Navvye/Olympiad-Problems-CSV huggingface dataset:
2009 Tuymaada Olympiad
Determine the maximum number $ h$ satisfying the following condition: for every $ a\in [0,h]$ and every polynomial $ P(x)$ of degree 99 such that $ P(0)=P(1)=0$, there exist $ x_1,x_2\in [0,1]$ such that $ P(x_1)=P(x_2)$ and $ x_2-x_1=a$.



Proposed by F. Petrov, D. Rostovsky, A. Khrabrov
2008 Tuymaada Olympiad
Portraits of famous scientists hang on a wall. The scientists lived between 1600 and 2008, and none of them lived longer than 80 years. Vasya multiplied the years of birth of these scientists, and Petya multiplied the years of their death. Petya's result is exactly $ 5\over 4$ times greater than  Vasya's result. What minimum number of portraits can be on the wall?



Author: V. Frank
2008 Tuymaada Olympiad
Is it possible to arrange on a circle all composite positive integers not exceeding $ 10^6$, so that no two neighbouring numbers are coprime?



Author: L. Emelyanov



Tuymaada 2008, Junior League, First Day, Problem 2.Prove that all composite positive integers not exceeding $ 10^6$

may be arranged on a circle so that no two neighbouring numbers are coprime.
2008 Tuymaada Olympiad
100 unit squares of an infinite squared plane form a $ 10\times 10$ square.  Unit segments forming these squares are coloured in several colours.  It is known that the border of every square with sides on grid lines contains segments of at most two colours. (Such square is not necessarily contained in the original $ 10\times 10$ square.) What maximum number of colours may appear in this colouring?



Author: S. Berlov
2008 Tuymaada Olympiad
Point $ I_1$ is the reflection of incentre $ I$ of triangle $ ABC$ across the side $ BC$. The circumcircle of $ BCI_1$ intersects the line $ II_1$ again at point $ P$. It is known that $ P$ lies outside the incircle of the triangle $ ABC$. Two tangents drawn from $ P$ to the latter circle touch it at points $ X$ and $ Y$. Prove that the line $ XY$ contains a medial line of the triangle $ ABC$.



Author: L. Emelyanov
2008 Tuymaada Olympiad
A loader has a waggon and a little cart. The waggon can carry up to 1000 kg, and the cart can carry only up to 1 kg.  A finite number of sacks with sand lie in a storehouse. It is known that their total weight is more than 1001 kg, while each sack weighs not more than 1 kg. What maximum weight of sand can the loader carry in the waggon and the cart, regardless of particular weights of sacks?



Author: M.Ivanov, D.Rostovsky, V.Frank
2008 Tuymaada Olympiad
Let $ ABCD$ be an isosceles trapezoid with $ AD \parallel BC$. Its diagonals $ AC$ and $ BD$ intersect at point $ M$. Points $ X$ and $ Y$ on the segment $ AB$ are such that $ AX = AM$, $ BY = BM$. Let $ Z$ be the midpoint of $ XY$ and $ N$ is the point of intersection of the segments $ XD$ and $ YC$. Prove that the line $ ZN$ is parallel to the bases of the trapezoid.



Author: A. Akopyan, A. Myakishev
2008 Tuymaada Olympiad
A set $ X$ of positive integers is called nice if for each pair $ a$, $ b\in X$ exactly one of the numbers $ a + b$ and $ |a - b|$ belongs to $ X$ (the numbers $ a$ and $ b$ may be equal). Determine the number of nice sets containing the number 2008.



Author: Fedor Petrov
2008 Tuymaada Olympiad
250 numbers are chosen among positive integers not exceeding 501. Prove that for every integer $ t$ there are four chosen numbers $ a_1$, $ a_2$, $ a_3$, $ a_4$, such that $ a_1 + a_2 + a_3 + a_4 - t$ is divisible by 23.



Author: K. Kokhas
2008 Tuymaada Olympiad
Several irrational numbers are written on a blackboard. It is known that for every two numbers $ a$ and $ b$ on the blackboard, at least one of the numbers $ a\over b+1$ and $ b\over a+1$ is rational. What maximum number of irrational numbers can be on the blackboard?



Author: Alexander Golovanov
2008 Tuymaada Olympiad
A group of persons is called good if its members can be distributed to several rooms so that nobody is acquainted with any person in the same room

but it is possible to choose a person from each room so that all the chosen persons are acquainted with each other.



A group is called perfect if it is good and every set of its members is also good.



