Navvye/Olympiad-Problems-CSV huggingface dataset:
1995 Belarus National Olympiad
Some students of a group were friends of some others. One day all students of the group take part in a picnic. During the picnic some friends had a quarrel with each other, but some other students became friends. After the picnic, the number of friends for each student changed by $1$. Prove that the number of students in the group was even.
1995 Belarus National Olympiad
Given a triangle $ABC$, let $K$ be the midpoint of $AB$ and $L$ be the point on the side $AC$ such that $AL = LC + CB$. Show that if $\angle KLB = 90^o$ then $AC = 3 CB$ and conversely, if $AC = 3 CB$ then $\angle KLB = 90^o$.
1995 Belarus National Olympiad
Two circles touch in $M$, and lie inside a rectangle $ABCD$. One of them touches the sides $AB$ and $AD$, and the other one touches $AD,BC,CD$. The radius of the second circle is four times that of the first circle. Find the ratio in which the common tangent of the circles in $M$ divides $AB$ and $CD$.
1995 Belarus National Olympiad
Let $p$ and $q$ be distinct positive integers. Prove that at least one of the equations $x^2+px+q=0$ and $x^2+qx+p=0$ has a real root.
1995 Belarus National Olympiad
The expression $1\oplus2\oplus3\oplus4\oplus5\oplus6\oplus7\oplus8\oplus9$ is written on a blackboard. Bill and Peter play the following game. They replace $\oplus$ by $+$ or $\cdot$, making their moves in turn, and one of them can use only $+$, while the other one can use only $\cdot$. At the beginning, Bill selects the sign he will use, and he tries to make the result an even number. Peter tries to make the result an odd number. Prove that Peter can always win.



Original WordingThe expression $1*2*3*4*5*6*7*8*9$ is written on a blackboard. Bill and Peter play the following game. They replace $*$ by $+$ or $\cdot$, making their moves in turn, and one of them can use only $+$, while the other one can use only $\cdot$. At the beginning Bill selects the sign he will use, and he tries to make the result an even number. Peter tries to make the result an odd number. Prove that Peter can always win.
1995 Belarus National Olympiad
Five numbers 1,2,3,4,5 are written on a blackboard. A student may

erase any two of the numbers a and b on the board and write the

numbers a+b and ab replacing them. If this operation is performed repeatedly, can the numbers 21,27,64,180,540 ever appear on the board?
1997 Belarusian National Olympiad
$$Problem1:$$A two-digit number which is not a multiple of $10$ is given. Assuming it is divisible

by the sum of its digits, prove that it is also divisible by $3$. Does the statement hold for three-digit numbers as well?
1997 Belarusian National Olympiad
$$Problem 2:$$Points $D$ and $E$ are taken on side $CB$ of triangle $ABC$, with $D$ between $C$ and $E$,

such that $\angle BAE =\angle CAD$. If $AC < AB$, prove that $AC.AE < AB.AD$.
1997 Belarusian National Olympiad
$$Problem 3:$$Is it possible to mark 10 red, 10 blue and 10 green points on a plane such that:

For each red point A, the point (among the marked ones) closest to A is blue; for each blue point B, the point closest to B is green; and for each green point C, the point closest to C is red?
1997 Belarusian National Olympiad
$$Problem 4:$$The sum of $5$ positive numbers equals $2$. Let $S_k$ be the sum of the $k-th$ powers of

these numbers. Determine which of the numbers $2,S_2,S_3,S_4$ can be the greatest among them.
1997 Belarusian National Olympiad
$$Problem 1$$;Find all composite numbers $n$ with the following property: For every proper divisor $d$ of $n$ (i.e. $1 < d < n$), it holds that $n-12 \geq d \geq n-20$.
1997 Belarusian National Olympiad
$$Problem 2 :$$If ABCD is as convex quadrilateral with $\angle ADC = 30$

and $BD = AB+BC+CA$,

prove that $BD$ bisects $\angle ABC$.
1997 Belarusian National Olympiad
$$Problem3;$$If distinct real numbers x,y satisfy $\{x\} = \{y\}$ and  $\{x^3\}=\{y^3\}$

prove that $x$ is a root of a quadratic equation with integer coefficients.
1997 Belarusian National Olympiad
$$Problem 4 $$Straight lines $k,l,m$ intersecting each other in three different points are drawn on a classboard. Bob remembers that in some coordinate system the lines$ k,l,m$ have the equations $y = ax, y = bx$ and $y = c +2\frac{ab}{a+b}x$ (where $ab(a + b)$ is non zero).

Misfortunately, both axes are erased. Also, Bob remembers that there is missing a line $n$ ($y = -ax + c$), but he has forgotten $a,b,c$. How can he reconstruct the line $n$?
1999 Belarusian National Olympiad
Evaluate the product $\prod_{k=0}^{2^{1999}}(4\sin^2 \frac{k\pi}{2^{2000}}-3)$
1999 Belarusian National Olympiad
Let $m, n$ be positive integers. Starting with all positive integers written in a line, we can form a list of numbers in two ways:



$(1)$ Erasing every $m$-th and then, in the obtained list, erasing every $n$-th number;

$(2)$ Erasing every $n$-th number and then, in the obtained list, erasing every $m$-th number.



