Navvye/Olympiad-Problems-CSV huggingface dataset:
2020 Tuymaada Olympiad
Does the system of equation

\begin{align*}

\begin{cases}

x_1 + x_2 &= y_1 + y_2 + y_3 + y_4 \\
x_1^2 + x_2^2 &= y_1^2 + y_2^2 + y_3^2 + y_4^2 \\
x_1^3 + x_2^3 &= y_1^3 + y_2^3 + y_3^3 + y_4^3

\end{cases}

\end{align*}admit a solution in integers such that the absolute value of each of these integers is greater than $2020$?
2020 Tuymaada Olympiad
Given positive real numbers $a_1, a_2, \dots, a_n$. Let

$$ m = \min \left( a_1 + \frac{1}{a_2}, a_2 + \frac{1}{a_3}, \dots, a_{n - 1} + \frac{1}{a_n} , a_n + \frac{1}{a_1} \right). $$Prove the inequality

$$ \sqrt[n]{a_1 a_2 \dots a_n} + \frac{1}{\sqrt[n]{a_1 a_2 \dots a_n}} \ge m. $$
2020 Tuymaada Olympiad
For each positive integer $k$, let $g(k)$ be the maximum possible number of points in the plane such that pairwise distances between these points have only $k$ different values. Prove that there exists $k$ such that $g(k) > 2k + 2020$.
2020 Tuymaada Olympiad
An isosceles triangle $ABC$ ($AB = BC$) is given. Circles $\omega_1$ and $\omega_2$ with centres $O_1$ and $O_2$ lie in the angle $ABC$ and touch the sides $AB$ and $CB$ at $A$ and $C$ respectively, and touch each other externally at point $X$. The side $AC$ meets the circles again at points $Y$ and $Z$. $O$ is the circumcenter of the triangle $XYZ$. Lines $O_2 O$ and $O_1 O$ intersect lines $AB$ and $BC$ at points $C_1$ and $A_1$ respectively. Prove that $B$ is the circumcentre of the triangle $A_1 OC_1$.
2020 Tuymaada Olympiad
Several policemen try to catch a thief who has $2m$ accomplices. To that end they place the accomplices under surveillance. In the beginning, the policemen shadow nobody. Every morning each policeman places under his surveillance one of the accomplices. Every evening the thief stops trusting one of his accomplices The thief is caught if by the $m$-th evening some policeman shadows exactly those $m$ accomplices who are still trusted by the thief. Prove that to guarantee the capture of the thief at least $2^m$ policemen are needed.
2020 Tuymaada Olympiad
The degrees of polynomials $P$ and $Q$ with real coefficients do not exceed $n$. These polynomials satisfy the identity

