Navvye/Olympiad-Problems-CSV huggingface dataset:
2014 Tuymaada Olympiad
A $k\times \ell$ 'parallelogram' is drawn on a paper with hexagonal cells (it consists of $k$ horizontal rows of $\ell$ cells each). In this parallelogram a set of non-intersecting sides of hexagons is chosen; it divides all the vertices into pairs.



Juniors) How many vertical sides can there be in this set?

Seniors) How many ways are there to do that?



[asy]

size(120);

defaultpen(linewidth(0.8));

path hex = dir(30)--dir(90)--dir(150)--dir(210)--dir(270)--dir(330)--cycle;

for(int i=0;i<=3;i=i+1)

{

for(int j=0;j<=2;j=j+1)

{

real shiftx=j*sqrt(3)/2+i*sqrt(3),shifty=j*3/2;

draw(shift(shiftx,shifty)*hex);

}

}

[/asy]

(T. Doslic)
2014 Tuymaada Olympiad
There is an even number of cards on a table; a positive integer is written on each card. Let $a_k$ be the number of cards having $k$ written on them. It is known that

$$a_n-a_{n-1}+a_{n-2}- \cdots \ge 0 $$

for each positive integer $n$. Prove that the cards can be partitioned into pairs so that the numbers in each pair differ by $1$.



(A. Golovanov)
2014 Tuymaada Olympiad
Each of $n$ black squares and $n$ white squares can be obtained by a translation from each other. Every two squares of different colours have a common point. Prove that ther is a point belonging at least to $n$ squares.



(V. Dolnikov)
2014 Tuymaada Olympiad
A parallelogram $ABCD$ is given. The excircle of triangle $\triangle{ABC}$ touches the sides $AB$ at $L$ and the extension of $BC$ at $K$. The line $DK$ meets the diagonal $AC$ at point $X$; the line $BX$ meets the median $CC_1$ of trianlge $\triangle{ABC}$ at ${Y}$. Prove that the line $YL$, median $BB_1$ of triangle $\triangle{ABC}$ and its bisector $CC^\prime$ have a common point.



(A. Golovanov)
2014 Tuymaada Olympiad
Let positive integers $a,\ b,\ c$ be pairwise coprime. Denote by $g(a, b, c)$ the maximum integer not representable in the form $xa+yb+zc$ with positive integral $x,\ y,\ z$. Prove that

$$ g(a, b, c)\ge \sqrt{2abc}$$



(M. Ivanov)



Remarks (containing spoilers!)1. It can be proven that $g(a,b,c)\ge \sqrt{3abc}$.

2. The constant $3$ is the best possible, as proved by the equation $g(3,3k+1,3k+2)=9k+5$.
2014 Tuymaada Olympiad
Given are three different primes. What maximum number of these primes can divide their sum?



(A. Golovanov)
2014 Tuymaada Olympiad
For two quadratic trinomials $P(x)$ and $Q(x)$ there is a linear function $\ell(x)$ such that $P(x)=Q(\ell(x))$ for all real $x$. How many such linear functions $\ell(x)$ can exist?



(A. Golovanov)
2014 Tuymaada Olympiad
Radius of the circle $\omega_A$ with centre at vertex $A$ of a triangle $\triangle{ABC}$ is equal to the radius of the excircle tangent to $BC$. The circles $\omega_B$ and $\omega_C$ are defined similarly. Prove that if two of these circles are tangent then every two of them are tangent to each other.



(L. Emelyanov)
2014 Tuymaada Olympiad
There are $m$ villages on the left bank of the Lena, $n$ villages on the right bank and one village on an island. It is known that $(m+1,n+1)>1$. Every two villages separated by water are connected by ferry with positive integral number.

The inhabitants of each village say that all the ferries operating in their village have different numbers and these numbers form a segment of the series of the integers. Prove that at least some of them are wrong.



(K. Kokhas)
2013 Tuymaada Olympiad
$100$ heaps of stones lie on a table. Two players make moves in turn. At each move, a player can remove any non-zero number of stones from the table, so that at least one heap is left untouched. The player that cannot move loses. Determine, for each initial position, which of the players, the first or the second, has a winning strategy.



K. Kokhas



EDIT. It is indeed confirmed by the sender that empty heaps are still heaps, so the third post contains the right guess of an interpretation.
2013 Tuymaada Olympiad
Points $X$ and $Y$ inside the rhombus $ABCD$ are such that $Y$ is inside the convex quadrilateral $BXDC$ and $2\angle XBY = 2\angle XDY = \angle ABC$. Prove that the lines $AX$ and $CY$ are parallel.



