Olympiad problems

2004 All-Russian Olympiad Regional Round, 8.4

The cells of the $11 \times 111 \times11$ cube contain the numbers $ 1, 2, , . .. . . 1331$, once each number. Two worms are sent from one corner cube to the opposite corner. Each of them can crawl into a cube adjacent to the edge, while the first can crawl if the number in the adjacent cube differs by $8$, the second - if they differ by $ 9$. Is there such an arrangement of numbers that both worms can get to the opposite corner cube?

1981 Bulgaria National Olympiad, Problem 2

Tags: geometry , Triangles , angle

Let $ABC$ be a triangle such that the altitude $CH$ and the sides $CA,CB$ are respectively equal to a side and two distinct diagonals of a regular heptagon. Prove that $\angle ACB<120^\circ$.

1978 Chisinau City MO, 166

Tags: analytic geometry , circle , irrational , rational

It is known that at least one coordinate of the center $(x_0, y_0)$ of the circle $(x -x_0)^2+ (y -y_0)^2 = R^2$ is irrational. Prove that on the circle itself there are at most two points with rational coordinates.

2004 Korea Junior Math Olympiad, 1

Tags: geometry , KMO , inequalities

For positive reals $a_1, a_2, ..., a_5$ such that $a^2_1+a^2_2+...+a^2_5=2$, consider five squares with sides $a_1, a_2, ..., a_5$ respectively. Show that these squares can be placed inside (including boundaries) a square with side length of $2$ so that the square themselves do not overlap each other.

2020-21 KVS IOQM India, 16

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If $x$ and $y$ are positive integers such that $(x-4)(x-10)=2^y$, then Find maximum value of $x+y$

2023 Malaysia IMONST 2, 4

Tags: geometry

Given a right angled triangle $ABC$ with $\angle BAC = 90^{\circ}$. The points $D,E,F$ lie on sides $BC,CA,AB$ respectively so that $AD$ is perpendicular to $BC$ and $EF$ is parallel to $BC$. A point $G$ lies on side $AC$ such that $AG=CE$. Prove that $\angle GDF = 90^{\circ}$.

2018 All-Russian Olympiad, 2

Tags: geometry , inscribed circles , homothety

Circle $\omega$ is tangent to sides $AB, AC$ of triangle $ABC$. A circle $\Omega$ touches the side $AC$ and line $AB$ (produced beyond $B$), and touches $\omega$ at a point $L$ on side $BC$. Line $AL$ meets $\omega, \Omega$ again at $K, M$. It turned out that $KB \parallel CM$. Prove that $\triangle LCM$ is isosceles.

2016 Bosnia And Herzegovina - Regional Olympiad, 1

Tags: limit , algebra , Sequence

Let $a_1=1$ and $a_{n+1}=a_{n}+\frac{1}{2a_n}$ for $n \geq 1$. Prove that $a)$ $n \leq a_n^2 < n + \sqrt[3]{n}$ $b)$ $\lim_{n\to\infty} (a_n-\sqrt{n})=0$

2016 CMIMC, 4

Tags: 2016 , CMIMC , geometry

Andrew the Antelope canters along the surface of a regular icosahedron, which has twenty equilateral triangle faces and edge length 4. If he wants to move from one vertex to the opposite vertex, the minimum distance he must travel can be expressed as $\sqrt{n}$ for some integer $n$. Compute $n$.

2009 May Olympiad, 1

Tags: combinatorics

Initially, the number $1$ is written on the blackboard. At each step, the number on the blackboard is erased and another is written, which is obtained by applying any of the following operations: Operation A: Multiply the number on the board with $\frac12$. Operation B: Subtract the number on the board from $1$. For example, if the number $\frac38$ is on the board, it can be replaced by $\frac12 \frac38=\frac{3}{16}$ or by $1-\frac38=\frac58$ . Give a sequence of steps after which the number on the board is $\frac{2009}{2^{20009}}$ .

1955 AMC 12/AHSME, 5

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$ y$ varies inversely as the square of $ x$. When $ y\equal{}16$, $ x\equal{}1$. When $ x\equal{}8$, $ y$ equals: $ \textbf{(A)}\ 2 \qquad \textbf{(B)}\ 128 \qquad \textbf{(C)}\ 64 \qquad \textbf{(D)}\ \frac{1}{4} \qquad \textbf{(E)}\ 1024$

1984 IMO Longlists, 55

Tags: modular arithmetic , number theory , number theory unsolved

Let $a, b, c$ be natural numbers such that $a+b+c = 2pq(p^{30}+q^{30}), p > q$ being two given positive integers. $(a)$ Prove that $k = a^3 + b^3 + c^3$ is not a prime number. $(b)$ Prove that if $a\cdot b\cdot c$ is maximum, then $1984$ divides $k$.