$$ P(x) x^{n + 1} + Q(x) (x+1)^{n + 1} = 1. $$Determine all possible values of $Q \left( - \frac{1}{2} \right)$.
2019 Tuymaada Olympiad
In a sequence $a_1, a_2, ..$ of real numbers the product $a_1a_2$ is negative, and to define $a_n$ for $n > 2$ one pair $(i, j)$ is chosen among all the pairs $(i, j), 1 \le i < j < n$, not chosen before, so that $a_i +a_j$ has minimum absolute value, and then $a_n$ is set equal to $a_i + a_j$ . Prove that $|a_i| < 1$ for some $i$.
2019 Tuymaada Olympiad
A triangle $ABC$ with $AB < AC$ is inscribed in a circle $\omega$. Circles  $\gamma_1$ and $\gamma_2$ touch the lines $AB$ and $AC$, and their centres lie on the circumference of $\omega$. Prove that $C$ lies on a common external tangent to  $\gamma_1$ and $\gamma_2$.
2019 Tuymaada Olympiad
The plan of a picture gallery is a chequered figure where each square is a room, and every room can be reached from each other by moving to rooms adjacent by side. A custodian in a room can watch all the rooms that can be reached from this room by one move of a chess rook (without leaving the gallery). What minimum number of custodians is sufficient to watch all the rooms in every gallery of $n$ rooms ($n > 1$)?
2019 Tuymaada Olympiad
A quota of diplomas at the All-Russian Olympiad should be strictly less than $45\%$. More than $20$ students took part in the olympiad. After the olympiad the Authorities declared the results low because the quota of diplomas was significantly less than $45\%$. The Jury responded that the quota was already maximum possible on this olympiad or any other olympiad with smaller number of participants. Then the Authorities ordered to increase the number of participants for the next olympiad so that the quota of diplomas became at least two times closer to $45\%$. Prove that the number of participants should be at least doubled.
2019 Tuymaada Olympiad
Is it possible to draw in the plane the graph presented in the figure so that all the vertices are different points and all the edges are unit segments? (The segments can intersect at points different from vertices.)
2019 Tuymaada Olympiad
Let $\mathbb{S}$ is the set of prime numbers that less or equal to 26. Is there any $a_1, a_2, a_3, a_4, a_5, a_6 \in \mathbb{N}$ such that $$ gcd(a_i,a_j) \in \mathbb{S} \qquad \text {for }  1\leq i \ne j \leq 6$$and for every element $p$ of $\mathbb{S}$ there exists a pair of $ 1\leq k \ne l \leq 6$ such that $$s=gcd(a_k,a_l)?$$
2019 Tuymaada Olympiad
A circle $\omega$ touches the sides $A$B and $BC$ of a triangle $ABC$ and intersects its side $AC$ at $K$. It is known that the tangent to $\omega$ at $K$ is symmetrical to the line $AC$ with respect to the line $BK$. What can be the difference $AK -CK$ if $AB = 9$ and $BC = 11$?
2019 Tuymaada Olympiad
Andy, Bess, Charley and Dick play on a $1000 \times 1000$ board. They make moves in turn: Andy first, then Bess, then Charley and finally Dick, after that Andy moves again and so on. At each move a player must paint several unpainted squares forming $2 \times  1, 1 \times  2, 1 \times  3$, or $3 \times  1$ rectangle. The player that cannot move loses. Prove that some three players can cooperate to make the fourth player lose.
2019 Tuymaada Olympiad
A trapezoid $ABCD$ with $BC // AD$ is given. The points $B'$ and $C'$ are symmetrical to $B$ and $C$ with respect to $CD$ and $AB$, respectively. Prove that the midpoint of the segment joining the circumcentres of $ABC'$ and $B'CD$ is equidistant from $A$ and $D$.
2019 Tuymaada Olympiad
The plan of a picture gallery is a chequered figure where each square is a room, and every room can be reached from each other by moving to adjacent rooms. A custodian in a room can watch all the rooms that can be reached from this room by one move of a chess queen (without leaving the gallery). What minimum number of custodians is sufficient to watch all the rooms in every gallery of $n$ rooms ($n > 2$)?
2019 Tuymaada Olympiad
A calculator can square a number or add $1$ to it. It cannot add $1$ two times in a row. By several operations it transformed a number $x$ into a number $S > x^n + 1$ ($x, n,S$ are positive integers). Prove that $S > x^n + x - 1$.
2019 Tuymaada Olympiad
Prove that the expression

$$ (1^4+1^2+1)(2^4+2^2+1)\dots(n^4+n^2+1)$$is not square for all $n \in \mathbb{N}$
2019 Tuymaada Olympiad
$N$ cells chosen on a rectangular grid. Let $a_i$ is number of chosen cells in $i$-th row, $b_j$ is number of chosen cells in $j$-th column. Prove that

