Navvye/Olympiad-Problems-CSV huggingface dataset:
1996 All-Russian Olympiad
Show that in the arithmetic progression with first term 1 and ratio 729, there are infinitely many powers of 10.



L. Kuptsov
1996 All-Russian Olympiad
In the isosceles triangle $ABC$ ($AC = BC$) point $O$ is the circumcenter, $I$ the incenter, and $D$ lies on $BC$ so that lines $OD$ and $BI$ are perpendicular. Prove that $ID$ and $AC$ are parallel.



M. Sonkin
1996 All-Russian Olympiad
Two piles of coins lie on a table. It is known that the sum of the weights of the coins in the two piles are equal, and for any natural number $k$, not exceeding the number of coins in either pile, the sum of the weights of the $k$ heaviest coins in the first pile is not more than that of the second pile. Show that for any natural number $x$, if each coin (in either pile) of weight not less than $x$ is replaced by a coin of weight $x$, the first pile will not be lighter than the second.



D. Fon-der-Flaas
1996 All-Russian Olympiad
Can a $5\times 7$ checkerboard be covered by L's (figures formed from a $2\times2$ square by removing one of its four $1\times1$ corners), not crossing its borders, in several layers so that each square of the board is covered by the same number of L's?



M. Evdokimov
1996 All-Russian Olympiad
Points $E$ and $F$ are given on side $BC$ of convex quadrilateral $ABCD$ (with $E$ closer than $F$ to $B$). It is known that $\angle BAE = \angle CDF$ and $\angle EAF = \angle FDE$. Prove that $\angle FAC = \angle EDB$.



M. Smurov
1996 All-Russian Olympiad
On a coordinate plane are placed four counters, each of whose centers has integer coordinates. One can displace any counter by the vector joining the centers of two of the other counters. Prove that any two preselected counters can be made to coincide by a finite sequence of moves.



Р. Sadykov
1996 All-Russian Olympiad
Find all natural numbers $n$, such that there exist relatively prime integers $x$ and $y$ and an integer $k > 1$ satisfying the equation $3^n =x^k + y^k$.



A. Kovaldji, V. Senderov
1996 All-Russian Olympiad
Show that if the integers $a_1$; $\dots$ $a_m$ are nonzero and for each $k =0; 1; \dots ;n$ ($n < m - 1$),

$a_1 + a_22^k + a_33^k + \dots + a_mm^k = 0$; then the sequence $a_1, \dots, a_m$ contains at least $n+1$ pairs of consecutive terms having opposite signs.



O. Musin
1996 All-Russian Olympiad
At the vertices of a cube are written eight pairwise distinct natural numbers, and on each of its edges is written the greatest common divisor of the numbers at the endpoints of the edge. Can the sum of the numbers written at the vertices be the same as the sum of the numbers written at the edges?



A. Shapovalov
1996 All-Russian Olympiad
Three sergeants and several solders serve in a platoon. The sergeants take turns on duty. The commander has given the following orders:

(a) Each day, at least one task must be issued to a soldier.

(b) No soldier may have more than two task or receive more than one tasks in a single day.

(c) The lists of soldiers receiving tasks for two different days must not be the same.

(d) The first sergeant violating any of these orders will be jailed.

Can at least one of the sergeants, without conspiring with the others, give tasks according to these rules and avoid being jailed?



M. Kulikov
1996 All-Russian Olympiad
A convex polygon is given, no two of whose sides are parallel. For each side we consider the angle the side subtends at the vertex farthest from the side. Show that the sum of these angles equals $180^\circ$.



M. Smurov
1996 All-Russian Olympiad
Goodnik writes 10 numbers on the board, then Nogoodnik writes 10 more numbers, all 20 of the numbers being positive and distinct. Can Goodnik choose his 10 numbers so that no matter what Nogoodnik writes, he can form 10 quadratic trinomials of the form $x^2 +px+q$, whose coeficients $p$ and $q$ run through all of the numbers written, such that the real roots of these trinomials comprise exactly 11 values?



I. Rubanov
1996 All-Russian Olympiad
Can the number obtained by writing the numbers from 1 to $n$ in order ($n > 1$) be the same when read left-to-right and right-to-left?