S. Berlov
2013 Tuymaada Olympiad
The vertices of a connected graph cannot be coloured with less than $n+1$ colours (so that adjacent vertices have different colours).

Prove that $\dfrac{n(n-1)}{2}$ edges can be removed from the graph so that it remains connected.



V. Dolnikov



EDIT. It is confirmed by the official solution that the graph is tacitly assumed to be finite.
2013 Tuymaada Olympiad
Prove that if $x$, $y$, $z$ are positive real numbers and $xyz = 1$ then

$$\frac{x^3}{x^2+y}+\frac{y^3}{y^2+z}+\frac{z^3}{z^2+x}\geq \dfrac {3} {2}.$$



A. Golovanov
2013 Tuymaada Olympiad
Prove that every polynomial of fourth degree can be represented  in the form $P(Q(x))+R(S(x))$, where $P,Q,R,S$ are quadratic trinomials.



A. Golovanov



EDIT. It is confirmed that assuming the coefficients to be real, while solving the problem, earned a maximum score.
2013 Tuymaada Olympiad
Solve the equation $p^2-pq-q^3=1$ in prime numbers.



A. Golovanov
2013 Tuymaada Olympiad
Points $A_1$, $A_2$, $A_3$, $A_4$ are the vertices of a regular tetrahedron of edge length $1$. The points $B_1$ and $B_2$ lie inside the figure bounded by the plane $A_1A_2A_3$ and the spheres of radius $1$ and centres $A_1$, $A_2$, $A_3$.

Prove that $B_1B_2 < \max\{B_1A_1, B_1A_2, B_1A_3, B_1A_4\}$.



 A. Kupavsky
2013 Tuymaada Olympiad
Cards numbered from 1 to $2^n$ are distributed among $k$ children, $1\leq k\leq 2^n$, so that each child gets at least one card. Prove that the number of ways to do that is divisible by $2^{k-1}$ but not by $2^k$.



 M. Ivanov
2013 Tuymaada Olympiad
$ABCDEF$ is a convex hexagon, such that in it $AC \parallel DF$, $BD \parallel AE$ and $CE \parallel BF$. Prove that

$$AB^2+CD^2+EF^2=BC^2+DE^2+AF^2.$$



N. Sedrakyan
2013 Tuymaada Olympiad
For every positive real numbers $a$ and $b$ prove the inequality

$$\displaystyle \sqrt{ab} \leq \dfrac{1}{3} \sqrt{\dfrac{a^2+b^2}{2}}+\dfrac{2}{3} \dfrac{2}{\dfrac{1}{a}+\dfrac{1}{b}}.$$



A. Khabrov
2013 Tuymaada Olympiad
Each face of a $7 \times 7 \times 7$ cube is divided into unit squares. What is the maximum number of squares that can be chosen so that no two chosen squares have a common point?



A. Chukhnov
2013 Tuymaada Olympiad
Quadratic trinomials with positive leading coefficients are arranged in the squares of a $6 \times 6$ table. Their $108$  coefficients are all integers from $-60$ to $47$ (each number is used once). Prove that at least in one column the sum of all trinomials has a real root.



K. Kokhas & F. Petrov
2013 Tuymaada Olympiad
The point $A_1$ on the perimeter of a convex quadrilateral $ABCD$ is such that the line $AA_1$  divides the quadrilateral into two parts of equal area. The points $B_1$, $C_1$, $D_1$ are defined similarly.

Prove that the area of the quadrilateral $A_1B_1C_1D_1$ is greater than a quarter of the area of $ABCD$.



L. Emelyanov
2012 Tuymaada Olympiad
Tanya and Serezha take turns putting chips in empty squares of a chessboard. Tanya starts with a chip in an arbitrary square. At every next move, Serezha must put a chip in the column where Tanya put her last chip, while Tanya must put a chip in the row where Serezha put his last chip. The player who cannot make a move loses. Which of the players has a winning strategy?



Proposed by A. Golovanov
2012 Tuymaada Olympiad
Let $P(x)$ be a real quadratic trinomial, so that for all $x\in \mathbb{R}$ the inequality $P(x^3+x)\geq P(x^2+1)$ holds. Find the sum of the roots of $P(x)$.



