Navvye/Olympiad-Problems-CSV huggingface dataset:
2010 All-Russian Olympiad
There are $100$ random, distinct real numbers corresponding to $100$ points on a circle. Prove that you can always choose $4$ consecutive points in such a way that the sum of the two numbers corresponding to the points on the outside is always greater than the sum of the two numbers corresponding to the two points on the inside.
2010 All-Russian Olympiad
Lines tangent to circle $O$ in points $A$ and $B$, intersect in point $P$. Point $Z$ is the center of $O$. On the minor arc $AB$, point $C$ is chosen not on the midpoint of the arc. Lines $AC$ and $PB$ intersect at point $D$. Lines $BC$ and $AP$ intersect at point $E$. Prove that the circumcentres of triangles $ACE$, $BCD$, and $PCZ$ are collinear.
2010 All-Russian Olympiad
There are 100 apples on the table with total weight of 10 kg. Each apple weighs no less than 25 grams. The apples need to be cut for 100 children so that each of the children gets 100 grams. Prove that you can do it in such a way that each piece weighs no less than 25 grams.
2010 All-Russian Olympiad
Let $a \neq b a,b \in \mathbb{R}$ such that $(x^2+20ax+10b)(x^2+20bx+10a)=0$ has no roots for $x$. Prove that $20(b-a)$ is not an integer.
2010 All-Russian Olympiad
Each of $1000$ elves has a hat, red on the inside and blue on the outside or vise versa. An elf with a hat that is red outside can only lie, and an elf with a hat that is blue outside can only tell the truth. One day every elf tells every other elf, “Your hat is red on the outside.” During that day, some of the elves turn their hats inside out at any time during the day. (An elf can do that more than once per day.) Find the smallest possible number of times any hat is turned inside out.
2010 All-Russian Olympiad
Let us call a natural number $unlucky$ if it cannot be expressed as $\frac{x^2-1}{y^2-1} $ with natural numbers $x,y >1$. Is the number of $unlucky$ numbers finite or infinite?
2010 All-Russian Olympiad
In a acute triangle $ABC$, the median, $AM$, is longer than side $AB$. Prove that you can cut triangle $ABC$ into $3$ parts out of which you can construct a rhombus.
2010 All-Russian Olympiad
Let $O$ be the circumcentre of the acute non-isosceles triangle $ABC$. Let $P$ and $Q$ be points on the altitude $AD$ such that $OP$ and $OQ$ are perpendicular to $AB$ and $AC$ respectively. Let $M$ be the midpoint of $BC$ and $S$ be the circumcentre of triangle $OPQ$. Prove that $\angle BAS =\angle CAM$.
2010 All-Russian Olympiad
In each unit square of square $100*100$ write any natural number. Called rectangle with sides parallel sides of square $good$ if sum of number inside rectangle divided by $17$. We can painted all unit squares in $good$ rectangle. One unit square cannot painted twice or more.

Find maximum $d$ for which we can guaranteed paint at least $d$ points.
2010 All-Russian Olympiad
Into triangle $ABC$ gives point $K$ lies on bisector of $ \angle BAC$. Line $CK$ intersect circumcircle $ \omega$ of triangle $ABC$ at $M \neq C$. Circle $ \Omega$ passes through $A$, touch $CM$ at $K$ and intersect segment $AB$ at $P \neq A$ and $\omega $ at $Q \neq A$.

Prove, that $P$, $Q$, $M$ lies at one line.
2010 All-Russian Olympiad
Given $n \geq 3$ pairwise different prime numbers $p_1, p_2, ....,p_n$. Given, that for any $k \in \{ 1,2,....,n \}$ residue by division of $ \prod_{i \neq k} p_i$ by $p_k$ equals one number $r$. Prove, that $r \leq n-2 $.
2010 All-Russian Olympiad
In the county some pairs of towns connected by two-way non-stop flight. From any town we can flight to any other (may be not on one flight). Gives, that if we consider any cyclic (i.e. beginning and finish towns match) route, consisting odd number of flights, and close all flights of this route, then we can found two towns, such that we can't fly from one to other.