$$ \prod_{i} a_i! \cdot \prod_{j} b_j! \leq N! $$
2019 Tuymaada Olympiad
In $\triangle ABC$ $\angle B$ is obtuse and $AB \ne BC$. Let $O$ is the circumcenter and $\omega$ is the circumcircle of this triangle. $N$ is the midpoint of arc $ABC$. The circumcircle of $\triangle BON$ intersects $AC$ on points $X$ and $Y$. Let $BX \cap \omega = P \ne B$ and  $BY \cap \omega = Q \ne B$. Prove that $P, Q$ and reflection of $N$ with respect to line $AC$ are collinear.
2018 Tuymaada Olympiad
Real numbers $a \neq 0, b, c$ are given. Prove that there is a polynomial $P(x)$ with real coefficients such that the polynomial $x^2+1$ divides the polynomial $aP(x)^2+bP(x)+c$.



Proposed by A. Golovanov
2018 Tuymaada Olympiad
A circle touches the side $AB$ of the triangle $ABC$ at $A$, touches the side $BC$ at $P$ and intersects the side $AC$ at $Q$. The line symmetrical to $PQ$ with respect to $AC$ meets the line $AP$ at $X$. Prove that $PC=CX$.



Proposed by S. Berlov
2018 Tuymaada Olympiad
$n$ rooks and $k$ pawns are arranged on a $100 \times 100$ board. The rooks cannot leap over pawns. For which minimum $k$ is it possible that no rook can capture any other rook?



Junior League: $n=2551$ (Proposed by A. Kuznetsov)

Senior League: $n=2550$ (Proposed by N. Vlasova)
2018 Tuymaada Olympiad
Prove that for every positive integer $d > 1$ and $m$ the sequence $a_n=2^{2^n}+d$ contains two terms $a_k$ and $a_l$ ($k \neq l$) such that their greatest common divisor is greater than $m$.



Proposed by T. Hakobyan
2018 Tuymaada Olympiad
$99$ identical balls lie on a table. $50$ balls are made of copper, and $49$ balls are made of zinc. The assistant numbered the balls. Once spectrometer test is applied to $2$ balls and allows to determine whether they are made of the same metal or not. However, the results of the test can be obtained only the next day. What minimum number of tests is required to determine the material of each ball if all the tests should be performed today?



Proposed by N. Vlasova, S. Berlov
2018 Tuymaada Olympiad
The numbers $1, 2, 3, \dots, 1024$ are written on a blackboard. They are divided into pairs. Then each pair is wiped off the board and non-negative difference of its numbers is written on the board instead. $512$ numbers obtained in this way are divided into pairs and so on. One number remains on the blackboard after ten such operations. Determine all its possible values.



Proposed by A. Golovanov
2018 Tuymaada Olympiad
Prove the inequality $$(x^3+2y^2+3z)(4y^3+5z^2+6x)(7z^3+8x^2+9y)\geq720(xy+yz+xz)$$for $x, y, z \geq 1$.



Proposed by K. Kokhas
2018 Tuymaada Olympiad
Quadrilateral $ABCD$ with perpendicular diagonals is inscribed in a circle with centre $O$. The tangents to this circle at $A$ and $C$ together with line $BD$ form the triangle $\Delta$. Prove that the circumcircles of $BOD$ and $\Delta$ are tangent.



Additional information for Junior LeagueShow that this point lies belongs to $\omega$, the circumcircle of $OAC$



Proposed by A. Kuznetsov
2018 Tuymaada Olympiad
Do there exist three different quadratic trinomials $f(x), g(x), h(x)$ such that the roots of the equation $f(x)=g(x)$ are $1$ and $4$, the roots of the equation $g(x)=h(x)$ are $2$ and $5$, and the roots of the equation $h(x)=f(x)$ are $3$ and $6$?



Proposed by A. Golovanov
2018 Tuymaada Olympiad
A point $P$ on the side $AB$ of a triangle $ABC$ and points $S$ and $T$ on the sides $AC$ and $BC$ are such that $AP=AS$ and $BP=BT$. The circumcircle of $PST$ meets the sides $AB$ and $BC$ again at $Q$ and $R$, respectively. The lines $PS$ and $QR$ meet at $L$. Prove that the line $CL$ bisects the segment $PQ$.



