Navvye/Olympiad-Problems-CSV huggingface dataset:
2000 All-Russian Olympiad
Two circles are internally tangent at $N$. The chords $BA$ and $BC$ of the larger circle are tangent to the smaller circle at $K$ and $M$ respectively. $Q$ and $P$ are midpoint of arcs $AB$ and $BC$ respectively. Circumcircles of triangles $BQK$ and $BPM$  are intersect at $L$. Show that $BPLQ$ is a parallelogram.
2000 All-Russian Olympiad
Some paper squares of $k$ distinct colors are placed on a rectangular table, with sides parallel to the sides of the table.  Suppose that for any $k$ squares of distinct colors, some two of them can be nailed on the table with only one nail.  Prove that there is a color such that all squares of that color can be nailed with $2k-2$ nails.
2000 All-Russian Olympiad
Find all functions $ f: \mathbb{R}\longrightarrow \mathbb{R}$ such that

$$f(x+y)+f(y+z)+f(z+x)\ge 3f(x+2y+3z)$$

for all $x, y, z \in \mathbb R$.
2000 All-Russian Olympiad
Prove that one can partition the set of natural numbers into $100$ nonempty subsets such that among any three natural numbers $a$, $b$, $c$ satisfying $a+99b=c$, there are two that belong to the same subset.
2000 All-Russian Olympiad
A convex pentagon $ABCDE$ is given in the coordinate plane with all vertices in lattice points.  Prove that there must be at least one lattice point in the pentagon determined by the diagonals $AC$, $BD$, $CE$, $DA$, $EB$ or on its boundary.
2000 All-Russian Olympiad
Let $a_1, a_2, \cdots, a_n$ be a sequence of nonnegative integers.  For $k=1,2,\cdots,n$ denote $$ m_k = \max_{1 \le l \le k} \frac{a_{k-l+1} + a_{k-l+2} + \cdots + a_k}{l}. $$ Prove that for every $\alpha > 0$ the number of values of $k$ for which $m_k > \alpha$ is less than $\frac{a_1+a_2+ \cdots +a_n}{\alpha}.$
2000 All-Russian Olympiad
Prove the inequality $$ \sin^n (2x) + \left( \sin^n x - \cos^n x \right)^2 \le 1. $$
2000 All-Russian Olympiad
A perfect number, greater than $28$ is divisible by $7$. Prove that it is also divisible by $49$.
2000 All-Russian Olympiad
A quadrilateral $ABCD$ is circumscribed about a circle $\omega$.  The lines $AB$ and $CD$ meet at $O$.  A circle $\omega_1$ is tangent to side $BC$ at $K$ and to the extensions of sides $AB$ and $CD$, and a circle $\omega_2$ is tangent to side $AD$ at $L$ and to the extensions of sides $AB$ and $CD$.  Suppose that points $O$, $K$, $L$ lie on a line.  Prove that the midpoints of $BC$ and $AD$ and the center of $\omega$ also lie on a line.
2000 All-Russian Olympiad
All points in a $100 \times 100$ array are colored in one of four colors red, green, blue or yellow in such a way that there are $25$ points of each color in each row and in any column. Prove that there are two rows and two columns such that their four intersection points are all in different colors.
1999 All-Russian Olympiad
The decimal digits of a natural number $A$ form an increasing sequence (from left to right).  Find the sum of the digits of $9A$.
1999 All-Russian Olympiad
There are several cities in a country. Some pairs of the cities are connected by a two-way airline of one of the $N$ companies, so that each company serves exactly one airline from each city, and one can travel between any two cities, possibly with transfers. During a financial crisis, $N-1$ airlines have been canceled, all from different companies. Prove that it is still possible to travel between any two cities.
1999 All-Russian Olympiad
A triangle $ABC$ is inscribed in a circle $S$.  Let $A_0$ and $C_0$ be the midpoints of the arcs $BC$ and $AB$ on $S$, not containing the opposite vertex, respectively.  The circle $S_1$ centered at $A_0$ is tangent to $BC$, and the circle $S_2$ centered at $C_0$ is tangent to $AB$.  Prove that the incenter $I$ of $\triangle ABC$ lies on a common tangent to $S_1$ and $S_2$.
1999 All-Russian Olympiad
Initially numbers from 1 to 1000000 are all colored black. A move consists of picking one number, then change the color (black to white or white to black) of itself and all other numbers NOT coprime with the chosen number. Can all numbers become white after finite numbers of moves?