Proved, that we can divided all country on $4$ regions, such that any flight connected towns from other regions.
2010 All-Russian Olympiad
ِDo there exist non-zero reals numbers $a_1, a_2, ....., a_{10}$ for which $$(a_1+\frac{1}{a_1})(a_2+\frac{1}{a_2}) \cdots(a_{10}+\frac{1}{a_{10}})= (a_1-\frac{1}{a_1})(a_2-\frac{1}{a_2})\cdots(a_{10}-\frac{1}{a_{10}}) \ ? $$
2010 All-Russian Olympiad
On an $n\times n$ chart, where $n \geq 4$, stand "$+$" signs in the cells of the main diagonal and "$-$" signs in all the other cells. You can change all the signs in one row or in one column, from $-$ to $+$ or from $+$ to $-$. Prove that you will always have $n$ or more $+$ signs after finitely many operations.
2010 All-Russian Olympiad
Quadrilateral $ABCD$ is inscribed into circle $\omega$, $AC$ intersect $BD$ in point $K$. Points $M_1$, $M_2$, $M_3$, $M_4$-midpoints of arcs $AB$, $BC$, $CD$, and $DA$ respectively. Points $I_1$, $I_2$, $I_3$, $I_4$-incenters of triangles $ABK$, $BCK$, $CDK$, and $DAK$ respectively. Prove that lines $M_1I_1$, $M_2I_2$, $M_3I_3$, and $M_4I_4$ all intersect in one point.
2010 All-Russian Olympiad
Given is a natural number $n \geq 3$. What is the smallest possible value of $k$ if the following statements are true?

For every $n$ points $ A_i = (x_i, y_i) $ on a plane, where no three points are collinear, and for any real numbers $ c_i$ ($1 \le i \le n$) there exists such polynomial $P(x, y)$, the degree of which is no more than $k$, where $ P(x_i, y_i) = c_i $ for every $i = 1, \dots, n$.



(The degree of a nonzero monomial $ a_{i,j} x^{i}y^{j} $ is $i+j$, while the degree of polynomial $P(x, y)$ is the greatest degree of the degrees of its monomials.)
2010 All-Russian Olympiad
If $n \in \mathbb{N} n > 1$ prove that for every $n$ you can find $n$ consecutive natural numbers the product of which is divisible by all primes not exceeding $2n+1$, but is not divisible by any other primes.
2010 All-Russian Olympiad
Could the four centers of the circles inscribed into the faces of a tetrahedron be coplanar?



(vertexes of tetrahedron not coplanar)
2010 All-Russian Olympiad
Polynomial $P(x)$ with degree $n \geq 3$ has $n$ real roots  $x_1 < x_2 < x_3 <...< x_n$, such that $x_2-x_1<x_3-x_2<....<x_n-x_{n-1}$. Prove that the maximum of the function $y=|P(x)|$ where $x$ is on the interval $[ x_1, x_n ]$, is in the interval $[x_n-1, x_n]$.
2010 All-Russian Olympiad
In a board school, there are 9 subjects, 512 students, and 256 rooms (two people in each room.) For every student there is a set (a subset of the 9 subjects) of subjects the student is interested in. Each student has a different set of subjects, (s)he is interested in, from all other students. (Exactly one student has no subjects (s)he is interested in.)