Proposed by A. Golovanov, M. Ivanov, K. Kokhas
2012 Tuymaada Olympiad
Point $P$ is taken in the interior of the triangle $ABC$, so that

$$\angle PAB = \angle PCB = \dfrac {1} {4} (\angle A + \angle C).$$

Let $L$ be the foot of the angle bisector of $\angle B$. The line $PL$ meets the circumcircle of $\triangle APC$ at point $Q$. Prove that $QB$ is the angle bisector of $\angle AQC$.



Proposed by S. Berlov
2012 Tuymaada Olympiad
Let $p=4k+3$ be a prime. Prove that if

$$\dfrac {1} {0^2+1}+\dfrac{1}{1^2+1}+\cdots+\dfrac{1}{(p-1)^2+1}=\dfrac{m} {n}$$

(where the fraction $\dfrac {m} {n}$ is in reduced terms), then $p \mid 2m-n$.



Proposed by A. Golovanov
2012 Tuymaada Olympiad
Solve in positive integers the following equation:

$${1\over n^2}-{3\over 2n^3}={1\over m^2}$$



Proposed by A. Golovanov
2012 Tuymaada Olympiad
Quadrilateral $ABCD$ is both cyclic and circumscribed. Its incircle touches its sides $AB$ and $CD$ at points $X$ and $Y$, respectively. The perpendiculars to $AB$ and $CD$ drawn at $A$ and $D$, respectively, meet at point $U$; those drawn at $X$ and $Y$ meet at point $V$, and finally, those drawn at $B$ and $C$ meet at point $W$. Prove that points $U$, $V$ and $W$ are collinear.



Proposed by A. Golovanov
2012 Tuymaada Olympiad
Prove that for any real numbers $a,b,c$ satisfying $abc = 1$ the following inequality holds

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



Proposed by V. Aksenov
2012 Tuymaada Olympiad
Integers not divisible by $2012$ are arranged on the arcs of an oriented graph. We call the weight of a vertex the difference between the sum of the numbers on the arcs coming into it and the sum of the numbers on the arcs going away from it. It is known that the weight of each vertex is divisible by $2012$. Prove that non-zero integers with absolute values not exceeding $2012$ can be arranged on the arcs of this graph, so that the weight of each vertex is zero.



Proposed by W. Tutte
2012 Tuymaada Olympiad
A rectangle $ABCD$ is given. Segment $DK$ is equal to $BD$ and lies on the half-line $DC$. $M$ is the midpoint of $BK$. Prove that $AM$ is the angle bisector of $\angle BAC$.



Proposed by S. Berlov
2012 Tuymaada Olympiad
Prove that $N^2$ arbitrary distinct positive integers ($N>10$) can be arranged in a $N\times N$ table, so that all $2N$ sums in rows and columns are distinct.



Proposed by S. Volchenkov
2012 Tuymaada Olympiad
Let $p=1601$. Prove that if

$$\dfrac {1} {0^2+1}+\dfrac{1}{1^2+1}+\cdots+\dfrac{1}{(p-1)^2+1}=\dfrac{m} {n},$$

where we only sum over terms with denominators not divisible by $p$ (and the fraction $\dfrac {m} {n}$ is in reduced terms) then $p \mid 2m+n$.



Proposed by A. Golovanov
2012 Tuymaada Olympiad
The vertices of a regular $2012$-gon are labeled $A_1,A_2,\ldots, A_{2012}$ in some order. It is known that if $k+\ell$ and $m+n$ leave the same remainder when divided by $2012$, then the chords $A_kA_{\ell}$ and $A_mA_n$ have no common points. Vasya walks around the polygon and sees that the first two vertices are labeled $A_1$ and $A_4$. How is the tenth vertex labeled?



Proposed by A. Golovanov
2012 Tuymaada Olympiad
A circle is contained in a quadrilateral with successive sides of lengths $3,6,5$ and $8$. Prove that the length of its radius is less than $3$.



Proposed by K. Kokhas
2012 Tuymaada Olympiad
$25$ little donkeys stand in a row; the rightmost of them is Eeyore. Winnie-the-Pooh wants to give a balloon of one of the seven colours of the rainbow to each donkey, so that successive donkeys receive balloons of different colours, and so that at least one balloon of each colour is given to some donkey. Eeyore wants to give to each of the $24$ remaining donkeys a pot of one of six colours of the rainbow (except red), so that at least one pot of each colour is given to some donkey (but successive donkeys can receive pots of the same colour). Which of the two friends has more ways to get his plan implemented, and how many times more?