Edited by pbornsztein
1999 All-Russian Olympiad
An equilateral triangle of side $n$ is divided into equilateral triangles of side $1$.  Find the greatest possible number of unit segments with endpoints at vertices of the small triangles that can be chosen so that no three of them are sides of a single triangle.
1999 All-Russian Olympiad
Prove that for all natural numbers $n$, $$ \sum_{k=1}^{n^2} \left\{ \sqrt{k} \right\} \le \frac{n^2-1}{2}. $$ Here, $\{x\}$ denotes the fractional part of $x$.
1999 All-Russian Olympiad
A circle through vertices $A$ and $B$ of triangle $ABC$ meets side $BC$ again at $D$. A circle through $B$ and $C$ meets side $AB$ at $E$ and the first circle again at $F$.  Prove that if points $A$, $E$, $D$, $C$ lie on a circle with center $O$ then $\angle BFO$ is right.
1999 All-Russian Olympiad
There are $2000$ components in a circuit, every two of which were initially joined by a wire.  The hooligans Vasya and Petya cut the wires one after another.  Vasya, who starts, cuts one wire on his turn, while Petya cuts one or three.  The hooligan who cuts the last wire from some component loses.  Who has the winning strategy?
1999 All-Russian Olympiad
There are three empty jugs on a table.  Winnie the Pooh, Rabbit, and Piglet put walnuts in the jugs one by one.  They play successively, with the initial determined by a draw.  Thereby Winnie the Pooh plays either in the first or second jug, Rabbit in the second or third, and Piglet in the first or third.  The player after whose move there are exactly 1999 walnuts loses the games.  Show that Winnie the Pooh and Piglet can cooperate so as to make Rabbit lose.
1999 All-Russian Olympiad
Find all bounded sequences $(a_n)_{n=1}^\infty$ of natural numbers such that for all $n \ge 3$, $$ a_n = \frac{a_{n-1} + a_{n-2}}{\gcd(a_{n-1}, a_{n-2})}. $$
1999 All-Russian Olympiad
The incircle of $\triangle ABC$ touch $AB$,$BC$,$CA$ at $K$,$L$,$M$.  The common external tangents to the incircles of $\triangle AMK$,$\triangle BKL$,$\triangle CLM$, distinct from the sides of $\triangle ABC$, are drawn.  Show that these three lines are concurrent.
1999 All-Russian Olympiad
A frog is placed on each cell of a $n \times n$ square inside an infinite chessboard (so initially there are a total of $n \times n$ frogs). Each move consists of a frog $A$ jumping over a frog $B$ adjacent to it with $A$ landing in the next cell and $B$ disappearing (adjacent means two cells sharing a side). Prove that at least $ \left[\frac{n^2}{3}\right]$ moves are needed to reach a configuration where no more moves are possible.
1999 All-Russian Olympiad
The sum of the (decimal) digits of a natural number $n$ equals $100$, and the sum of digits of $44n$ equals $800$.  Determine the sum of digits of $3n$.
1999 All-Russian Olympiad
In triangle $ABC$, a circle passes through $A$ and $B$ and is tangent to $BC$. Also, a circle that passes through $B$ and $C$ is tangent to $AB$. These two circles intersect at a point $K$ other than $B$. If $O$ is the circumcenter of $ABC$, prove that $\angle{BKO}=90^\circ$.
1999 All-Russian Olympiad
Positive numbers $x,y$ satisfy $x^2+y^2 \ge x^3+y^4$.  Prove that $x^3+y^3 \le 2$.
1999 All-Russian Olympiad
In a group of 12 persons, among any 9 there are 5 which know each other. Prove