Prove that the whole school can line up in a circle in such a way that every pair of the roommates has the two people standing next to each other, and those pairs of students standing next to each other that are not roommates, have the following properties. One of the two students is interested in all the subjects that the other student is interested in, and also exactly one more subject.
2009 All-Russian Olympiad
The denominators of two irreducible fractions are 600 and 700. Find the minimum value of the denominator of their sum (written as an irreducible fraction).
2009 All-Russian Olympiad
Let be given a triangle $ ABC$ and its internal angle bisector $ BD$ $ (D\in BC)$. The line $ BD$ intersects the circumcircle $ \Omega$ of triangle $ ABC$ at $ B$ and $ E$. Circle $ \omega$ with diameter $ DE$ cuts $ \Omega$ again at $ F$. Prove that $ BF$ is the symmedian line of triangle $ ABC$.
2009 All-Russian Olympiad
Given are positive integers $ n>1$ and $ a$ so that $ a>n^2$, and among the integers $ a+1, a+2, \ldots, a+n$ one can find a multiple of each of the numbers $ n^2+1, n^2+2, \ldots, n^2+n$. Prove that $ a>n^4-n^3$.
2009 All-Russian Olympiad
There are n cups arranged on the circle. Under one of cups is hiden a coin. For every move, it is allowed to choose 4 cups and verify if the coin lies under these cups. After that, the cups are returned into its former places and the coin moves to one of two neigbor cups. What is the minimal number of moves we need in order to eventually find where the coin is?
2009 All-Russian Olympiad
Let $ a$, $ b$, $ c$ be three real numbers satisfying that $$ \left\{\begin{array}{c c c} \left(a+b\right)\left(b+c\right)\left(c+a\right)&=&abc\\ \left(a^3+b^3\right)\left(b^3+c^3\right)\left(c^3+a^3\right)&=&a^3b^3c^3\end{array}\right.$$ Prove that $ abc=0$.
2009 All-Russian Olympiad
Can be colored the positive integers with 2009 colors if we know that each color paints infinitive integers and that we can not find three numbers colored by three different colors for which the product of two numbers equal to the third one?
2009 All-Russian Olympiad
We call any eight squares in a diagonal of a chessboard as a fence. The rook is moved on the chessboard in such way that he stands neither on each square over one time nor on the squares of the fences (the squares which the rook passes is not considered ones it has stood on). Then what is the maximum number of times which the rook jumped over the fence?
2009 All-Russian Olympiad
Triangles $ ABC$ and $ A_1B_1C_1$ have the same area. Using compass and ruler, can we always construct triangle $ A_2B_2C_2$ equal to triangle $ A_1B_1C_1$ so that the lines $ AA_2$, $ BB_2$, and $ CC_2$ are parallel?
2009 All-Russian Olympiad
Find all value of $ n$ for which there are nonzero real numbers $ a, b, c, d$ such that after expanding and collecting similar terms, the polynomial $ (ax + b)^{100} - (cx + d)^{100}$ has exactly $ n$ nonzero coefficients.
2009 All-Russian Olympiad
How many times changes the sign of the function $$ f(x)=\cos x\cos\frac{x}{2}\cos\frac{x}{3}\cdots\cos\frac{x}{2009}$$ at the interval $ \left[0, \frac{2009\pi}{2}\right]$?
2009 All-Russian Olympiad
On a circle there are 2009 nonnegative integers not greater than 100. If two numbers sit next to each other, we can increase both of them by 1. We can do this at most $ k$ times. What is the minimum $ k$ so that we can make all the numbers on the circle equal?
2009 All-Russian Olympiad
Given strictly increasing sequence $ a_1<a_2<\dots$ of positive integers such that each its term $ a_k$ is divisible either by 1005 or 1006, but neither term is divisible by $ 97$. Find the least possible value of maximal difference of consecutive terms $ a_{i+1}-a_i$.
2009 All-Russian Olympiad
Given a finite tree $ T$ and isomorphism $ f: T\rightarrow T$. Prove that either there exist a vertex $ a$ such that $ f(a)=a$ or there exist two neighbor vertices $ a$, $ b$ such that $ f(a)=b$, $ f(b)=a$.
2009 All-Russian Olympiad
The incircle $ (I)$ of a given scalene triangle $ ABC$ touches its sides $ BC$, $ CA$, $ AB$ at $ A_1$, $ B_1$, $ C_1$, respectively. Denote $ \omega_B$, $ \omega_C$ the incircles of quadrilaterals $ BA_1IC_1$ and $ CA_1IB_1$, respectively. Prove that the internal common tangent of $ \omega_B$ and $ \omega_C$ different from $ IA_1$ passes through $ A$.
2009 All-Russian Olympiad
Let $ x$, $ y$ be two integers with $ 2\le x, y\le 100$. Prove that $ x^{2^n} + y^{2^n}$ is not a prime for some positive integer $ n$.
2009 All-Russian Olympiad
In a country, there are some cities linked together by roads. The roads just meet each other inside the cities. In each city, there is a board which showing the shortest length of the road originating in that city and going through all other cities (the way can go through some cities more than one times and is not necessary to turn back to the originated city). Prove that 2 random numbers in the boards can't be greater or lesser than 1.5 times than each other.
2009 All-Russian Olympiad
Consider the sequence of numbers $(a_n)$ ($n = 1, 2, \ldots$) defined as follows: $ a_1\in (1, 2)$, $ a_{k + 1} = a_k + \frac{k}{a_k}$ ($k = 1, 2, \ldots$). Prove that there exists at most one pair of distinct positive integers $(i, j)$ such that $a_i + a_j$ is an integer.
2009 All-Russian Olympiad
Let $ ABCD$ be a triangular pyramid such that no face of the pyramid is a right triangle and the orthocenters of triangles $ ABC$, $ ABD$, and $ ACD$ are collinear. Prove that the center of the sphere circumscribed to the pyramid lies on the plane passing through the midpoints of $ AB$, $ AC$ and $ AD$.
2009 All-Russian Olympiad
Given a set $ M$ of points $ (x,y)$ with integral coordinates satisfying $ x^2 + y^2\leq 10^{10}$. Two players play a game. One of them marks a point on his first move. After this, on each move the moving player marks a point, which is not yet marked and joins it with the previous marked point. Players are not allowed to mark a point symmetrical to the one just chosen. So, they draw a broken line. The requirement is that lengths of edges of this broken line must strictly increase. The player, which can not make a move, loses. Who have a winning strategy?
2009 All-Russian Olympiad
Prove that $$ \log_ab+\log_bc+\log_ca\le \log_ba+\log_cb+\log_ac$$ for all $ 1<a\le b\le c$.
2009 All-Russian Olympiad
There are $ k$ rooks on a $ 10 \times 10$ chessboard. We mark all the squares that at least one rook can capture (we consider the square where the rook stands as captured by the rook). What is the maximum value of $ k$ so that the following holds for some arrangement of $ k$ rooks: after removing any rook from the chessboard, there is at least one marked square not captured by any of the remaining rooks.
2009 All-Russian Olympiad
Let be given a parallelogram $ ABCD$ and two points $ A_1$, $ C_1$ on its sides $ AB$, $ BC$, respectively. Lines $ AC_1$ and $ CA_1$ meet at $ P$. Assume that the circumcircles of triangles $ AA_1P$ and $ CC_1P$ intersect at the second point $ Q$ inside triangle $ ACD$. Prove that $ \angle PDA = \angle QBA$.
2008 All-Russian Olympiad
Do there exist $ 14$ positive integers, upon increasing each of them by $ 1$,their product increases exactly $ 2008$ times?
2008 All-Russian Olympiad
Numbers $ a,b,c$ are such that the equation $ x^3 + ax^2 + bx + c$ has three real roots.Prove that if $ - 2\leq a + b + c\leq 0$,then at least one of these roots belongs to the segment $ [0,2]$
2008 All-Russian Olympiad
In a scalene triangle $ ABC, H$ and $ M$ are the orthocenter an centroid respectively. Consider the triangle formed by the lines through $ A,B$ and $ C$ perpendicular to $ AM,BM$ and $ CM$ respectively. Prove that the centroid of this triangle lies on the line $ MH$.
2008 All-Russian Olympiad
There are several scientists collaborating in Niichavo. During an $ 8$-hour working day, the scientists went to cafeteria, possibly several times.It is known that for every two scientist, the total time in which exactly one of them was in cafeteria is at least $ x$ hours ($ x>4$).