Proposed by A. Antropov
2018 Tuymaada Olympiad
A prime $p$ and a positive integer $n$ are given. The product $$(1^3+1)(2^3+1)...((n-1)^3+1)(n^3+1)$$is divisible by $p^3$. Prove that $p \leq n+1$.



Proposed by Z. Luria
2018 Tuymaada Olympiad
A school has three senior classes of $M$ students each. Every student knows at least $\frac{3}{4}M$ people in each of the other two classes. Prove that the school can send $M$ non-intersecting teams to the olympiad so that each team consists of $3$ students from different classes who know each other.



Proposed by C. Magyar, R. Martin
2017 Tuymaada Olympiad
Functions $f$ and $g$ are defined on the set of all integers in the interval $[-100; 100]$ and take integral values. Prove that for some integral $k$ the number of solutions of the equation $f(x)-g(y)=k$ is odd.

( A. Golovanov)
2017 Tuymaada Olympiad
$ABCD $ is a cyclic quadrilateral such that the diagonals  $AC $ and $BD $ are perpendicular and their intersection is $P $. Point $Q $ on the segment $CP$ is such that $CQ=AP $. Prove that the perimeter of triangle $BDQ $ is at least  $2AC $.



Tuymaada 2017 Q2 Juniors
2017 Tuymaada Olympiad
In a country every 2 cities are connected either by a direct bus route or a direct plane flight. A $clique$ is a set of cities such that every 2 cities in the set are connected by a direct flight. A $cluque$ is a set of cities such that every 2 cities in the set are connected by a direct flight, and every 2 cities in the set are connected to the same number of cities by a bus route. A $claque$ is a set of cities such that every 2 cities in the set are connected by a direct flight, and every 2 numbers of bus routes from a city in the set are different. Prove that the number of cities of any clique is at most the product of the biggest possible number of cities in a cluque and the the biggest possible number of cities in a claque.



Tuymaada 2017 Q3 Juniors
2017 Tuymaada Olympiad
A right triangle has all its sides rational numbers and the area $S $. Prove that there exists a right triangle, different from the original one, such that all its sides are rational numbers and its area is $S $.



Tuymaada 2017 Q4 Juniors
2017 Tuymaada Olympiad
$BL $ is the bisector of an isosceles triangle $ABC $. A point $D $ is chosen on the Base  $BC $ and a point $E $ is chosen on the lateral side $AB $ so that $AE=\frac {1}{2}AL=CD $. Prove that $LE=LD $.



Tuymaada 2017 Q5 Juniors
2017 Tuymaada Olympiad
Let $\sigma(n) $ denote the sum of positive divisors of a number  $n $. A positive integer $N=2^rb $ is given,where $r $ and $b $ are positive integers and $b $ is odd. It is known that $\sigma(N)=2N-1$. Prove that $b$ and $\sigma (b) $ are coprime.



Tuymaada Q6 Juniors
2017 Tuymaada Olympiad
An equilateral triangle with side $20$ is divided by there series of parallel lines into  $400$ equilateral triangles with side $1$. What maximum number of these small triangles can be crossed  (internally) by one line?



Tuymaada 2017 Q7 Juniors
2017 Tuymaada Olympiad
We consider the graph with vertices  $A_1,A_2,\dots A_{2015}$ , $B_1,B_2,\dots B_{2015}$ and edges $A_iA _{i+1}, A_iB_i, B_iB_{i+17} $, taken cyclicaly. Is it true that 4 cops can catch a robber on this graph for every initial position?( First the 4 cops make a move, then the robber makes a move, then the cops make a move etc. A move consists of jumping from the vertex you stay on an adiacent vertex or by staying on your current vertex. Everyone knows the position of everyone everytime. The cops can coordinate their moves. The robber is caught when he shares the same vertex with a cop.)