that there are 6 persons in this group which know each other
1999 All-Russian Olympiad
Do there exist $19$ distinct natural numbers with equal sums of digits, whose sum equals $1999$?
1999 All-Russian Olympiad
Each rational point on a real line is assigned an integer.  Prove that there is a segment such that the sum of the numbers at its endpoints does not exceed twice the number at its midpoint.
1999 All-Russian Olympiad
A circle touches sides $DA$, $AB$, $BC$, $CD$ of a quadrilateral $ABCD$ at points $K$, $L$, $M$, $N$, respectively. Let $S_1$, $S_2$, $S_3$, $S_4$ respectively be the incircles of triangles $AKL$, $BLM$, $CMN$, $DNK$. The external common tangents distinct from the sides of $ABCD$ are drawn to $S_1$ and $S_2$, $S_2$ and $S_3$, $S_3$ and $S_4$, $S_4$ and $S_1$. Prove that these four tangents determine a rhombus.
1999 All-Russian Olympiad
Four natural numbers are such that the square of the sum of any two of them is divisible by the product of the other two numbers.  Prove that at least three of these numbers are equal.
1999 All-Russian Olympiad
Three convex polygons are given on a plane.  Prove that there is no line cutting all the polygons if and only if each of the polygons can be separated from the other two by a line.
1999 All-Russian Olympiad
Through vertex $A$ of a tetrahedron $ABCD$ passes a plane tangent to the circumscribed sphere of the tetrahedron. Show that the lines of intersection of the plane with the planes $ABC$, $ABD$, $ACD$, form six equal angles if and only if: $$AB\cdot CD=AC\cdot BD=AD\cdot BC$$
1999 All-Russian Olympiad
There are $2000$ components in a circuit, every two of which were initially joined by a wire.  The hooligans Vasya and Petya cut the wires one after another.  Vasya, who starts, cuts one wire on his turn, while Petya cuts two or three.  The hooligan who cuts the last wire from some component loses.  Who has the winning strategy?
1998 All-Russian Olympiad
The angle formed by the rays $y=x$ and $y=2x$ ($x \ge 0$) cuts off two arcs from a given parabola $y=x^2+px+q$.  Prove that the projection of one arc onto the $x$-axis is shorter by $1$ than that of the second arc.
1998 All-Russian Olympiad
A convex polygon is partitioned into parallelograms.  A vertex of the polygon is called good if it belongs to exactly one parallelogram.  Prove that there are more than two good vertices.
1998 All-Russian Olympiad
Let $S(x)$ denote the sum of the decimal digits of $x$.  Do there exist natural numbers $a,b,c$ such that $$ S(a+b)<5, \quad S(b+c)<5, \quad S(c+a)<5, \quad S(a+b+c)> 50? $$
1998 All-Russian Olympiad
A maze is an $8 \times 8$ board with some adjacent squares separated by walls, so that any two squares can be connected by a path not meeting any wall.  Given a command LEFT, RIGHT, UP, DOWN, a pawn makes a step in the corresponding direction unless it encounters a wall or an edge of the chessboard.  God writes a program consisting of a finite sequence of commands and gives it to the Devil, who then constructs a maze and places the pawn on one of the squares.  Can God write a program which guarantees the pawn will visit every square despite the Devil's efforts?
1998 All-Russian Olympiad
We are given five watches which can be winded forward.  What is the smallest sum of winding intervals which allows us to set them to the same time, no matter how they were set initially?
1998 All-Russian Olympiad
In triangle $ABC$ with $AB>BC$, $BM$ is a median and $BL$ is an angle bisector.  The line through $M$ and parallel to $AB$ intersects $BL$ at point $D$, and the line through $L$ and parallel to $BC$ intersects $BM$ at point $E$.  Prove that $ED$ is perpendicular to $BL$.
1998 All-Russian Olympiad
A jeweller makes a chain consisting of $N>3$ numbered links.  A querulous customer then asks him to change the order of the links, in such a way that the number of links the jeweller must open is maximized.  What is the maximum number?
1998 All-Russian Olympiad
Two distinct positive integers $a,b$ are written on the board. The smaller of them is erased and replaced with the number $\frac{ab}{|a-b|}$. This process is repeated as long as the two numbers are not equal. Prove that eventually the two numbers on the board will be equal.
1998 All-Russian Olympiad
Two lines parallel to the $x$-axis cut the graph of $y=ax^3+bx^2+cx+d$ in points $A,C,E$ and $B,D,F$ respectively, in that order from left to right.  Prove that the length of the projection of the segment $CD$ onto the $x$-axis equals the sum of the lengths of the projections of $AB$ and $EF$.
1998 All-Russian Olympiad
Two polygons are given on the plane. Assume that the distance between any two vertices of the same polygon is at most 1, and that the distance between any two vertices of different polygons is at least $ 1/\sqrt{2}$. Prove that these two polygons have no common interior points.