A perfect group planned a party. However one of its members, Alice, brought here acquaintance Bob, who was not originally expected, and introduced him to all her other acquaintances. Prove that the new group is also perfect.





Author: C. Berge
2008 Tuymaada Olympiad
Every street in the city of Hamiltonville connects two squares, and every square may be reached by streets from every other. The governor discovered that if he closed all squares of any route not passing any square more than once, every remained square would be reachable from each other. Prove that there exists a circular route passing every square of the city exactly once.



Author: S. Berlov
2008 Tuymaada Olympiad
A loader has two carts. One of them can carry up to 8 kg, and another can carry up to 9 kg. A finite number of sacks with sand lie in a storehouse. It is known that their total weight is more than 17 kg, while each sack weighs not more than 1 kg. What maximum weight of sand can the loader carry on his two carts, regardless of particular weights of sacks?





Author: M.Ivanov, D.Rostovsky, V.Frank
2008 Tuymaada Olympiad
A convex hexagon is given. Let $ s$ be the sum of the lengths of the three segments connecting the midpoints of its opposite sides. Prove that there is a point in the hexagon such that the sum of its distances to the lines containing the sides of the hexagon does not exceed $ s.$



Author: N. Sedrakyan
2007 Tuymaada Olympiad
Positive integers $ a<b$ are given. Prove that among every $ b$ consecutive positive integers there are two numbers whose product is divisible by $ ab$.
2007 Tuymaada Olympiad
Two polynomials $ f(x)=a_{100}x^{100}+a_{99}x^{99}+\dots+a_{1}x+a_{0}$ and $ g(x)=b_{100}x^{100}+b_{99}x^{99}+\dots+b_{1}x+b_{0}$ of degree $ 100$ differ from each other by a permutation of coefficients. It is known that $ a_{i}\ne b_{i}$ for $ i=0, 1, 2, \dots, 100$. Is it possible that $ f(x)\geq g(x)$ for all real $ x$?
2007 Tuymaada Olympiad
$ AA_{1}$, $ BB_{1}$, $ CC_{1}$ are altitudes of an acute triangle $ ABC$. A circle passing through $ A_{1}$ and $ B_{1}$ touches the arc $ AB$ of its circumcircle at $ C_{2}$. The points $ A_{2}$, $ B_{2}$ are defined similarly. Prove that the lines $ AA_{2}$, $ BB_{2}$, $ CC_{2}$ are concurrent.
2007 Tuymaada Olympiad
Determine maximum real $ k$ such that there exist a set $ X$ and its subsets $ Y_{1}$, $ Y_{2}$, $ ...$, $ Y_{31}$ satisfying the following conditions:

(1) for every two elements of $ X$ there is an index $ i$ such that $ Y_{i}$ contains neither of these elements;



(2) if any non-negative numbers $ \alpha_{i}$ are assigned to the subsets $ Y_{i}$ and $ \alpha_{1}+\dots+\alpha_{31}=1$ then there is an element $ x\in X$ such that the sum of $ \alpha_{i}$ corresponding to all the subsets $ Y_{i}$ that contain $ x$ is at least $ k$.
2007 Tuymaada Olympiad
What minimum number of colours is sufficient to colour all positive real numbers so that every two numbers whose ratio is 4 or 8 have different colours?
2007 Tuymaada Olympiad
Point $ D$ is chosen on the side $ AB$ of triangle $ ABC$. Point $ L$ inside the triangle $ ABC$ is such that $ BD=LD$ and $ \angle LAB=\angle LCA=\angle DCB$. It is known that $ \angle ALD+\angle ABC=180^\circ$. Prove that $ \angle BLC=90^\circ$.
2007 Tuymaada Olympiad
Several knights are arranged on an infinite chessboard. No square is attacked by more than one knight (in particular, a square occupied by a knight can be attacked by one knight but not by two). Sasha outlined a $ 14\times 16$ rectangle. What maximum number of knights can this rectangle contain?
2007 Tuymaada Olympiad
Prove that there exists a positive $ c$ such that for every positive integer $ N$ among any $ N$ positive integers not exceeding $ 2N$ there are two numbers whose greatest common divisor is greater than $ cN$.
2006 Tuymaada Olympiad
Seven different odd primes are given. Is it possible that for any two of them, the difference of their eight powers to be divisible by all the remaining ones ?