N. Agakhanov
1996 All-Russian Olympiad
Several hikers travel at fixed speeds along a straight road. It is known that over some period of time, the sum of their pairwise distances is monotonically decreasing. Show that there is a hiker, the sum of whose distances to the other hikers is monotonically decreasing over the same period.



A. Shapovalov
1996 All-Russian Olympiad
Show that for $n\ge 5$, a cross-section of a pyramid whose base is a regular $n$-gon cannot be a regular $(n + 1)$-gon.



N. Agakhanov, N. Tereshin
1996 All-Russian Olympiad
Do there exist three natural numbers greater than 1, such that the square of each, minus one, is divisible by each of the others?



A. Golovanov
1996 All-Russian Olympiad
In isosceles triangle $ABC$ ($AB = BC$) one draws the angle bisector $CD$. The perpendicular to $CD$ through the center of the circumcircle of $ABC$ intersects $BC$ at $E$. The parallel to $CD$ through $E$ meets $AB$ at $F$. Show that $BE$ = $FD$.



M. Sonkin
1996 All-Russian Olympiad
Does there exist a finite set $M$ of nonzero real numbers, such that for any natural number $n$ a polynomial of degree no less than $n$ with coeficients in $M$, all of whose roots are real and belong to $M$?



E. Malinnikova
1996 All-Russian Olympiad
The numbers from 1 to 100 are written in an unknown order. One may ask about any 50 numbers and find out their relative order. What is the fewest questions needed to find the order of all 100 numbers?



S. Tokarev
1995 All-Russian Olympiad
A freight train departed from Moscow at $x$ hours and $y$ minutes and arrived at Saratov at $y$ hours and $z$ minutes. The length of its trip was $z$ hours and $x$ minutes. Find all possible values of $x$.



S. Tokarev
1995 All-Russian Olympiad
A chord $CD$ of a circle with center $O$ is perpendicular to a diameter $AB$. A chord  $AE$ bisects the radius $OC$. Show that the line $DE$ bisects the chord $BC$

V. Gordon
1995 All-Russian Olympiad
Can the equation $f(g(h(x))) = 0$, where $f$, $g$, $h$ are quadratic polynomials, have the solutions $1, 2, 3, 4, 5, 6, 7, 8$?



S. Tokarev
1995 All-Russian Olympiad
Can the numbers from 1 to 81 be written in a 9×9 board, so that the sum of numbers in each 3×3 square is the same?

S. Tokarev
1995 All-Russian Olympiad
We call natural numbers  similar if they are written with the same (decimal) digits. For example, 112, 121, 211 are similar numbers having the digits 1, 1, 2. Show that there exist three similar 1995-digit numbers with no zero digits, such that the sum of two of them equals the third.

S. Dvoryaninov
1995 All-Russian Olympiad
In an acute-angled triangle ABC, points $A_2$, $B_2$, $C_2$ are the midpoints of the altitudes $AA_1$, $BB_1$, $CC_1$, respectively. Compute the sum of angles $B_2A_1C_2$, $C_2B_1A_2$ and $A_2C_1B_2$.

D. Tereshin
1995 All-Russian Olympiad
There are three boxes of stones. Sisyphus moves stones one by one between the boxes. Whenever he moves a stone, Zeus gives him the number of coins that is equal to the difference between the number of stones in the box the stone was put in, and that in the box the stone was taken from (the moved stone does not count). If this difference is negative, then Sisyphus returns the corresponding amount to Zeus (if Sisyphus cannot pay, generous Zeus allows him to make the move and pay later).

After some time all the stones lie in their initial boxes. What is the greatest possible earning of Sisyphus at that moment?

I. Izmest’ev
1995 All-Russian Olympiad
Numbers 1 and −1 are written in the cells of a board 2000×2000. It is known that the sum of all the numbers in the board is positive. Show that one can select 1000 rows and 1000 columns such that the sum of numbers written in their intersection cells is at least 1000.

D. Karpov
1995 All-Russian Olympiad
Does there exist a sequence of natural numbers in which every natural number occurs exactly once, such that for each $k = 1, 2, 3, \dots$ the sum of the first $k$ terms of the sequence is divisible by $k$?