Eeyore is a character in the Winnie-the-Pooh books by A. A. Milne. He is generally depicted as a pessimistic, gloomy, depressed, old grey stuffed donkey, who is a friend of the title character, Winnie-the-Pooh. His name is an onomatopoeic representation of the braying sound made by a normal donkey. Of course, Winnie-the-Pooh is a fictional anthropomorphic bear.



Proposed by F. Petrov
2011 Tuymaada Olympiad
Red, blue, and green children are arranged in a circle. When a teacher asked the red children that have a green neighbor to raise their hands, $20$ children raised their hands. When she asked the blue children that have a green neighbor to raise their hands, $25$ children raised their hands. Prove that some child that raised her hand had two green neighbors.
2011 Tuymaada Olympiad
How many ways are there to remove an $11\times11$ square from a $2011\times2011$ square so that the remaining part can be tiled with dominoes ($1\times 2$ rectangles)?
2011 Tuymaada Olympiad
An excircle of triangle $ABC$ touches the side $AB$ at $P$ and the extensions of sides $AC$ and $BC$ at $Q$ and $R$, respectively. Prove that if the midpoint of $PQ$ lies on the circumcircle of $ABC$, then the midpoint of $PR$ also lies on that circumcircle.
2011 Tuymaada Olympiad
Prove that, among $100000$ consecutive $100$-digit positive integers, there is an integer $n$ such that the length of the period of the decimal expansion of $\frac1n$ is greater than $2011$.
2011 Tuymaada Olympiad
Each real number greater than $1$ is coloured red or blue with both colours being used. Prove that there exist real numbers $a$ and $b$ such that the numbers $a+b$ and $ab$ are of different colours.
2011 Tuymaada Olympiad
A circle passing through the vertices $A$ and $B$ of a cyclic quadrilateral $ABCD$ intersects diagonals $AC$ and $BD$ at $E$ and $F$, respectively. The lines $AF$ and $BC$ meet at a point $P$, and the lines $BE$ and $AD$ meet at a point $Q$. Prove that $PQ$ is parallel to $CD$.
2011 Tuymaada Olympiad
In a word of more than $10$ letters, any two consecutive letters are different. Prove that one can change places of two consecutive letters so that the resulting word is not periodic, that is, cannot be divided into equal subwords.
2011 Tuymaada Olympiad
The Duke of Squares left to his three sons a square estate, $100\times 100$ square miles, made up of ten thousand $1\times 1$ square mile square plots. The whole estate was divided among his sons as follows. Each son was assigned a point inside the estate. A $1\times 1$ square plot was bequeathed to the son whose assigned point was closest to the center of this square plot. Is it true that, irrespective of the choice of assigned points, each of the regions bequeathed to the sons is connected (that is, there is a path between every two of its points, never leaving the region)?
2011 Tuymaada Olympiad
Circles $\omega_1$ and $\omega_2$ intersect at points $A$ and $B$, and $M$ is the midpoint of $AB$. Points $S_1$ and $S_2$ lie on the line $AB$ (but not between $A$ and $B$). The tangents drawn from $S_1$ to $\omega_1$ touch it at $X_1$ and $Y_1$, and the tangents drawn from $S_2$ to $\omega_2$ touch it at $X_2$ and $Y_2$. Prove that if the line $X_1X_2$ passes through $M$, then line $Y_1Y_2$ also passes through $M$.
2011 Tuymaada Olympiad
Written in each square of an infinite chessboard is the minimum number of moves needed for a knight to reach that square from a given square $O$. A square is called singular if $100$ is written in it and $101$ is written in all four squares sharing a side with it. How many singular squares are there?
2011 Tuymaada Olympiad
In a set of consecutive positive integers, there are exactly $100$ perfect cubes and $10$ perfect fourth powers. Prove that there are at least $2000$ perfect squares in the set.
2011 Tuymaada Olympiad
Each real number greater than 1 is colored red or blue with both colors being used. Prove that there exist real numbers $a$ and $b$ such that the numbers $a+\frac1b$ and $b+\frac1a$ are different colors.
2011 Tuymaada Olympiad
In a convex hexagon $AC'BA'CB'$, every two opposite sides are equal. Let $A_1$ denote the point of intersection of $BC$ with the perpendicular bisector of $AA'$. Define $B_1$ and $C_1$ similarly. Prove that $A_1$, $B_1$, and $C_1$ are collinear.
2011 Tuymaada Olympiad
Let $P(n)$ be a quadratic trinomial with integer coefficients.  For each positive integer $n$, the number $P(n)$ has a proper divisor $d_n$, i.e., $1<d_n<P(n)$, such that the sequence $d_1,d_2,d_3,\ldots$ is increasing. Prove that either $P(n)$ is the product of two linear polynomials with integer coefficients or all the values of $P(n)$, for positive integers $n$, are divisible by the same integer $m>1$.
2010 Tuymaada Olympiad
Misha and Sahsa play a game on a $100\times 100$ chessboard. First, Sasha places $50$ kings on the board, and Misha places a rook, and then they move in turns, as following (Sasha begins):



