Olympiad problems

2023 All-Russian Olympiad Regional Round, 9.10

A $100 \times 100 \times 100$ cube is divided into a million unit cubes and in each small cube there is a light bulb. Three faces $100 \times 100$ of the large cube having a common vertex are painted: one in red, one in blue and the other in green. Call a $\textit{column}$ a set of $100$ cubes forming a block $1 \times 1 \times 100$. Each of the $30 000$ columns have one painted end cell, on which there is a switch. After pressing a switch, the states of all light bulbs of this column are changed. Petya pressed several switches, getting a situation with exactly $k$ lamps on. Prove that Vasya can press several switches so that all lamps are off, but by using no more than $\frac {k} {100}$ switches on the red face.

Kvant 2023, M2741

Tags: combinatorics , Russia

Given is a positive integer $k$. There are $n$ points chosen on a line, such the distance between any two adjacent points is the same. The points are colored in $k$ colors. For each pair of monochromatic points such that there are no points of the same color between them, we record the distance between these two points. If all distances are distinct, find the largest possible $n$.

2023 All-Russian Olympiad Regional Round, 9.6

Tags: number theory , least common multiple , Russia

Does there exist a positive integer $m$, such that if $S_n$ denotes the lcm of $1,2, \ldots, n$, then $S_{m+1}=4S_m$?

2005 All-Russian Olympiad, 3

Tags: geometry , incenter , parallelogram , trigonometry , complex numbers , Russia

A quadrilateral $ABCD$ without parallel sides is circumscribed around a circle with centre $O$. Prove that $O$ is a point of intersection of middle lines of quadrilateral $ABCD$ (i.e. barycentre of points $A,\,B,\,C,\,D$) iff $OA\cdot OC=OB\cdot OD$.

2023 All-Russian Olympiad Regional Round, 9.3

Tags: combinatorics , Russia

Given is a positive integer $n$. There are $2n$ mutually non-attacking rooks placed on a grid $2n \times 2n$. The grid is splitted into two connected parts, symmetric with respect to the center of the grid. What is the largest number of rooks that could lie in the same part?

2022 All-Russian Olympiad, 3

Tags: geometry , geometry solved , Russia , All Russian Olympiad , fixed circle , perpendicular lines , Russian

An acute-angled triangle $ABC$ is fixed on a plane with largest side $BC$. Let $PQ$ be an arbitrary diameter of its circumscribed circle, and the point $P$ lies on the smaller arc $AB$, and the point $Q$ is on the smaller arc $AC$. Points $X, Y, Z$ are feet of perpendiculars dropped from point $P$ to the line $AB$, from point $Q$ to the line $AC$ and from point $A$ to line $PQ$. Prove that the center of the circumscribed circle of triangle $XYZ$ lies on a fixed circle.

1962 All-Soviet Union Olympiad, 4

Tags: algebra , Russia

Prove that there are no integers $a, b, c, d$ such that the polynomial $ax^3+bx^2+cx+d$ equals $1$ at $x=19$ and $2$ at $x=62$.

2004 All-Russian Olympiad, 4

Tags: inequalities , induction , Russia , 2004 , cyclic inequality , n-variable inequality

Let $n > 3$ be a natural number, and let $x_1$, $x_2$, ..., $x_n$ be $n$ positive real numbers whose product is $1$. Prove the inequality \[ \frac {1}{1 + x_1 + x_1\cdot x_2} + \frac {1}{1 + x_2 + x_2\cdot x_3} + ... + \frac {1}{1 + x_n + x_n\cdot x_1} > 1. \]

2001 All-Russian Olympiad, 4

Tags: number theory , relatively prime , Russia , Divisors , MONT

Find all odd positive integers $ n > 1$ such that if $ a$ and $ b$ are relatively prime divisors of $ n$, then $ a\plus{}b\minus{}1$ divides $ n$.

2023 All-Russian Olympiad Regional Round, 11.9

Tags: absolute value , algebra , Russia , inequalities proposed

If $a, b, c$ are non-zero reals, prove that $|\frac{b} {a}-\frac{b} {c}|+|\frac{c} {a}-\frac{c}{b}|+|bc+1|>1$.

2018 All-Russian Olympiad, 4

Tags: Russia , combinatorics

On the $n\times n$ checker board, several cells were marked in such a way that lower left ($L$) and upper right($R$) cells are not marked and that for any knight-tour from $L$ to $R$, there is at least one marked cell. For which $n>3$, is it possible that there always exists three consective cells going through diagonal for which at least two of them are marked?

1962 All-Soviet Union Olympiad, 8

Tags: Russia , geometry

Given is a fixed regular pentagon $ABCDE$ with side $1$. Let $M$ be an arbitrary point inside or on it. Let the distance from $M$ to the closest vertex be $r_1$, to the next closest be $r_2$ and so on, so that the distances from $M$ to the five vertices satisfy $r_1\le r_2\le r_3\le r_4\le r_5$. Find (a) the locus of $M$ which gives $r_3$ the minimum possible value, and (b) the locus of $M$ which gives $r_3$ the maximum possible value.