Tuymaada 2017 Q8 Juniors
2017 Tuymaada Olympiad
There are 25 masks of different colours. k sages play the following game. They are shown all the masks. Then the sages agree on their strategy. After that the masks are put on them so that each sage sees the masks on the others but can not see who wears each mask and does not see his own mask. No communication is allowed. Then each of them simultaneously names one colour trying to guess the colour of his mask. Find the minimum k  for which the sages can agree so that at least one of them surely guesses the colour of his mask.

( S. Berlov )
2017 Tuymaada Olympiad
Does there exist a quadratic trinomial $f(x)$ such that  $f(1/2017)=1/2018$, $f(1/2018)=1/2017$, and two of its coefficients  are integers?



(A. Khrabrov)
2017 Tuymaada Olympiad
Let $\sigma(n)$ denote the sum of positive divisors of a number $n$.  A positive integer $N=2^r b$ is given, where $r$ and $b$ are positive integers  and $b$ is odd. It is known that $\sigma(N)=2N-1$. Prove that $b$ and  $\sigma(b)$ are coprime.



(J. Antalan, J. Dris)
2017 Tuymaada Olympiad
A point $E$ lies on the extension of the side $AD$ of the rectangle $ABCD$ over $D$. The ray $EC$ meets the circumcircle $\omega$ of $ABE$ at the point $F\ne E$. The rays $DC$ and $AF$ meet at $P$. $H$ is the foot of the perpendicular drawn from $C$ to the line $\ell$ going through  $E$ and parallel to $AF$. Prove that the line $PH$ is tangent to $\omega$.



(A. Kuznetsov)
2017 Tuymaada Olympiad
Two points $A$ and $B$ are given in the plane. A point $X$ is called their  preposterous midpoint if there is a Cartesian coordinate system in the plane such that the coordinates of $A$ and $B$ in this system are non-negative, the abscissa of $X$ is the geometric mean of the abscissae of $A$ and $B$, and the ordinate of $X$ is the geometric mean of the ordinates  of $A$ and $B$. Find the locus of all the preposterous midpoints of $A$ and $B$.



(K. Tyschu)
2016 Tuymaada Olympiad
The sequence $(a_n)$ is defined by $a_1=0$,

$$

a_{n+1}={a_1+a_2+\ldots+a_n\over n}+1.

$$Prove that $a_{2016}>{1\over 2}+a_{1000}$.
2016 Tuymaada Olympiad
A cube stands on one of the squares of an infinite chessboard.

On each face of the cube there is an arrow pointing in one of the four directions

parallel to the sides of the face. Anton looks on the cube from above and

rolls it over an edge in the direction pointed by the arrow on the top face.

Prove that the cube cannot cover any $5\times 5$ square.
2016 Tuymaada Olympiad
Altitudes $AA_1$, $BB_1$, $CC_1$ of an acute triangle $ABC$ meet at $H$.

$A_0$, $B_0$, $C_0$ are the midpoints of $BC$, $CA$, $AB$ respectively. Points $A_2$, $B_2$, $C_2$ on the segments $AH$, $BH$, $HC_1$ respectively are such that $\angle A_0B_2A_2 = \angle B_0C_2B_2 = \angle C_0A_2C_2 =90^\circ$.

Prove that the lines $AC_2$, $BA_2$, $CB_2$ are concurrent.
2016 Tuymaada Olympiad
For each positive integer $k$ find the number of solutions in nonnegative integers $x,y,z$ with $x\le y \le z$ of the equation

$$8^k=x^3+y^3+z^3-3xyz$$
2016 Tuymaada Olympiad
The ratio of prime numbers $p$ and $q$ does not exceed 2 ($p\ne q$).