At his move, Sasha moves each of the kings one square in any direction, and Misha can move the rook on the horizontal or vertical any number of squares. The kings cannot be captured or stepped over. Sasha's purpose is to capture the rook, and Misha's is to avoid capture.



Is there a winning strategy available for Sasha?
2010 Tuymaada Olympiad
Let $ABC$ be an acute triangle, $H$ its orthocentre, $D$ a point on the side $[BC]$, and $P$ a point such that $ADPH$ is a parallelogram.

Show that $\angle BPC > \angle BAC$.
2010 Tuymaada Olympiad
Let $f(x) = ax^2+bx+c$ be a quadratic trinomial with $a$,$b$,$c$ reals such that any quadratic trinomial obtained by a permutation of $f$'s coefficients has an integer root (including $f$ itself).

Show that $f(1)=0$.
2010 Tuymaada Olympiad
On a blackboard there are $2010$ natural nonzero numbers. We define a "move" by erasing $x$ and $y$ with $y\neq0$ and replacing them with $2x+1$ and $y-1$, or we can choose to replace them by $2x+1$ and $\frac{y-1}{4}$ if $y-1$ is divisible by 4.



Knowing that in the beginning the numbers $2006$ and $2008$ have been erased, show that the original set of numbers cannot be attained again by any sequence of moves.
2010 Tuymaada Olympiad
We have a set $M$ of real numbers with $|M|>1$ such that for any $x\in M$ we have either $3x-2\in M$ or $-4x+5\in M$.

Show that $M$ is infinite.
2010 Tuymaada Olympiad
We have a number $n$ for which we can find 5 consecutive numbers, none of which is divisible by $n$, but their product is.

Show that we can find 4 consecutive numbers, none of which is divisible by $n$, but their product is.
2010 Tuymaada Olympiad
Let $ABC$ be a triangle, $I$ its incenter, $\omega$ its incircle, $P$ a point such that $PI\perp BC$ and $PA\parallel BC$, $Q\in (AB), R\in (AC)$ such that $QR\parallel BC$ and $QR$ tangent to $\omega$.

Show that $\angle QPB = \angle CPR$.
2010 Tuymaada Olympiad
Direct one-way flights run between some cities in a certain country. Prove that there is a nonempty set $A$ of cities such that



there is no flight between any two cities of $A$; and

from every city not in $A$, one can reach some city of $A$ either by a direct flight or by two flights with one change.
2010 Tuymaada Olympiad
In acute triangle $ABC$, let $H$ denote its orthocenter and let $D$ be a point on side $BC$. Let $P$ be the point so that $ADPH$ is a parallelogram. Prove that $\angle DCP<\angle BHP$.
2010 Tuymaada Olympiad
Arranged in a circle are $2010$ digits, each of them equal to $1$, $2$, or $3$. For each positive integer $k$, it's known that in any block of $3k$ consecutive digits, each of the digits appears at most $k+10$ times. Prove that there is a block of several consecutive digits with the same number of $1$s, $2$s, and $3$s.
2010 Tuymaada Olympiad
Prove that for any positive real number $\alpha$, the number $\lfloor\alpha n^2\rfloor$ is even for infinitely many positive integers $n$.
2010 Tuymaada Olympiad
Baron Münchausen boasts that he knows a remarkable quadratic triniomial with positive coefficients. The trinomial has an integral root; if all of its coefficients are increased by $1$, the resulting trinomial also has an integral root; and if all of its coefficients are also increased by $1$, the new trinomial, too, has an integral root. Can this be true?
2010 Tuymaada Olympiad
For a given positive integer $n$, it's known that there exist $2010$ consecutive positive integers such that none of them is divisible by $n$ but their product is divisible by $n$. Prove that there exist $2004$ consecutive positive integers such that none of them is divisible by $n$ but their product is divisible by $n$.
2010 Tuymaada Olympiad
In a country there are $4^9$ schoolchildren living in four cities. At the end of the school year a state examination was held in 9 subjects. It is known that any two students have different marks at least in one subject.  However, every two students from the same city got equal marks at least in one subject. Prove that there is a subject such that every two children living in the same city have equal marks in this subject.