Proposed by F. Petrov, K. Sukhov
2006 Tuymaada Olympiad
We call a sequence of integers a Fibonacci-type sequence if it is infinite in both ways and $a_{n}=a_{n-1}+a_{n-2}$ for any $n\in\mathbb{Z}$. How many Fibonacci-type sequences can we find, with the property that in these sequences there are two consecutive terms, strictly positive, and less or equal than $N$ ? (two sequences are considered to be the same if they differ only by shifting of indices)



Proposed by I. Pevzner
2006 Tuymaada Olympiad
A line $d$ is given in the plane. Let $B\in d$ and $A$ another point, not on $d$, and such that $AB$ is not perpendicular on $d$. Let $\omega$ be a variable circle touching $d$ at $B$ and letting $A$ outside, and $X$ and $Y$ the points on $\omega$ such that $AX$ and $AY$ are tangent to the circle. Prove that the line $XY$ passes through a fixed point.



Proposed by F. Bakharev
2006 Tuymaada Olympiad
Find all functions $f: (0,\infty)\rightarrow(0,\infty)$ with the following properties: $f(x+1)=f(x)+1$  and $f\left(\frac{1}{f(x)}\right)=\frac{1}{x}$.



Proposed by P. Volkmann
2006 Tuymaada Olympiad
There are 100 boxers, each of them having different strengths, who participate in a tournament. Any of them fights each other only once. Several boxers form a plot. In one of their matches, they hide in their glove a horse shoe. If in a fight, only one of the boxers has a horse shoe hidden, he wins the fight; otherwise, the stronger boxer wins. It is known that there are three boxers who obtained (strictly) more wins than the strongest three boxers. What is the minimum number of plotters ?



Proposed by N. Kalinin
2006 Tuymaada Olympiad
Let $ABC$ be a triangle, $G$ it`s centroid, $H$ it`s orthocenter, and $M$ the midpoint of the arc $\widehat{AC}$ (not containing $B$). It is known that $MG=R$, where $R$ is the radius of the circumcircle. Prove that $BG\geq BH$.



Proposed by F. Bakharev
2006 Tuymaada Olympiad
From a $n\times (n-1)$ rectangle divided into unit squares, we cut the corner, which consists of the first row and the first column. (that is, the corner has $2n-2$ unit squares). For the following, when we say corner we reffer to the above definition, along with rotations and symmetry. Consider an infinite lattice of unit squares. We will color the squares with $k$ colors, such that for any corner, the squares in that corner are coloured differently (that means that there are no squares coloured with the same colour). Find out the minimum of $k$.



Proposed by S. Berlov
2006 Tuymaada Olympiad
For a positive integer, we define it's set of exponents the unordered list of all the exponents of the primes, in it`s decomposition. For example, $18=2\cdot 3^{2}$ has it`s set of exponents $1,2$ and $300=2^{2}\cdot 3\cdot 5^{2}$ has it`s set of exponents $1,2,2$. There are given two arithmetical progressions $\big(a_{n}\big)_{n}$ and $\big(b_{n}\big)_{n}$, such that for any positive integer $n$, $a_{n}$ and $b_{n}$ have the same set of exponents. Prove that the progressions are proportional (that is, there is $k$ such that $a_{n}=kb_{n}$ for any $n$).



Proposed by A. Golovanov
2005 Tuymaada Olympiad
The positive integers $1,2,...,121$ are arranged in the squares of a $11 \times 11$ table. Dima found the product of numbers in each row and Sasha found the product of the numbers in each column. Could they get the same set of $11$ numbers?



Proposed by S. Berlov
2005 Tuymaada Olympiad
Six members of the team of Fatalia for the International Mathematical Olympiad are selected from $13$ candidates. At the TST the candidates got $a_1,a_2, \ldots, a_{13}$ points with $a_i \neq a_j$ if $i \neq j$.



The team leader has already $6$ candidates and now wants to see them and nobody other in the team. With that end in view he constructs a polynomial $P(x)$ and finds the creative potential of each candidate by the formula $c_i = P(a_i)$.



For what minimum $n$ can he always find a polynomial $P(x)$ of degree not exceeding $n$ such that the creative potential of all $6$ candidates is strictly more than that of the $7$ others?



Proposed by F. Petrov, K. Sukhov
2005 Tuymaada Olympiad
The organizers of a mathematical congress found that if they accomodate any participant in a room the rest can be accomodated in double rooms so that 2 persons living in each room know each other. Prove that every participant can organize a round table on graph theory for himself and an even number of other people so that each participant of the round table knows both his neigbours.