What is the largest possible number of scientist that could work in Niichavo that day,in terms of $ x$?
2008 All-Russian Olympiad
The distance between two cells of an infinite chessboard is defined as the minimum nuber to moves needed for a king for move from one to the other.One the board are chosen three cells on a pairwise distances equal to $ 100$. How many cells are there that are on the distance $ 50$ from each of the three cells?
2008 All-Russian Olympiad
The incircle of a triangle $ABC$ touches the side $AB$ and $AC$ at respectively at $X$ and $Y$. Let $K$ be the midpoint of the arc $\widehat{AB}$ on the circumcircle of $ABC$. Assume that $XY$ bisects the segment $AK$. What are the possible measures of angle $BAC$?
2008 All-Russian Olympiad
A natural number is written on the blackboard. Whenever number $ x$ is written, one can write any of the numbers $ 2x + 1$ and $ \frac {x}{x + 2}$. At some moment the number $ 2008$ appears on the blackboard. Show that it was there from the very beginning.
2008 All-Russian Olympiad
We are given $ 3^{2k}$ apparently identical coins,one of which is fake,being lighter than the others. We also dispose of three apparently identical balances without weights, one of which is broken (and yields outcomes unrelated to the actual situations). How can we find the fake coin in $ 3k+1$ weighings?
2008 All-Russian Olympiad
The columns of an $ n\times n$ board are labeled $ 1$ to $ n$. The numbers $ 1,2,...,n$ are arranged in the board so that the numbers in each row and column are pairwise different. We call a cell "good" if the number in it is greater than the label of its column. For which $ n$ is there an arrangement in which each row contains equally many good cells?
2008 All-Russian Olympiad
A circle $ \omega$ with center $ O$ is tangent to the rays of an angle $ BAC$ at $ B$ and $ C$. Point $ Q$ is taken inside the angle $ BAC$. Assume that point $ P$ on the segment $ AQ$ is such that $ AQ\perp OP$. The line $ OP$ intersects the circumcircles $ \omega_{1}$ and $ \omega_{2}$ of triangles $ BPQ$ and $ CPQ$ again at points $ M$ and $ N$. Prove that $ OM = ON$.
2008 All-Russian Olympiad
The sequences $ (a_n),(b_n)$ are defined by $ a_1=1,b_1=2$ and $$a_{n + 1} = \frac {1 + a_n + a_nb_n}{b_n}, \quad b_{n + 1} = \frac {1 + b_n + a_nb_n}{a_n}.$$