By the way, can two sides of a polygon intersect?
1998 All-Russian Olympiad
In scalene $\triangle ABC$, the tangent from the foot of the bisector of $\angle A$ to the incircle of $\triangle ABC$, other than the line $BC$, meets the incircle at point $K_a$.  Points $K_b$ and $K_c$ are analogously defined.  Prove that the lines connecting $K_a$, $K_b$, $K_c$ with the midpoints of $BC$, $CA$, $AB$, respectively, have a common point on the incircle.
1998 All-Russian Olympiad
Let $k$ be a positive integer.  Some of the $2k$-element subsets of a given set are marked.  Suppose that for any subset of cardinality less than or equal to $(k+1)^2$ all the marked subsets contained in it (if any) have a common element.  Show that all the marked subsets have a common element.
1998 All-Russian Olympiad
Initially the numbers $19$ and $98$ are written on a board.  Every minute, each of the two numbers is either squared or increased by $1$.  Is it possible to obtain two equal numbers at some time?
1998 All-Russian Olympiad
A binary operation $*$ on real numbers has the property that $(a * b) * c = a+b+c$ for all $a$, $b$, $c$. Prove that $a * b = a+b$.
1998 All-Russian Olympiad
Let n be an integer at least 4. In a convex n-gon, there is NO four vertices lie on a same circle. A circle is called circumscribed if it passes through 3 vertices of the n-gon and contains all other vertices. A circumscribed circle is called boundary if it passes through 3 consecutive vertices, a circumscribed circle is called inner if it passes through 3 pairwise non-consecutive points. Prove the number of boundary circles is 2 more than the number of inner circles.
1998 All-Russian Olympiad
Each square of a $(2^n-1) \times (2^n-1)$ board contains either $1$ or $-1$.  Such an arrangement is called successful if each number is the product of its neighbors.  Find the number of successful arrangements.
1998 All-Russian Olympiad
Let $ABC$ be a triangle with circumcircle $w$. Let $D$ be the midpoint of arc $BC$ that contains $A$. Define $E$ and $F$ similarly. Let the incircle of $ABC$ touches $BC,CA,AB$ at $K,L,M$ respectively. Prove that $DK,EL,FM$ are concurrent.
1998 All-Russian Olympiad
A set $\mathcal S$ of translates of an equilateral triangle is given in the plane, and any two have nonempty intersection.  Prove that there exist three points such that every triangle in $\mathcal S$ contains one of these points.
1998 All-Russian Olympiad
A sequence of distinct circles $\omega_1, \omega_2, \cdots$ is inscribed in the parabola $y=x^2$ so that $\omega_n$ and $\omega_{n+1}$ are tangent for all $n$.  If $\omega_1$ has diameter $1$ and touches the parabola at $(0,0)$, find the diameter of $\omega_{1998}$.
1998 All-Russian Olympiad
Are there $1998$ different positive integers, the product of any two being divisible by the square of their difference?
1998 All-Russian Olympiad
A tetrahedron $ABCD$ has all edges of length less than $100$, and contains two nonintersecting spheres of diameter $1$.  Prove that it contains a sphere of diameter $1.01$.
1998 All-Russian Olympiad
A figure $\Phi$ composed of unit squares has the following property: if the squares of an $m \times n$ rectangle ($m,n$ are fixed) are filled with numbers whose sum is positive, the figure $\Phi$ can be placed within the rectangle (possibly after being rotated) so that the sum of the covered numbers is also positive. Prove that a number of such figures can be put on the $m\times n$ rectangle so that each square is covered by the same number of figures.
1997 All-Russian Olympiad
Let $P(x)$ be a quadratic polynomial with nonnegative coeficients. Show that for any real numbers $x$ and $y$, we have the inequality $P(xy)^2 \leqslant P(x^2)P(y^2)$.