Proposed by S. Berlov, S. Ivanov
2005 Tuymaada Olympiad
In a triangle $ABC$, let $A_{1}$, $B_{1}$, $C_{1}$ be the points where the excircles touch the sides $BC$, $CA$ and $AB$ respectively. Prove that $A A_{1}$, $B B_{1}$ and $C C_{1}$ are the sidelenghts of a triangle.



Proposed by L. Emelyanov
2005 Tuymaada Olympiad
You have $2$ columns of $11$ squares in the middle, in the right and in the left you have columns of $9$ squares (centered on the ones of $11$ squares), then columns of $7,5,3,1$ squares. (This is the way it was explained in the original thread,  ; anyway, i think you can understand how it looks)



Several rooks stand on the table and beat all the squares ( a rook beats the square it stands in, too). Prove that one can remove several rooks such that not more than $11$ rooks are left and still beat all the table.



Proposed by D. Rostovsky, based on folklore
2005 Tuymaada Olympiad
Given are a positive integer $n$ and an infinite sequence of proper fractions $x_0 = \frac{a_0}{n}$, $\ldots$, $x_i=\frac{a_i}{n+i}$, with $a_i < n+i$.  Prove that there exist a positive integer $k$ and integers $c_1$, $\ldots$, $c_k$ such that $$ c_1 x_1 + \ldots + c_k x_k = 1.  $$



Proposed by M. Dubashinsky
2005 Tuymaada Olympiad
Let $I$ be the incentre of triangle $ABC$. A circle containing the points $B$ and $C$ meets the segments $BI$ and $CI$ at points $P$ and $Q$ respectively. It is known that $BP\cdot CQ=PI\cdot QI$. Prove that the circumcircle of the triangle $PQI$ is tangent to the circumcircle of $ABC$.



Proposed by S. Berlov
2005 Tuymaada Olympiad
Let $a,b,c$ be positive reals s.t. $a^2+b^2+c^2=1$. Prove the following inequality $$ \sum \frac{a}{a^3+bc} >3 . $$



Proposed by A. Khrabrov
2004 Tuymaada Olympiad
Do there exist a sequence $a_{1}, a_{2}, a_{3}, \ldots$ of real numbers and a non-constant polynomial $P(x)$ such that $a_{m}+a_{n}=P(mn)$ for every positive integral $m$ and $n?$



Proposed by A. Golovanov
2004 Tuymaada Olympiad
In the plane are given 100 lines such that no 2 are parallel and no 3 meet in a point. The intersection points are marked. Then all the lines and k of the marked points are erased. Given the remained points of intersection for what max k one can reconstruct the lines?



Proposed by A. Golovanov
2004 Tuymaada Olympiad
An acute triangle $ABC$ is inscribed in a circle of radius 1 with centre $O;$ all the angles of $ABC$ are greater than $45^\circ.$

$B_{1}$ is the foot of perpendicular from $B$ to $CO,$ $B_{2}$ is the foot of perpendicular from $B_{1}$ to $AC.$

Similarly, $C_{1}$ is the foot of perpendicular from $C$ to $BO,$ $C_{2}$ is the foot of perpendicular from $C_{1}$ to $AB.$

The lines $B_{1}B_{2}$ and $C_{1}C_{2}$ intersect at $A_{3}.$ The points $B_{3}$ and $C_{3}$ are defined in the same way.

Find the circumradius of triangle $A_{3}B_{3}C_{3}.$



Proposed by F.Bakharev, F.Petrov
2004 Tuymaada Olympiad
There are many opposition societies in the city of N.

Each society consists of $10$ members. It is known that for every $2004$ societies there is a person belonging to at least $11$ of them.

Prove that the government can arrest $2003$ people so that at least one member of each society is arrested.



Proposed by V.Dolnikov, D.Karpov
2004 Tuymaada Olympiad
50 knights of King Arthur sat at the Round Table. A glass of white or red wine stood before each of them. It is known that at least one glass of red wine and at least one glass of white wine stood on the table.  The king clapped his hands twice. After the first clap every knight with a glass of red wine before him took a glass from his left neighbour. After the second clap every knight with a glass of white wine (and possibly something more) before him gave this glass to the left neughbour of his left neighbour. Prove that some knight was left without wine.



Proposed by A. Khrabrov, incorrect translation from Hungarian
2004 Tuymaada Olympiad
The incircle of triangle $ABC$ touches its sides $AB$ and $BC$ at points $P$ and $Q.$ The line $PQ$ meets the circumcircle of triangle $ABC$ at points $X$ and $Y.$ Find $\angle XBY$ if $\angle ABC = 90^\circ.$



Proposed by A. Smirnov
2004 Tuymaada Olympiad
Zeroes and ones are arranged in all the squares of $n\times n$ table.