Show that $ a_{2008} < 5$.
2008 All-Russian Olympiad
Determine all triplets of real numbers $ x,y,z$ satisfying $$1+x^4\leq 2(y-z)^2,\quad  1+y^4\leq 2(x-z)^2,\quad  1+z^4\leq 2(x-y)^2.$$
2008 All-Russian Olympiad
In a scalene triangle $ ABC$ the altitudes $ AA_{1}$ and $ CC_{1}$ intersect at $ H, O$ is the circumcenter, and $ B_{0}$ the midpoint of side $ AC$. The line $ BO$ intersects side $ AC$ at $ P$, while the lines $ BH$ and $ A_{1}C_{1}$ meet at $ Q$. Prove that the lines $ HB_{0}$ and $ PQ$ are parallel.
2008 All-Russian Olympiad
For which integers $ n>1$ do there exist natural numbers $ b_1,b_2,\ldots,b_n$ not all equal such that the number $ (b_1+k)(b_2+k)\cdots(b_n+k)$ is a power of an integer for each natural number $ k$? (The exponents may depend on $ k$, but must be greater than $ 1$)
2008 All-Russian Olympiad
On the cartesian plane are drawn several rectangles with the sides parallel to the coordinate axes. Assume that any two rectangles can be cut by a vertical or a horizontal line. Show that it's possible to draw one horizontal and one vertical line such that each rectangle is cut by at least one of these two lines.
2008 All-Russian Olympiad
Petya and Vasya are given equal sets of $ N$ weights, in which the masses of any two weights are in ratio at most $ 1.25$. Petya succeeded to divide his set into $ 10$ groups of equal masses, while Vasya succeeded to divide his set into $ 11$ groups of equal masses. Find the smallest possible $ N$.
2008 All-Russian Olympiad
Given a finite set $ P$ of prime numbers, prove that there exists a positive integer $ x$ such that it can be written in the form $ a^p + b^p$ ($ a,b$ are positive integers), for each $ p\in P$, and cannot be written in that form for each $ p$ not in $ P$.
2008 All-Russian Olympiad
Each face of a tetrahedron can be placed in a circle of radius $ 1$. Show that the tetrahedron can be placed in a sphere of radius $ \frac{3}{2\sqrt2}$.
2008 All-Russian Olympiad
The numbers from $ 51$ to $ 150$ are arranged in a $ 10\times 10$ array. Can this be done in such a way that, for any two horizontally or vertically adjacent numbers $ a$ and $ b$, at least one of the equations $ x^2 - ax + b = 0$ and $ x^2 - bx + a = 0$ has two integral roots?
2008 All-Russian Olympiad
A magician should determine the area of a hidden convex $ 2008$-gon $ A_{1}A_{2}\cdots A_{2008}$. In each step he chooses two points on the perimeter, whereas the chosen points can be vertices or points dividing selected sides in selected ratios. Then his helper divides the polygon into two parts by the line through these two points and announces the area of the smaller of the two parts. Show that the magician can find the area of the polygon in $ 2006$ steps.
2008 All-Russian Olympiad
In convex quadrilateral $ ABCD$, the rays $ BA,CD$ meet at $ P$, and the rays $ BC,AD$ meet at $ Q$. $ H$ is the projection of $ D$ on $ PQ$. Prove that there is a circle inscribed in $ ABCD$ if and only if the incircles of triangles $ ADP,CDQ$ are visible from $ H$ under the same angle.
2008 All-Russian Olympiad
In a chess tournament $ 2n+3$ players take part. Every two play exactly one match. The schedule is such that no two matches are played at the same time, and each player, after taking part in a match, is free in at least $ n$ next (consecutive) matches. Prove that one of the players who play in the opening match will also play in the closing match.
2007 All-Russian Olympiad
Given reals numbers $a$, $b$, $c$. Prove that at least one of three equations $x^{2}+(a-b)x+(b-c)=0$, $x^{2}+(b-c)x+(c-a)=0$, $x^{2}+(c-a)x+(a-b)=0$  has a real root.