1980 Bundeswettbewerb Mathematik, 2

Tags: geometry , bisectors , distance , Triangle

In a triangle $ABC$, the bisectors of angles $A$ and $B$ meet the opposite sides of the triangle at points $D$ and $E$, respectively. A point $P$ is arbitrarily chosen on the line $DE$. Prove that the distance of $P$ from line $AB$ equals the sum or the difference of the distances of $P$ from lines $AC$ and $BC$.

Estonia Open Junior - geometry, 2012.1.5

Tags: geometry , circle

A hiking club wants to hike around a lake along an exactly circular route. On the shoreline they determine two points, which are the most distant from each other, and start to walk along the circle, which has these two points as the endpoints of its diameter. Can they be sure that, independent of the shape of the lake, they do not have to swim across the lake on any part of their route?

2001 Croatia Team Selection Test, 3

Tags: Diophantine equation , diophantine , number theory

Find all solutions of the equation $(a^a)^5 = b^b$ in positive integers.

2024 Macedonian Balkan MO TST, Problem 3

Tags: number theory , prime numbers

Let $p \neq 5$ be a prime number. Prove that $p^5-1$ has a prime divisor of the form $5x+1$.

2009 China Team Selection Test, 3

Tags: algebra , polynomial , induction , modular arithmetic , binomial coefficients , algebra proposed

Let $ f(x)$ be a $ n \minus{}$degree polynomial all of whose coefficients are equal to $ \pm 1$, and having $ x \equal{} 1$ as its $ m$ multiple root. If $ m\ge 2^k (k\ge 2,k\in N)$, then $ n\ge 2^{k \plus{} 1} \minus{} 1.$

2013 May Olympiad, 4

Tags: algebra

Pablo wrote $5$ numbers on one sheet and then wrote the numbers $6,7,8,8,9,9,10,10,11$ and $ 12$ on another sheet that he gave Sofia, indicating that those numbers are the possible sums of two of the numbers that he had hidden. Decide if with this information Sofia can determine the five numbers Pablo wrote .

1990 IMO Shortlist, 2

Tags: combinatorics , Set systems , graph theory , Extremal combinatorics , IMO Shortlist

Given $ n$ countries with three representatives each, $ m$ committees $ A(1),A(2), \ldots, A(m)$ are called a cycle if [i](i)[/i] each committee has $ n$ members, one from each country; [i](ii)[/i] no two committees have the same membership; [i](iii)[/i] for $ i \equal{} 1, 2, \ldots,m$, committee $ A(i)$ and committee $ A(i \plus{} 1)$ have no member in common, where $ A(m \plus{} 1)$ denotes $ A(1);$ [i](iv)[/i] if $ 1 < |i \minus{} j| < m \minus{} 1,$ then committees $ A(i)$ and $ A(j)$ have at least one member in common. Is it possible to have a cycle of 1990 committees with 11 countries?

2003 IMO Shortlist, 6

Tags: number theory , polynomial , Divisibility , IMO , IMO 2003 , IMO Shortlist , Hi

Let $p$ be a prime number. Prove that there exists a prime number $q$ such that for every integer $n$, the number $n^p-p$ is not divisible by $q$.

2001 Manhattan Mathematical Olympiad, 4

Tags:

You have a four-liter jug and a six-liter pot (both of cylindrical shape), and a big barrel of water. Can you measure exactly one liter of water?

1993 China Team Selection Test, 2

Tags: quadratics , function , algebra unsolved , algebra , inequalities

Let $n \geq 2, n \in \mathbb{N}$, $a,b,c,d \in \mathbb{N}$, $\frac{a}{b} + \frac{c}{d} < 1$ and $a + c \leq n,$ find the maximum value of $\frac{a}{b} + \frac{c}{d}$ for fixed $n.$

2020/2021 Tournament of Towns, P4

Tags: geometry , Tournament of Towns

[list=a] [*]Is it possible to split a square into 4 isosceles triangles such that no two are congruent? [*]Is it possible to split an equilateral triangle into 4 isosceles triangles such that no two are congruent? [/list] [i]Vladimir Rastorguev[/i]

2020 Kazakhstan National Olympiad, 4

Tags: Arithmetic Progression , game strategy , combinatorics , arithmetic sequence

Alice and Bob play a game on the infinite side of a checkered strip, in which the cells are numbered with consecutive integers from left to right (..., $-2$, $-1$, $0$, $1$, $2$, ...). Alice in her turn puts one cross in any free cell, and Bob in his turn puts zeros in any 2020 free cells. Alice will win if he manages to get such 4 cells marked with crosses, the corresponding cell numbers will form an arithmetic progression. Bob's goal in this game is to prevent Alice from winning. They take turns and Alice moves first. Will Alice be able to win no matter how Bob plays?

1995 Miklós Schweitzer, 1

Tags: complex analysis , function

Prove that a harmonic function that is not identically zero in the plane cannot vanish on a two-dimensional positive-measure set.