Fedor Petrov
2009 Tuymaada Olympiad
All squares of a $ 20\times 20$ table are empty. Misha* and Sasha** in turn put chips in free squares (Misha* begins). The player after whose move there are four chips on the intersection of two rows and two columns wins. Which of the players has a winning strategy?



Proposed by A. Golovanov





US Name Conversions: 



Misha*: Naoki

Sasha**: Richard
2009 Tuymaada Olympiad
$ P(x)$ is a quadratic trinomial. What maximum number of terms equal to the sum of the two preceding terms can occur in the sequence $ P(1)$, $ P(2)$, $ P(3)$, $ \dots?$



Proposed by A. Golovanov
2009 Tuymaada Olympiad
In a cyclic quadrilateral $ ABCD$ the sides $ AB$ and $ AD$ are equal, $ CD>AB+BC$. Prove that $ \angle ABC>120^\circ$.
2009 Tuymaada Olympiad
Each of the subsets $ A_1$, $ A_2$, $ \dots,$ $ A_n$ of a 2009-element set $ X$ contains at least 4 elements. The intersection of every two of these subsets contains at most 2 elements. Prove that in $ X$ there is a 24-element subset $ B$ containing neither of the sets $ A_1$, $ A_2$, $ \dots,$ $ A_n$.
2009 Tuymaada Olympiad
A magician asked a spectator to think of a three-digit number $ \overline{abc}$ and then to tell him the sum of numbers $ \overline{acb}$, $ \overline{bac}$, $ \overline{bca}$, $ \overline{cab}$, and $ \overline{cba}$. He claims that when he knows this sum he can determine the original number. Is that so?
2009 Tuymaada Olympiad
$ M$ is the midpoint of base $ BC$ in a trapezoid $ ABCD$. A point $ P$ is chosen on the base $ AD$. The line $ PM$ meets the line $ CD$ at a point $ Q$ such that $ C$ lies between $ Q$ and $ D$. The perpendicular to the bases drawn through $ P$ meets the line $ BQ$ at $ K$. Prove that $ \angle QBC = \angle KDA$.



Proposed by S. Berlov
2009 Tuymaada Olympiad
An arrangement of chips in the squares of $ n\times n$ table is called sparse if every $ 2\times 2$ square contains at most 3 chips. Serge put chips in some squares of the table (one in a square) and obtained a sparse arrangement. He noted however that if any chip is moved to any free square then the arrangement is no more sparce. For what $ n$ is this possible?



Proposed by S. Berlov
2009 Tuymaada Olympiad
The sum of several non-negative numbers is not greater than 200, while the sum of their squares is not less than 2500. Prove that among them there are four numbers whose sum is not less than 50.



Proposed by A. Khabrov
2009 Tuymaada Olympiad
Three real numbers are given. Fractional part of the product of every two of them is $ 1\over 2$. Prove that these numbers are irrational.



Proposed by A. Golovanov
2009 Tuymaada Olympiad
A necklace consists of 100 blue and several red beads. It is known that every segment of the necklace containing 8 blue beads contain also at least 5 red beads. What minimum number of red beads can be in the necklace?



Proposed by A. Golovanov
2009 Tuymaada Olympiad
On the side $ AB$ of a cyclic quadrilateral $ ABCD$ there is a point $ X$ such that diagonal $ BD$ bisects $ CX$ and diagonal $ AC$ bisects $ DX$. What is the minimum possible value of $ AB\over CD$?



Proposed by S. Berlov
2009 Tuymaada Olympiad
Is there a positive integer $ n$ such that among 200th digits after decimal point in the decimal representations of $ \sqrt{n}$, $ \sqrt{n+1}$, $ \sqrt{n+2}$, $ \ldots,$ $ \sqrt{n+999}$ every digit occurs 100 times?



Proposed by A. Golovanov
2009 Tuymaada Olympiad
A triangle $ ABC$ is given. Let $ B_1$ be the reflection of $ B$ across the line $ AC$, $ C_1$ the reflection of $ C$ across the line $ AB$, and $ O_1$ the reflection of the circumcentre of $ ABC$ across the line $ BC$. Prove that the circumcentre of $ AB_1C_1$ lies on the line $ AO_1$.



Proposed by A. Akopyan