Prove that there are two consecutive positive integers such that

the largest prime divisor of one of them is $p$ and that of the other is $q$.
2016 Tuymaada Olympiad
The numbers $a$, $b$, $c$, $d$ satisfy $0<a \leq b \leq d \leq c$ and

${a+c=b+d}$. Prove that for every internal point $P$ of a segment with

length $a$ this segment is a side of a circumscribed quadrilateral with consecutive sides

$a$, $b$, $c$, $d$, such that its incircle contains~$P$.
2016 Tuymaada Olympiad
For every $x$, $y$, $z>{3\over 2}$ prove the inequality

$$

x^{24} + \root 5\of {y^{60}+z^{40}}  \geq  

\left(x^4 y^3 + {1\over 3} y^2 z^2 + {1\over 9} x^3 z^3 \right)^2.

$$
2016 Tuymaada Olympiad
A connected graph is given. Prove that its vertices can be coloured

blue and green and some of its edges marked so that every two vertices

are connected by a path of marked edges, every marked edge connects

two vertices of different colour and no two green vertices are connected

by an edge of the original graph.
2016 Tuymaada Olympiad
Tanya and Serezha have a heap of $2016$ candies. They make moves in turn, Tanya moves first. At each move a player can eat either one candy or (if the number of candies is even at the moment) exactly half of all candies. The player that cannot move loses. Which of the players has a winning strategy?
2016 Tuymaada Olympiad
The point $D$ on the altitude $AA_1$ of an acute triangle $ABC$ is such that

$\angle BDC=90^\circ$; $H$ is the orthocentre of $ABC$. A circle

with diameter $AH$ is constructed. Prove that the tangent drawn from $B$

to this circle is equal to $BD$.
2016 Tuymaada Olympiad
Non-negative numbers $a$, $b$, $c$ satisfy

$a^2+b^2+c^2\geq 3$. Prove the inequality

$$

(a+b+c)^3\geq 9(ab+bc+ca).

$$
2016 Tuymaada Olympiad
Positive numbers are written in the squares of a 10 × 10 table. Frogs sit in five squares and cover the numbers in these squares. Kostya found the sum of all visible numbers and got 10. Then each frog jumped to an adjacent square and Kostya’s sum changed to $10^2$. Then the frogs jumped again, and the sum changed to $10^3$ and so on: every new sum was 10 times greater than the previous one. What maximum sum can Kostya obtain?
2016 Tuymaada Olympiad
Is there a positive integer $N>10^{20}$ such that all its decimal digits are odd,

the numbers of digits 1, 3, 5, 7, 9 in its decimal representation are equal,

and it is divisible by each 20-digit number obtained from it by deleting

digits? (Neither deleted nor remaining digits must be consecutive.)
2016 Tuymaada Olympiad
The flights map of air company  $K_{r,r}$ presents several cities. Some cities are connected by a direct (two way) flight, the total number of flights is m. One must choose two non-intersecting groups of r cities each so that every city of the first group is connected by a flight with every city of the second group. Prove that number of possible choices does not exceed $2*m^r$ .
2015 Tuymaada Olympiad
On the football training there was $n$ footballers - forwards and goalkeepers. They made $k$ goals. Prove that main trainer can give for every footballer squad number from $1$ to $n$ such, that for every goal  the difference between squad number of  forward and squad number of goalkeeper is more than $n-k$.



(S. Berlov)
2015 Tuymaada Olympiad
$D$ is midpoint of $AC$ for $\triangle ABC$. Bisectors of $\angle ACB,\angle ABD$ are perpendicular. Find max value for $\angle BAC$



(S. Berlov)
2015 Tuymaada Olympiad
$P(x,y)$ is polynomial with real coefficients and $P(x+2y,x+y)=P(x,y)$. Prove that exists polynomial $Q(t)$ such that $P(x,y)=Q((x^2-2y^2)^2)$



A. Golovanov
2015 Tuymaada Olympiad
Let $n!=ab^2$ where $a$ is free from squares. Prove, that for every $\epsilon>0$ for every big enough $n$ it is true, that $$2^{(1-\epsilon)n}<a<2^{(1+\epsilon)n}$$