A. Shapovalov
1995 All-Russian Olympiad
Prove that if all angles of a convex $n$-gon are equal, then there are at least two of its sides that are not longer than their adjacent sides.

A. Berzin’sh, O. Musin
1995 All-Russian Olympiad
The sequence $a_1, a_2, ...$ of natural numbers satisfies $GCD(a_i, a_j)=GCD(i, j)$ for all $i \neq j$. Prove that $a_i=i$ for all $i$.
1995 All-Russian Olympiad
Let be given a semicircle with diameter $AB$ and center $O$, and a line intersecting the semicircle at $C$ and $D$ and the line $AB$ at $M$ ($MB < MA$, $MD < MC$). The circumcircles of the triangles $AOC$ and $DOB$ meet again at $L$. Prove that $\angle MKO$ is right.

L. Kuptsov
1995 All-Russian Olympiad
Let $P(x)$ and $Q(x)$ be monic polynomials. Prove that the sum of the squares of the coeficients of the polynomial $P(x)Q(x)$ is not smaller than the sum of the squares of the free coefficients of $P(x)$ and $Q(x)$.

A. Galochkin, O. Ljashko
1995 All-Russian Olympiad
Can the numbers $1,2,3,\ldots,100$ be covered with $12$ geometric progressions?



A. Golovanov
1995 All-Russian Olympiad
Prove that every real function, defined on all of $\mathbb R$, can be represented as a sum of two functions whose graphs both have an axis of symmetry.

D. Tereshin
1995 All-Russian Olympiad
Two points on the distance 1 are given in a plane. It is allowed to draw a line through two marked points, as well as a circle centered in a marked point with radius equal to the distance between some two marked points. By marked points we mean the two initial points and intersection points of two lines, two circles, or a line and a circle constructed so far. Let $C(n)$ be the minimum number of circles needed to construct two points on the distance $n$ if only a compass is used, and let $LC(n)$ be the minimum total number of circles and lines needed to do so if a ruler and a compass are used, where $n$ is a natural

number. Prove that the sequence $C(n)/LC(n)$ is not bounded.

A. Belov
1995 All-Russian Olympiad
Prove that for every natural number $a_1>1$ there exists an increasing sequence of natural numbers $a_n$ such that $a^2_1+a^2_2+\cdots+a^2_k$ is divisible by $a_1+a_2+\cdots+a_k$ for all $k \geq 1$.



A. Golovanov
1995 All-Russian Olympiad
A boy goes $n$ times at a merry-go-round with $n$ seats. After every time he moves in the clockwise direction and takes another seat, not making a full circle. The number of seats he passes by at each move is called the length of the move. For which $n$ can he sit at every seat, if the lengths of all the $n-1$ moves he makes have different lengths?

V. New
1995 All-Russian Olympiad
The altitudes of a tetrahedron intersect in a point. Prove that this point, the foot of one of the altitudes, and the points dividing the other three altitudes in the ratio $2 : 1$ (measuring from the vertices) lie on a sphere.

D. Tereshin
1994 All-Russian Olympiad
Prove that if $(x+\sqrt{x^2 +1}) (y+\sqrt{y^2 +1}) =  1$, then $x+y = 0$.
1994 All-Russian Olympiad
Two circles $S_1$ and $S_2$ touch externally at $F$. their external common tangent touches $S_1$ at $A$ and $S_2$ at $B$. A line, parallel to $AB$ and tangent to $S_2$ at $C$, intersects $S_1$ at $D$ and $E$. Prove that points $A,F,C$ are collinear.



(A. Kalinin)
1994 All-Russian Olympiad
There are three piles of matches on the table: one with $100$ matches, one with $200$, and one with $300$. Two players play the following game. They play alternatively, and a player on turn removes one of the piles and divides one of  the remaining piles into two nonempty piles. The player who cannot make a legal move loses. Who has a winning strategy?



(K. Kokhas’)
1994 All-Russian Olympiad
On a line are given $n$ blue and $n$ red points. Prove that the sum of distances between pairs of points of the same color does not exceed the sum of distances between pairs of points of different colors.