O. Podlipsky
2007 All-Russian Olympiad
The numbers $1,2,\ldots,100$ are written in the cells of a $10\times 10$ table, each number is written once. In one move, Nazar may interchange numbers in any two cells. Prove that he may get a table where the sum of the numbers in every two adjacent (by side) cells is composite after at most $35$ such moves.

N. Agakhanov
2007 All-Russian Olympiad
Given a rhombus $ABCD$. A point $M$ is chosen on its side $BC$. The lines, which pass through $M$ and are perpendicular to $BD$ and $AC$, meet line $AD$ in points $P$ and $Q$ respectively. Suppose that the lines $PB,QC,AM$ have a common point. Find all possible values of a ratio $\frac{BM}{MC}$.

S. Berlov, F. Petrov, A. Akopyan
2007 All-Russian Olympiad
A. Akopyan, A. Akopyan, A. Akopyan, I. Bogdanov



A conjurer Arutyun and his assistant Amayak are going to show following super-trick. A circle is drawn on the board in the room. Spectators mark $2007$ points on this circle, after that Amayak

removes one of them. Then Arutyun comes to the room and shows a semicircle, to which the removed point belonged. Explain, how Arutyun and Amayak may show this super-trick.
2007 All-Russian Olympiad
The distance between Maykop and Belorechensk is $24$ km. Two of three friends need to reach Belorechensk from Maykop and another friend wants to reach Maykop from Belorechensk. They have only one bike, which is initially in Maykop. Each guy may go on foot (with velocity at most $6$ kmph) or on a bike (with velocity at most $18$ kmph). It is forbidden to leave a bike on a road. Prove that all of them may achieve their goals after $2$ hours $40$ minutes. (Only one guy may seat on the bike simultaneously).



Folclore
2007 All-Russian Olympiad
A line, which passes through the incentre $I$ of the triangle $ABC$, meets its sides $AB$ and $BC$ at the points $M$ and $N$ respectively. The triangle $BMN$ is acute. The points $K,L$ are chosen on the side $AC$ such that $\angle ILA=\angle IMB$ and $\angle KC=\angle INB$. Prove that $AM+KL+CN=AC$.

S. Berlov
2007 All-Russian Olympiad
For an integer $n>3$ denote by $n?$ the product of all primes less than $n$. Solve the equation $n?=2n+16$.



V. Senderov
2007 All-Russian Olympiad
Given a matrix $\{a_{ij}\}_{i,j=0}^{9}$, $a_{ij}=10i+j+1$. Andrei is going to cover its entries by $50$ rectangles $1\times 2$ (each such rectangle contains two adjacent entries) so that the sum of $50$ products in these rectangles is minimal possible. Help him.

A. Badzyan
2007 All-Russian Olympiad
Unitary quadratic trinomials $ f(x)$ and $ g(x)$ satisfy the following interesting condition: $ f(g(x)) = 0$ and $ g(f(x)) = 0$ do not have real roots. Prove that at least one of equations $ f(f(x)) = 0$ and $ g(g(x)) = 0$ does not have real roots too.



S. Berlov
2007 All-Russian Olympiad
$100$ fractions are written on a board, their numerators are numbers from $1$ to $100$ (each once) and denominators are also numbers from $1$ to $100$ (also each once). It appears that the sum of these fractions equals to $a/2$ for some odd $a$. Prove that it is possible to interchange numerators of two fractions so that sum becomes a fraction with odd denominator.

N. Agakhanov, I. Bogdanov
2007 All-Russian Olympiad
Two players by turns draw diagonals in a regular $(2n+1)$-gon ($n>1$). It is forbidden to draw a diagonal, which was already drawn, or intersects an odd number of already drawn diagonals. The player, who has no legal move, loses. Who has a winning strategy?

K. Sukhov
2007 All-Russian Olympiad
$BB_{1}$ is a bisector of an acute triangle $ABC$. A perpendicular from $B_{1}$ to $BC$ meets a smaller arc $BC$ of a circumcircle of $ABC$ in a point $K$.  A perpendicular from $B$ to $AK$ meets $AC$ in a point $L$. $BB_{1}$ meets arc $AC$ in $T$. Prove that $K$, $L$, $T$ are collinear.

V. Astakhov