M. Ivanov
2015 Tuymaada Olympiad
There is some natural number $n>1$ on the board. Operation is adding to  number on the board it maximal non-trivial divisor. Prove, that after some some operations we get number, that is divisible by $3^{2000}$



A. Golovanov
2015 Tuymaada Olympiad
Let $0 \leq b \leq c \leq d \leq a$ and $a>14$ are integers. Prove, that there is such natural $n$ that can not be represented as $$n=x(ax+b)+y(ay+c)+z(az+d)$$where $x,y,z$ are some integers.



K. Kohas
2015 Tuymaada Olympiad
In $\triangle ABC$ points $M,O$ are midpoint of $AB$ and circumcenter. It is true, that $OM=R-r$. Bisector of external $\angle A$ intersect $BC$ at $D$ and bisector of external $\angle C$ intersect $AB$ at $E$.

Find possible values of $\angle CED$



D. Shiryaev
2015 Tuymaada Olympiad
There are $\frac{k(k+1)}{2}+1$ points on the  planes, some are connected by  disjoint segments ( also  point can not lies on segment, that connects two other points).  It is true, that plane is divided to some parallelograms and one infinite region.  What maximum number of segments can be drawn ?



 A.Kupavski, A. Polyanski
2015 Tuymaada Olympiad
There are $100$ different real numbers. Prove, that we can put it in $10 \times 10$ table, such that difference between two numbers in cells with common side are not equals $1$



A. Golovanov
2015 Tuymaada Olympiad
We call number as funny if it divisible by sum its digits $+1$.(for example $ 1+2+1|12$ ,so $12$ is funny) What is maximum number of consecutive funny numbers ?



 O. Podlipski
2015 Tuymaada Olympiad
Prove that there exists a positive integer $n$ such that in the decimal representation of each of the numbers $\sqrt{n}$,

$\sqrt[3]{n},..., \sqrt[10]{n}$ digits $2015$ stand immediately after the decimal point.



A.Golovanov
2015 Tuymaada Olympiad
Is there sequence $(a_n)$ of natural numbers, such that differences $\{a_{n+1}-a_n\}$ take every natural value and only one time and  differences $\{a_{n+2}-a_n\}$ take every natural value greater $2015$ and only one time ?



A. Golovanov
2015 Tuymaada Olympiad
$CL$ is bisector of $\angle C$ of $ABC$ and intersect circumcircle at $K$. $I$ - incenter of $ABC$. $IL=LK$.

Prove, that $CI=IK$



D. Shiryaev
2015 Tuymaada Olympiad
Four sages stand around a non-transparent baobab. Each of the sages wears red, blue, or green hat. A sage sees only his two neighbors. Each of them at the same time must make a guess about the color of his hat. If at least one sage guesses correctly, the sages win. They could consult before the game started. How should they act to win?
2014 Tuymaada Olympiad
Four consecutive three-digit numbers are divided respectively by four consecutive two-digit numbers. What minimum number of different remainders can be obtained?



(A. Golovanov)
2014 Tuymaada Olympiad
The points $K$ and $L$ on the side $BC$ of a triangle $\triangle{ABC}$ are such that $\widehat{BAK}=\widehat{CAL}=90^\circ$. Prove that the midpoint of the altitude drawn from $A$, the midpoint of $KL$ and the circumcentre of $\triangle{ABC}$ are collinear.



(A. Akopyan, S. Boev, P. Kozhevnikov)
2014 Tuymaada Olympiad
Positive numbers $a,\ b,\ c$ satisfy $\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}=3$. Prove the inequality

$$\dfrac{1}{\sqrt{a^3+1}}+\dfrac{1}{\sqrt{b^3+1}}+\dfrac{1}{\sqrt{c^3+1}}\le \dfrac{3}{\sqrt{2}}. $$



(N. Alexandrov)