(O. Musin)
1994 All-Russian Olympiad
Prove the equality

$\frac{a_1}{a_2(a_1+a_2)}+\frac{a_2}{a_3(a_2+a_3)}+...+\frac{a_n}{a_1(a_n+a_1)}$



=$\frac{a_2}{a_1(a_1+a_2)}+\frac{a_3}{a_2(a_2+a_3)}+...+\frac{a_1}{a_n(a_n+a_1)} $



(R. Zhenodarov)
1994 All-Russian Olympiad
Cards numbered with numbers $1$ to $1000$ are to be placed on the cells of a $1\times 1994$ rectangular board one by one, according to the following rule: If the cell next to the cell containing the card $n$ is free, then the card $n+1$ must be put on it. Prove that the number of possible arrangements is not more than half a mllion.
1994 All-Russian Olympiad
A trapezoid $ABCD$ ($AB ///CD$) has the property that there are points $P$ and $Q$ on sides $AD$ and $BC$ respectively such that $\angle APB = \angle CPD$ and  $\angle AQB = \angle CQD$. Show that the points $P$ and $Q$ are equidistant from the intersection point of the diagonals of the trapezoid.



(M. Smurov)
1994 All-Russian Olympiad
A plane is divided into unit squares by two collections of parallel lines. For any $n\times n$ square with sides on the division lines, we define its frame as the set of those unit squares which internally touch the boundary of the $n\times n$ square. Prove that there exists only one way of covering a given $100\times 100$ square whose sides are on the division lines with frames of $50$ squares (not necessarily contained in the $100\times 100$ square).



(A. Perlin)
1994 All-Russian Olympiad
Let be given three quadratic polynomials:

$P_1(x) = x^2 + p_1x+q_1, P_2(x) = x^2+ p_2x+q_2, P_3(x) = x^2 + p_3x+q_3$.

Prove that the equation $|P_1(x)|+|P_2(x)| = |P_3(x)|$ has at most eight real roots.
1994 All-Russian Olympiad
Let $a,b,c$ be the sides of a triangle, let $m_a,m_b,m_c$ be the corresponding medians, and let $D$ be the diameter of the circumcircle of the triangle.

Prove that $\frac{a^2+b^2}{m_c}+\frac{a^2+c^2}{m_b}+\frac{b^2+c^2}{m_a} \leq 6D$.
1994 All-Russian Olympiad
In a regular $ 6n+1$-gon, $ k$  vertices are painted in red and the others in blue. Prove that the number of isosceles triangles whose vertices are of the same color does not depend on the arrangement of the red vertices.
1994 All-Russian Olympiad
Prove that, for any natural numbers $k,m,n$: $[k,m] \cdot  [m,n] \cdot  [n,k] \ge  [k,m,n]^2$
1994 All-Russian Olympiad
I'll post some nice combinatorics problems here, taken from the wonderful training book "Les olympiades de mathmatiques" (in French) written by Tarik Belhaj Soulami.

Here goes the first one:



Let $\mathbb{I}$ be a non-empty subset of $\mathbb{Z}$ and let $f$ and $g$ be two functions defined on $\mathbb{I}$. Let $m$ be the number of pairs $(x,\;y)$ for which $f(x) = g(y)$, let $n$ be the number of pairs $(x,\;y)$ for which $f(x) = f(y)$ and let $k$ be the number of pairs $(x,\;y)$ for which $g(x) = g(y)$. Show that $$2m \leq n + k.$$
1994 All-Russian Olympiad
Let $ \Gamma_1,\Gamma_2$ and $ \Gamma_3$ be three non-intersecting circles,which are tangent to the circle $ \Gamma$ at points $ A_1,B_1,C_1$,respectively.Suppose that common tangent lines to $ (\Gamma_2,\Gamma_3)$,$ (\Gamma_1,\Gamma_3)$,$ (\Gamma_2,\Gamma_1)$ intersect in points $ A,B,C$.

Prove that lines $ AA_1,BB_1,CC_1$ are concurrent.
1994 All-Russian Olympiad
There are $30$ students in a class. In an examination, their results were all different from each other. It is given that everyone has the same number of friends. Find the maximum number of students such that each one of them has a better result than the majority of his friends.