E. Malinnikova
1997 All-Russian Olympiad
Given a convex polygon M invariant under a $90^\circ$ rotation, show that there exist two circles, the ratio of whose radii is $\sqrt2$, one containing M and the other contained in M.

A. Khrabrov
1997 All-Russian Olympiad
The lateral sides of a box with base $a\times b$ and height $c$ (where $a$; $b$;$ c$ are natural numbers) are completely covered without overlap by rectangles whose edges are parallel to the edges of the box, each containing an even number of unit squares. (Rectangles may cross the lateral edges of the box.) Prove that if $c$ is odd, then

the number of possible coverings is even.

D. Karpov, C. Gukshin, D. Fon-der-Flaas
1997 All-Russian Olympiad
The Judgment of the Council of Sages proceeds as follows: the king arranges the sages in a line and places either a white hat or a black hat on each sage's head. Each sage can see the color of the hats of the sages in front of him, but not of his own hat or of the hats of the sages behind him. Then one by one (in an order of their choosing), each sage guesses a color. Afterward, the king executes those sages who did not correctly guess the color of their own hat. The day before, the Council meets and decides to minimize the number of executions. What is the smallest number of sages guaranteed to survive in this case?



See also
1997 All-Russian Olympiad
Do there exist real numbers $b$ and $c$ such that each of the equations $x^2+bx+c = 0$ and $2x^2+(b+1)x+c+1 = 0$ have two integer roots?

N. Agakhanov
1997 All-Russian Olympiad
A class consists of 33 students. Each student is asked how many other students in the class have his first name, and how many have his last name. It turns out that each number from 0 to 10 occurs among the answers. Show that there are two students in the class with the same first and last name.

A. Shapovalov
1997 All-Russian Olympiad
The incircle of triangle $ABC$ touches sides $AB$;$BC$;$CA$ at $M$;$N$;$K$, respectively.

The line through $A$ parallel to $NK$ meets $MN$ at $D$.

The line through $A$ parallel to $MN$ meets $NK$ at $E$.

Show that the line $DE$ bisects sides $AB$ and $AC$ of triangle $ABC$.

M. Sonkin
1997 All-Russian Olympiad
The numbers from $1$ to $100$ are arranged in a $10\times 10$ table so that any two adjacent numbers have sum no larger than $S$. Find the least value of $S$ for which this is possible.

D. Hramtsov
1997 All-Russian Olympiad
Find all integer solutions of the equation $(x^2 - y^2)^2 = 1 + 16y$.

M. Sonkin
1997 All-Russian Olympiad
An $n\times n$ square grid ($n\geqslant 3$) is rolled into a cylinder. Some of the cells are then colored black. Show that there exist two parallel lines (horizontal, vertical or diagonal) of cells containing the same number of black cells.

E. Poroshenko
1997 All-Russian Olympiad
Two circles intersect at $A$ and $B$. A line through $A$ meets the first circle again at $C$ and the second circle again at $D$. Let $M$ and $N$ be the midpoints of the arcs $BC$ and $BD$ not containing $A$, and let $K$ be the midpoint of the segment $CD$. Show that $\angle MKN =\pi/2$.

(You may assume that $C$ and $D$ lie on opposite sides of $A$.)

D. Tereshin
1997 All-Russian Olympiad
A polygon can be divided into 100 rectangles, but not into 99. Prove that it cannot be divided into 100 triangles.

A. Shapovalov
1997 All-Russian Olympiad
Do there exist two quadratic trinomials $ax^2 +bx+c$ and $(a+1)x^2 +(b + 1)x + (c + 1)$ with integer coeficients, both of which have two integer roots?

N. Agakhanov
1997 All-Russian Olympiad
A circle centered at $O$ and inscribed in triangle $ABC$ meets sides $AC$;$AB$;$BC$ at $K$;$M$;$N$, respectively. The median $BB_1$ of the triangle meets $MN$ at $D$. Show that $O$;$D$;$K$ are collinear.