All the squares of the left column are filled by ones, and the sum of numbers in every figure of the form

[asy]size(50); draw((2,1)--(0,1)--(0,2)--(2,2)--(2,0)--(1,0)--(1,2));[/asy]

(consisting of a square and its neighbours from left and from below)

is even.

Prove that no two rows of the table are identical.



Proposed by O. Vanyushina
2004 Tuymaada Olympiad
It is known that $m$ and $n$ are positive integers, $m > n^{n-1}$, and all the numbers $m+1$, $m+2$, \dots, $m+n$ are composite. Prove that there exist such different primes $p_1$, $p_2$, \dots, $p_n$ that $p_k$ divides $m+k$ for $k = 1$, 2, \dots, $n$.



Proposed by C. A. Grimm
2003 Tuymaada Olympiad
A $2003\times 2004$ rectangle consists of unit squares. We consider rhombi formed by four diagonals of unit squares.

What maximum number of such rhombi can be arranged in this rectangle so that no two of them have any common points except vertices?



Proposed by A. Golovanov
2003 Tuymaada Olympiad
In a quadrilateral $ABCD$ sides $AB$ and $CD$ are equal, $\angle A=150^\circ,$ $\angle B=44^\circ,$ $\angle C=72^\circ.$

Perpendicular bisector of the segment $AD$ meets the side $BC$ at point $P.$

Find $\angle APD.$



Proposed by F. Bakharev
2003 Tuymaada Olympiad
Alphabet $A$ contains $n$ letters. $S$ is a set of words of finite length composed of letters of $A$. It is known that every infinite sequence of letters of $A$ begins with one and only one word of $S$.

Prove that the set $S$ is finite.



Proposed by F. Bakharev
2003 Tuymaada Olympiad
Find all continuous functions $f(x)$ defined for all $x>0$ such that for every $x$, $y > 0$

$$ f\left(x+{1\over x}\right)+f\left(y+{1\over y}\right)= f\left(x+{1\over y}\right)+f\left(y+{1\over x}\right) . $$





Proposed by F. Petrov
2003 Tuymaada Olympiad
Prove that for every $\alpha_{1}, \alpha_{2}, \ldots, \alpha_{n}$ in the interval $(0,\pi/2)$

$$\left({1\over \sin \alpha_{1}}+{1\over \sin \alpha_{2}}+\ldots+{1\over \sin \alpha_{n}}\right) \left({1\over \cos \alpha_{1}}+{1\over \cos \alpha_{2}}+\ldots+{1\over \cos \alpha_{n}}\right) \leq$$

$$\leq 2 \left({1\over \sin 2\alpha_{1}}+{1\over \sin 2\alpha_{2}}+\ldots+{1\over \sin 2\alpha_{n}}\right)^{2}.$$





Proposed by A. Khrabrov
2003 Tuymaada Olympiad
Which number is bigger : the number of positive integers not exceeding 1000000 that can be represented by the form $2x^{2}-3y^{2}$ with integral $x$ and $y$ or that of positive integers not exceeding 1000000 that can be represented by the form $10xy-x^{2}-y^{2}$ with integral $x$ and $y?$



Proposed by A. Golovanov
2003 Tuymaada Olympiad
In a convex quadrilateral $ABCD$ we have $AB\cdot CD=BC\cdot DA$ and $2\angle A+\angle C=180^\circ$. Point $P$ lies on the circumcircle of triangle $ABD$ and is the midpoint of the arc $BD$ not containing $A$. It is known that the point $P$ lies inside the quadrilateral $ABCD$. Prove that $\angle BCA=\angle DCP$



Proposed by S. Berlov
2003 Tuymaada Olympiad
Given are polynomial $f(x)$ with non-negative integral coefficients and positive integer $a.$ The sequence $\{a_{n}\}$ is defined by $a_{1}=a,$ $a_{n+1}=f(a_{n}).$ It is known that the set of primes dividing at least one of the terms of this sequence is finite.

Prove that $f(x)=cx^{k}$ for some non-negative integral $c$ and $k.$





Proposed by F. Petrov



For those of you who liked this problem.Check this thread out.
2002 Tuymaada Olympiad
Each of the points $G$ and $H$ lying from different sides of the plane of hexagon $ABCDEF$ is connected with all vertices of the hexagon.

