics , lcm , number theory

Let $n \ge 2$ be a natural number. For an $n$-element subset $F$ of $\{1, . . . , 2n\}$ we define $m(F)$ as the minimum of all $lcm \,\, (x, y)$ , where $x$ and $y$ are two distinct elements of $F$. Find the maximum value of $m(F)$.

IV Soros Olympiad 1997 - 98 (Russia), 9.8

Tags: combinatorics , combinatorial geometry

There is a king in the lower left corner of a chessboard of dimensions $6$ and $6$. In one move, he can move either one cell to the right, or one cell up, or one cell diagonally - to the right and up. How many different paths can the king take to the upper right corner of the board?

2023 Assara - South Russian Girl's MO, 4

Tags: combinatorics , coloring

In a $50 \times 50$ checkered square, each cell is painted in one of $100$ given colors so that all colors are present and it is impossible to cut a single-color domino from the square (i.e. a $1 \times 2$ rectangle). Galiia wants to recolor all the cells of one of the colors into another color (out of the given $100$ colors) so that this condition is preserved (i.e., it is still impossible to cut out a domino of the same color). Is it true that Galiia will definitely be able to do this?

2025 Kosovo National Mathematical Olympiad`, P3

Tags: number theory , decimal representation , power

Find all pairs of natural numbers $(m,n)$ such that the number $5^m+6^n$ has all same digits when written in decimal representation.

2022 AMC 8 -, 25

Tags:

A cricket randomly hops between $4$ leaves, on each turn hopping to one of the other $3$ leaves with equal probability. After $4$ hops what is the probability that the cricket has returned to the leaf where it started? $\textbf{(A)}~\displaystyle\frac{2}{9}\qquad\textbf{(B)}~\displaystyle\frac{19}{80}\qquad\textbf{(C)}~\displaystyle\frac{20}{81}\qquad\textbf{(D)}~\displaystyle\frac{1}{4}\qquad\textbf{(E)}~\displaystyle\frac{7}{27}$

2010 F = Ma, 21

Tags:

The gravitational self potential energy of a solid ball of mass density $\rho$ and radius $R$ is $E$. What is the gravitational self potential energy of a ball of mass density $\rho$ and radius $2R$? (A) $2E$ (B) $4E$ (C) $8E$ (D) $16E$ (E) $32E$

1989 Romania Team Selection Test, 2

Tags: system of equations , algebra , number theory , coprime

Let $a,b,c$ be coprime nonzero integers. Prove that for any coprime integers $u,v,w$ with $au+bv+cw = 0$ there exist integers $m,n, p$ such that $$\begin{cases} a = nw- pv \\ b = pu-mw \\ c = mv-nu \end{cases}$$

1966 AMC 12/AHSME, 36

Tags:

Let $(1+x+x^2)^n=a_0+a_1x+a_2x^2+...+a_{2n}x^{2n}$ be an identity in $x$. If we lt $s=a_0+a_2+a_4+...+a_{2n}$, then $s$ equals: $\text{(A)}\ 2^n\qquad \text{(B)}\ 2^n+1\qquad \text{(C)}\ \dfrac{3^n-1}{2}\qquad \text{(D)}\ \dfrac{3^n}{2}\qquad \text{(E)}\ \dfrac{3^n+1}{2}$

2019 LIMIT Category A, Problem 2

Tags: geometry

Let $ABCD$ be a quadrilateral with sides $\left|\overline{AB}\right|=2$, $\left|\overline{BC}\right|=\left|\overline{CD}\right|=4$ and $\left|\overline{DA}\right|=5$. The opposite angles, $\angle A$ and $\angle C$ are equal. The length of diagonal $BD$ equals $\textbf{(A)}~2\sqrt6$ $\textbf{(B)}~3\sqrt3$ $\textbf{(C)}~3\sqrt6$ $\textbf{(D)}~2\sqrt3$

1985 ITAMO, 2

Tags: rotation , geometry , 3d geometry

When a right triangle is rotated about one leg, the volume of the cone produced is $800 \pi$ $\text{cm}^3$. When the triangle is rotated about the other leg, the volume of the cone produced is $1920 \pi$ $\text{cm}^3$. What is the length (in cm) of the hypotenuse of the triangle?

2002 Brazil National Olympiad, 3

Tags: combinatorics

The squares of an $m\times n$ board are labeled from $1$ to $mn$ so that the squares labeled $i$ and $i+1$ always have a side in common. Show that for some $k$ the squares $k$ and $k+3$ have a side in common.

2012 Iran MO (3rd Round), 3

Tags: induction , pigeonhole principle , geometry , perpendicular bisector , algebra , binomial theorem , combinatorics

Prove that if $n$ is large enough, among any $n$ points of plane we can find $1000$ points such that these $1000$ points have pairwise distinct distances. Can you prove the assertion for $n^{\alpha}$ where $\alpha$ is a positive real number instead of $1000$?

2018 AMC 12/AHSME, 8

Tags: geometry

Line segment $\overline{AB}$ is a diameter of a circle with $AB=24$. Point $C$, not equal to $A$ or $B$, lies on the circle. As point $C$ moves around the circle, the centroid (center of mass) of $\triangle{ABC}$ traces out a closed curve missing two points. To the nearest positive integer, what is the area of the region bounded by this curve? $\textbf{(A)} \text{ 25} \qquad \textbf{(B)} \text{ 38} \qquad \textbf{(C)} \text{ 50} \qquad \textbf{(D)} \text{ 63} \qquad \textbf{(E)} \text{ 75}$

2023 Saint Petersburg Mathematical Olympiad, 5

Tags: function , algebra

Let $a>1$ be a positive integer and let $f(n)=n+[a\{n\sqrt{2}\}]$. Show that there exists a positive integer $n$, such that $f(f(n))=f(n)$, but $f(n) \neq n$.