2014 All-Russian Olympiad, 4

Tags: geometry , incenter , trigonometry , circumcircle , Russia

Given a triangle $ABC$ with $AB>BC$, $ \Omega $ is circumcircle. Let $M$, $N$ are lie on the sides $AB$, $BC$ respectively, such that $AM=CN$. $K(.)=MN\cap AC$ and $P$ is incenter of the triangle $AMK$, $Q$ is K-excenter of the triangle $CNK$ (opposite to $K$ and tangents to $CN$). If $R$ is midpoint of the arc $ABC$ of $ \Omega $ then prove that $RP=RQ$. M. Kungodjin

2019 All-Russian Olympiad, 8

Tags: algebra , inequalities , Russia , High school olympiad

For $a,b,c$ be real numbers greater than $1$, prove that \[\frac{a+b+c}{4} \geq \frac{\sqrt{ab-1}}{b+c}+\frac{\sqrt{bc-1}}{c+a}+\frac{\sqrt{ca-1}}{a+b}.\]

2006 All-Russian Olympiad, 4

Tags: geometry , circumcircle , Russia

Given a triangle $ABC$. Let a circle $\omega$ touch the circumcircle of triangle $ABC$ at the point $A$, intersect the side $AB$ at a point $K$, and intersect the side $BC$. Let $CL$ be a tangent to the circle $\omega$, where the point $L$ lies on $\omega$ and the segment $KL$ intersects the side $BC$ at a point $T$. Show that the segment $BT$ has the same length as the tangent from the point $B$ to the circle $\omega$.

1962 All-Soviet Union Olympiad, 5

Tags: Russia , combinatorics

An $n \times n$ array of numbers is given. $n$ is odd and each number in the array is $1$ or $-1$. Prove that the number of rows and columns containing an odd number of $-1$s cannot total $n$.

2022 Dutch IMO TST, 3

Tags: number theory , Russia , All Russian Olympiad

Let $n$ be a natural number. An integer $a>2$ is called $n$-decomposable, if $a^n-2^n$ is divisible by all the numbers of the form $a^d+2^d$, where $d\neq n$ is a natural divisor of $n$. Find all composite $n\in \mathbb{N}$, for which there's an $n$-decomposable number.

1962 All-Soviet Union Olympiad, 11

Tags: geometry , Russia

The triangle $ABC$ satisfies $0\le AB\le 1\le BC\le 2\le CA\le 3$. What is the maximum area it can have?

2023 All-Russian Olympiad Regional Round, 11.5

Tags: geometry , Russia

Given is a triangle $ABC$ with altitude $AH$ and median $AM$. The line $OH$ meets $AM$ at $D$. Let $AB \cap CD=E, AC \cap BD=F$. If $EH$ and $FH$ meet $(ABC)$ at $X, Y$, prove that $BY, CX, AH$ are concurrent.

2023 All-Russian Olympiad Regional Round, 10.8

Tags: geometry , parallelogram , Russia

The bisector of $\angle BAD$ of a parallelogram $ABCD$ meets $BC$ at $K$. The point $L$ lies on $AB$ such that $AL=CK$. The lines $AK$ and $CL$ meet at $M$. Let $(ALM)$ meet $AD$ after $D$ at $N$. Prove that $\angle CNL=90^{o}$

2017 Tournament Of Towns, 5

Tags: combinatorics , number theory , Russia

There is a set of control weights, each of them weighs a non-integer number of grams. Any integer weight from $1$ g to $40$ g can be balanced by some of these weights (the control weights are on one balance pan, and the measured weight on the other pan).What is the least possible number of the control weights? [i](Alexandr Shapovalov)[/i]

2000 All-Russian Olympiad, 4

Tags: combinatorics , Russia , weights , algorithm

We are given five equal-looking weights of pairwise distinct masses. For any three weights $A$, $B$, $C$, we can check by a measuring if $m(A) < m(B) < m(C)$, where $m(X)$ denotes the mass of a weight $X$ (the answer is [i]yes[/i] or [i]no[/i].) Can we always arrange the masses of the weights in the increasing order with at most nine measurings?

2023 All-Russian Olympiad Regional Round, 9.5

Tags: geometry , Russia

Let $ABCD$ be a cyclic quadrilateral such that the circles with diameters $AB$ and $CD$ touch at $S$. If $M, N$ are the midpoints of $AB, CD$, prove that the perpendicular through $M$ to $MN$ meets $CS$ on the circumcircle of $ABCD$.

2004 239 Open Mathematical Olympiad, 2

Tags: geometry , Russia , bisector

The incircle of a triangle $ABC$ has centre $I$ and touches sides $AB, BC, CA$ in points $C_1, A_1, B_1$ respectively. Denote by $L$ the foot of a bissector of angle $B$, and by $K$ the point of intersecting of lines $B_1I$ and $A_1C_1$. Prove that $KL\parallel BB_1$. [b]proposed by L. Emelyanov, S. Berlov[/b]

1961 All-Soviet Union Olympiad, 1

Tags: geometry , Equilateral Triangle , Russia

Points $A$ and $B$ move on circles centered at $O_A$ and $O_B$ such that $O_AA$ and $O_BB$ rotate at the same speed. Prove that vertex $C$ of the equilateral triangle $ABC$ moves along a certain circle at the same angular velocity. (The vertices of $ABC$ are oriented clockwise.)