M. Sonkin
1997 All-Russian Olympiad
Find all triples $m$; $n$; $l$ of natural numbers such that

$m + n = gcd(m; n)^2$; $m + l = gcd(m; l)^2$; $n + l = gcd(n; l)^2$:

S. Tokarev
1997 All-Russian Olympiad
On an infinite (in both directions) strip of squares, indexed by the integers, are placed several stones (more than one may be placed on a single square). We perform a sequence of moves of one of the following types:

(a) Remove one stone from each of the squares $n - 1$ and $n$ and place one stone on square $n + 1$.

(b) Remove two stones from square $n$ and place one stone on each of the squares $n + 1$, $n - 2$.

Prove that any sequence of such moves will lead to a position in which no further moves can be made, and moreover that this position is independent of the sequence of moves.

D. Fon-der-Flaas
1997 All-Russian Olympiad
The Judgment of the Council of Sages proceeds as follows: the king arranges the sages in a line and places either a white hat, black hat or a red hat on each sage's head. Each sage can see the color of the hats of the sages in front of him, but not of his own hat or of the hats of the sages behind him. Then one by one (in an order of their choosing), each sage guesses a color. Afterward, the king executes those sages who did not correctly guess the color of their own hat. The day before, the Council meets and decides to minimize the number of executions. What is the smallest number of sages guaranteed to survive in this case?

K. Knop



P.S. Of course,  the sages hear the previous guesses.



See also
1997 All-Russian Olympiad
An $n\times n\times n$ cube is divided into unit cubes. We are given a closed non-self-intersecting polygon (in space), each of whose sides joins the centers of two unit cubes sharing a common face. The faces of unit cubes which intersect the polygon are said to be distinguished. Prove that the edges of the unit cubes may be colored in two colors so that each distinguished face has an odd number of edges of each color, while each nondistinguished face has an even number of edges of each color.

M. Smurov
1997 All-Russian Olympiad
Of the quadratic trinomials $x^2 + px + q$ where $p$; $q$ are integers and $1\leqslant p, q \leqslant 1997$, which are there more of: those having integer roots or those not having real roots?

M. Evdokimov
1997 All-Russian Olympiad
We are given a polygon, a line $l$ and a point $P$ on $l$ in general position: all lines containing a side of the polygon meet $l$ at distinct points diering from $P$.

We mark each vertex of the polygon the sides meeting which, extended away from the vertex, meet the

line $l$ on opposite sides of $P$. Show that $P$ lies inside the polygon if and only if on each side of $l$ there are an odd number of marked vertices.

O. Musin
1997 All-Russian Olympiad
A sphere inscribed in a tetrahedron touches one face at the intersection of its angle bisectors, a second face at the intersection of its altitudes, and a third face at the intersection of its medians. Show that the tetrahedron is regular.

N. Agakhanov
1997 All-Russian Olympiad
In an $m\times n$ rectangular grid, where m and n are odd integers, $1\times 2$ dominoes are initially placed so as to exactly cover all but one of the $1\times 1$ squares at one corner of the grid.

It is permitted to slide a domino towards the empty square, thus exposing another square.

Show that by a sequence of such moves, we can move the empty square to any corner of the rectangle.

A. Shapovalov
1996 All-Russian Olympiad
Which are there more of among the natural numbers from 1 to 1000000, inclusive: numbers that can be represented as the sum of a perfect square and a (positive) perfect cube, or numbers that cannot be?



A. Golovanov
1996 All-Russian Olympiad
The centers $O_1$; $O_2$; $O_3$ of three nonintersecting circles of equal radius are positioned at the vertices of a triangle. From each of the points $O_1$; $O_2$; $O_3$ one draws tangents to the other two given circles. It is

known that the intersection of these tangents form a convex hexagon. The sides of the hexagon are alternately colored red and blue. Prove that the sum of the lengths of the red sides equals the sum of the lengths of the blue sides.



D. Tereshin
1996 All-Russian Olympiad
Let $x, y, p, n$, and $k$ be positive integers such that $x^n + y^n = p^k$.  Prove that if $n > 1$ is odd, and $p$ is an odd prime, then $n$ is a power of $p$.



A. Kovaldji, V. Senderov
1996 All-Russian Olympiad
In the Duma there are 1600 delegates, who have formed 16000 committees of 80 persons each. Prove that one can find two committees having no fewer than four common members.



A. Skopenkov