PS. Here majority means larger than half.
1994 All-Russian Olympiad
All Russian MO 1994 Grade 11

"Natural numbers $a$ and $b$ are such that $\frac{a+1}{b}+\frac{b+1}{a}$ is an integer. If $d$ is the greatest common divisor of $a$ and $b$, prove that $d^2 \le a + b$.
1994 All-Russian Olympiad
Inside a convex $100$-gon are selected $k$ points, $2 \leq k \leq 50$. Show that one can choose $2k$ vertices of the $100$-gon so that the convex $2k$-gon determined by these vertices contains all the selected points.
1994 All-Russian Olympiad
Two circles $S_1$ and $S_2$ touch externally at $F$. their external common tangent touches $S_1$ at A and $S_2$ at $B$. A line, parallel to $AB$ and tangent to $S_2$ at $C$, intersects $S_1$ at $D$ and $E$. Prove that the common chord of the circumcircles of  triangles $ABC$ and $BDE$ passes through point $F$.



(A. Kalinin)
1994 All-Russian Olympiad
Real numbers are written on the squares of an infinite grid. Two figures consisting of finitely many squares are given. They may be translated anywhere on the grid as long as their squares coincide with those of the grid. It is known that wherever the first figure is translated, the sum of numbers it covers is positive. Prove that the second figure can be translated so that the sum of the numbers it covers is also positive.
1994 All-Russian Olympiad
Let $a_1$ be a natural number not divisible by $5$. The sequence $a_1,a_2,a_3, . . .$ is defined by $a_{n+1} =a_n+b_n$, where $b_n$ is the last digit of $a_n$. Prove that the sequence contains infinitely many powers of two.



(N. Agakhanov)
1994 All-Russian Olympiad
The altitudes $AA_1,BB_1,CC_1,DD_1$ of a tetrahedron $ABCD$ intersect in the center $H$ of the sphere inscribed in the tetrahedron $A_1B_1C_1D_1$. Prove that the tetrahedron $ABCD$ is regular.