Is it possible to mark 18 segments thus formed by the numbers $1, 2, 3, \ldots, 18$ and arrange some real numbers at points $A, B, C, D, E, F, G, H$ so that each segment is marked with the difference of the numbers at its ends?



Proposed by A. Golovanov
2002 Tuymaada Olympiad
Let $a,b,c,d$ be positive real numbers such that $abcd=1$. Prove that

$$ \frac{1+ab}{1+a} + \frac{1+bc}{1+b} + \frac{1+cd}{1+c} + \frac{1+da}{1+d} \geq 4 .  $$





Proposed by A. Khrabrov
2002 Tuymaada Olympiad
A circle having common centre with the circumcircle of triangle $ABC$ meets the sides of the triangle at six points forming convex hexagon $A_{1}A_{2}B_{1}B_{2}C_{1}C_{2}$ ($A_{1}$ and $A_{2}$ lie on $BC$, $B_{1}$ and $B_{2}$ lie on $AC$, $C_{1}$ and $C_{2}$ lie on $AB$).

If $A_{1}B_{1}$ is parallel to the bisector of angle $B$, prove that $A_{2}C_{2}$ is parallel to the bisector of angle  $C$.



Proposed by S. Berlov
2002 Tuymaada Olympiad
A rectangular table with 2001 rows and 2002 columns is partitioned into $1\times 2$ rectangles. It is known that any other partition of the table into $1\times 2$ rectangles contains a rectangle belonging to the original partition.

Prove that the original partition contains two successive columns covered by 2001 horizontal rectangles.



Proposed by S. Volchenkov
2002 Tuymaada Olympiad
A positive integer $c$ is given. The sequence $\{p_{k}\}$ is constructed by the following rule: $p_{1}$ is arbitrary prime and for $k\geq 1$ the number $p_{k+1}$ is any prime divisor of $p_{k}+c$ not present among the numbers $p_{1}$, $p_{2}$, $\dots$, $p_{k}$. Prove that the sequence $\{p_{k}\}$ cannot be infinite.



Proposed by A. Golovanov
2002 Tuymaada Olympiad
Find all the functions $f(x),$ continuous on the whole real axis, such that for every real $x$ $$f(3x-2)\leq f(x)\leq f(2x-1).$$



Proposed by A. Golovanov
2002 Tuymaada Olympiad
The points $D$ and $E$ on the circumcircle of an acute triangle $ABC$ are such that $AD=AE = BC$. Let $H$ be the common point of the altitudes of triangle $ABC$.

It is known that $AH^{2}=BH^{2}+CH^{2}$.

Prove that $H$ lies on the segment $DE$.



Proposed by D. Shiryaev
2002 Tuymaada Olympiad
A real number $a$ is given. The sequence $n_{1}< n_{2}< n_{3}< ...$ consists of all the positive integral $n$ such that $\{na\}< \frac{1}{10}$. Prove that there are at most three different numbers among the numbers $n_{2}-n_{1}$, $n_{3}-n_{2}$, $n_{4}-n_{3}$, $\ldots$.



A corollary of a theorem from ergodic theory
2002 Tuymaada Olympiad
Points on the sides $ BC $, $ CA $ and $ AB $ of the triangle $ ABC $ are respectively $ A_1 $, $ B_1 $ and $ C_1 $ such that $ AC_1: C_1B = BA_1: A_1C = CB_1: B_1A = 2: 1 $. Prove that if triangle $ A_1B_1C_1 $ is equilateral, then triangle $ ABC $ is also equilateral.
2002 Tuymaada Olympiad
Is there a quadratic trinomial with integer coefficients, such that all values which are natural to be natural powers of two?
2002 Tuymaada Olympiad
Prove that for all $ x, y \in 0, 1 $ the inequality $ 5 (x^2+ y^2) ^2 \leq 4 + (x +y) ^4$ holds.
2002 Tuymaada Olympiad
In the cells of the table $ 100 \times100 $ are placed in pairs different numbers. Every minute each of the numbers changes to the largest of the numbers in the adjacent cells on the side. Can after $4$ hours all the numbers in the table be the same?
2002 Tuymaada Olympiad
The circle with the center of $ O $ touches the sides of the angle $ A $ at the points of $ K $ and $ M $. The tangent to the circle intersects the segments $ AK $ and $ AM $ at points $ B $ and $ C $ respectively, and the line $ KM $ intersects the segments $ OB $ and $ OC $ at the points $ D $ and $ E $. Prove that the area of the triangle $ ODE $ is equal to a quarter of the area of a triangle $ BOC $ if and only if the angle $ A $ is $ 60^\circ $.