(D. Tereshin)
1994 All-Russian Olympiad
Players $ A,B$ alternately move a knight on a $ 1994\times 1994$ chessboard. Player $ A$ makes only horizontal moves, i.e. such that the knight is moved to a neighboring row, while $ B$ makes only vertical moves. Initally player $ A$ places the knight to an arbitrary square and makes the first move. The knight cannot be moved to a square that was already visited during the game. A player who cannot make a move loses. Prove that player $ A$ has a winning strategy.
1993 All-Russian Olympiad
For a positive integer $n$, numbers $2n+1$ and $3n+1$ are both perfect squares. Is it possible for $5n+3$ to be prime?
1993 All-Russian Olympiad
Segments $AB$ and $CD$ of length $1$ intersect at point $O$ and angle $AOC$ is equal to sixty degrees. Prove that $AC+BD \ge 1$.
1993 All-Russian Olympiad
Quadratic trinomial $f(x)$ is allowed to be replaced by one of the trinomials $x^2f(1+\frac{1}{x})$ or $(x-1)^2f(\frac{1}{x-1})$. With the use of these operations, is it possible to go from $x^2+4x+3$ to $x^2+10x+9$?
1993 All-Russian Olympiad
In a family album, there are ten photos. On each of them, three people are pictured: in the middle stands a man, to the right of him stands his brother, and to the left of him stands his son. What is the least possible total number of people pictured, if all ten of the people standing in the middle of the ten pictures are different.
1993 All-Russian Olympiad
For integers $x$, $y$, and $z$, we have $(x-y)(y-z)(z-x)=x+y+z$. Prove that $27|x+y+z$.
1993 All-Russian Olympiad
A convex quadrilateral intersects a circle at points $A_1,A_2,B_1,B_2,C_1,C_2,D_1,$ and $D_2$. (Note that for some letter $N$, points $N_1$ and $N_2$ are on one side of the quadrilateral. Also, the points lie in that specific order on the circle.) Prove that if $A_1B_2=B_1C_2=C_1D_2= D_1A_2$, then quadrilateral formed by these four segments is cyclic.
1993 All-Russian Olympiad
What is the maximum number of checkers it is possible to put on a $ n \times n$ chessboard such that in every row and in every column there is an even number of checkers?
1993 All-Russian Olympiad
On a board, there are $n$ equations in the form $*x^2+*x+*$. Two people play a game where they take turns. During a turn, you are aloud to change a star into a number not equal to zero. After $3n$ moves, there will be $n$ quadratic equations. The first player is trying to make more of the equations not have real roots, while the second player is trying to do the opposite. What is the maximum number of equations that the first player can create without real roots no matter how the second player acts?
1993 All-Russian Olympiad
The lengths of the sides of a triangle are prime numbers of centimeters. Prove that its area cannot be an integer number of square centimeters.
1993 All-Russian Olympiad
From the symmetry center of two congruent intersecting circles, two rays are drawn that intersect the circles at four non-collinear points. Prove that these points lie on one circle.
1993 All-Russian Olympiad
Thirty people sit at a round table. Each of them is either smart or dumb. Each of them is asked: "Is your neighbor to the right smart or dumb?" A smart person always answers correctly, while a dumb person can answer both correctly and incorrectly. It is known that the number of dumb people does not exceed $F$. What is the largest possible value of $F$ such that knowing what the answers of the people are, you can point at at least one person, knowing he is smart?
1993 All-Russian Olympiad
Is it true that any two rectangles of equal area can be placed in the plane such that any horizontal line intersecting at least one of them will also intersect the other, and the segments of intersection will be equal?
1993 All-Russian Olympiad
A square is divided by horizontal and vertical lines that form $n^2$ squares each with side $1$. What is the greatest possible value of $n$ such that it is possible to select $n$ squares such that any rectangle with area $n$ formed by the horizontal and vertical lines would contain at least one of the selected $n$ squares.
1993 All-Russian Olympiad
If $ \{a_k\}$ is a sequence of real numbers, call the sequence $ \{a'_k\}$ defined by $ a_k' = \frac {a_k + a_{k + 1}}2$ the average sequence of $ \{a_k\}$. Consider the sequences $ \{a_k\}$; $ \{a_k'\}$ - average sequence of $ \{a_k\}$; $ \{a_k''\}$ - average sequence of $ \{a_k'\}$ and so on. If all these sequences consist only of integers, then $ \{a_k\}$ is called Good. Prove that if $ \{x_k\}$ is a good sequence, then $ \{x_k^2\}$ is also good.
1993 All-Russian Olympiad
Two right triangles are on a plane such that their medians (from the right angles to the hypotenuses) are parallel. Prove that the angle formed by one of the legs of one of the triangles and one of the legs of the other triangle is half the measure of the angle formed by the hypotenuses.
1993 All-Russian Olympiad
Find all functions $f(x)$ with the domain of all positive real numbers, such that for any positive numbers $x$ and $y$, we have $f(x^y)=f(x)^{f(y)}$.
1993 All-Russian Olympiad
Prove that there exists a positive integer $n$, such that if an equilateral triangle with side lengths $n$ is split into $n^2$ triangles with side lengths 1 with lines parallel to its sides, then among the vertices of the small triangles it is possible to choose $1993n$ points so that no three of them are vertices of an equilateral triangle.
1993 All-Russian Olympiad
Find all quadruples of real numbers such that each of them is equal to the product of some two other numbers in the quadruple.
1993 All-Russian Olympiad
The integers from $1$ to $1993$ are written in a line in some order. The following operation is performed with this line: if the first number is $k$ then the first $k$ numbers are rewritten in reverse order. Prove that after some finite number of these operations, the first number in the line of numbers will be $1$.
1993 All-Russian Olympiad
In a tennis tournament, $n$ players want to make $2$ vs $2$ matches such that each player has each of the other players as opponents exactly once. Find all possible values of $n$.
1993 All-Russian Olympiad
Prove that any two rectangular prisms with equal volumes can be placed in a space such that any horizontal plain that intersects one of the prisms will intersect the other forming a polygon with the same area.
1992 All Soviet Union Mathematical Olympiad
Show that $x^4 + y^4 + z^2\ge  xyz \sqrt8$ for all positive reals $x, y, z$.
1992 All Soviet Union Mathematical Olympiad
$E$ is a point on the diagonal $BD$ of the square $ABCD$. Show that the points $A, E$ and the circumcenters of $ABE$ and $ADE$ form a square.