A pair $(m,n)$ is called good if, whenever some positive integer $k$ occurs in both these lists, then it occurs in both lists on the same position.



(a) Show that the pair $(2, n)$ is good for any $n\in \mathbb{N}$.

(b) Is there a good pair $(m, n)$ with $2<m<n$?
1999 Belarusian National Olympiad
A sequence of numbers $a_1,a_2,...,a_{1999}$ is given. In each move it is allowed to choose two of the numbers, say $a_m,a_n$, and replace them by the numbers

$$\frac{a_n^2}{a_m^2}-\frac{n}{m}\left(\frac{a_m^2}{a_n}-a_m\right), \frac{a_m^2}{a_n^2}-\frac{m}{n}\left(\frac{a_n^2}{a_m}-a_n\right) $$respectively. Starting with the sequence $a_i = 1$ for $20 \nmid i$ and $a_i =\frac{1}{5}$ for $20 \mid i$, is it possible to obtain a sequence whose all terms are integers?
1999 Belarusian National Olympiad
A circle is inscribed in the trapezoid ABCD. Let K, L, M, N be the points of tangency of this circle with the diagonals AC and BD, respectively (K is between A and L, and M is between B and N). Given that $AK\cdot LC=16$ and $BM\cdot ND=\frac94$, find the radius of the circle.



[Moderator edit: A solution of this problem can be found on  , page 20 (the statement of the problem is on page 6). The author of the problem is I. Voronovich.]
1999 Belarusian National Olympiad
Determine the maximal value of $ k $, such that for positive reals $ a,b $ and $ c $ from inequality $ kabc >a^3+b^3+c^3 $ it follows that $ a,b $ and $ c $ are sides of a triangle.
1999 Belarusian National Olympiad
Solve in integer:$x^6+x^2y=y^3+2y^2$
1999 Belarusian National Olympiad
Let O be the center of circle W. Two equal chords AB and CD of W intersect at L such that AL>LB and DL>LC. Let M and N be points on AL and DL respectively such that (ALC)=2*(MON). Prove that the chord of W passing through M and N is equal to AB and CD.
1999 Belarusian National Olympiad
Let $n$ be an integer greater than 2. A positive integer is said to be attainable if it is 1 or can be obtained from 1 by a sequence of operations with the following properties:



1.)  The first operation is either addition or multiplication.



2.) Thereafter, additions and multiplications are used alternately.



3.)  In each addition, one can choose independently whether to add 2 or $n$



4.)  In each multiplication, one can choose independently whether to multiply by 2 or by $n$.



A positive integer which cannot be so obtained is said to be unattainable.





a.)  Prove that if $n\geq 9$, there are infinitely many unattainable positive integers.



b.)  Prove that if $n=3$, all positive integers except 7 are attainable.
2000 Belarusian National Olympiad
Pit and Bill play the following game. First Pit writes down a number $a$, then Bill writes a number $b$, then Pit writes a number $c$. Can Pit always play so that the three equations

$$x^3+ax^2+bx+c, x^3+bx^2+cx+a, x^3+cx^2+ax+b$$have (a) a common real root; (b) a common negative root?
2000 Belarusian National Olympiad
Find the number of pairs $(n, q)$, where $n$ is a positive integer and $q$ a non-integer rational number with $0 < q < 2000$, that satisfy $\{q^2\}=\left\{\frac{n!}{2000}\right\}$
2000 Belarusian National Olympiad
Let $N \ge 5$ be given. Consider all sequences $(e_1,e_2,...,e_N)$ with each $e_i$ equal to $1$ or $-1$. Per move one can choose any five consecutive terms and change their signs. Two sequences are said to be similar if one of them can be transformed into the other in finitely many moves. Find the maximum number of pairwise non-similar sequences of length $N$.
2000 Belarusian National Olympiad
The lateral sides and diagonals of a trapezoid intersect a line $l$, determining three equal segments on it. Must $l$ be parallel to the bases of the trapezoid?
2000 Belarusian National Olympiad
Nine points are given on a plane, no three of which lie on a line. Any two of these points are joined by a segment. Is it possible to color these segments by several colors in such a way that, for each color, there are exactly three segments

of that color and these three segments form a triangle?
2000 Belarusian National Olympiad
A vertex of a tetrahedron is called perfect if the three edges at this vertex are sides of a certain triangle. How many perfect vertices can a tetrahedron have?
2000 Belarusian National Olympiad
(a) Find all positive integers $n$ for which the equation $(a^a)^n = b^b$ has a solution

in positive integers $a,b$ greater than $1$.

(b) Find all positive integers $a, b$ satisfying $(a^a)^5=b^b$
2000 Belarusian National Olympiad
To any triangle with side lengths $a,b,c$ and the corresponding angles $\alpha, \beta, \gamma$ (measured in radians), the 6-tuple $(a,b,c,\alpha, \beta, \gamma)$ is assigned. Find the minimum possible number $n$ of distinct terms in the 6-tuple assigned to a scalene triangle.
2001 Belarusian National Olympiad
On the Cartesian coordinate plane, the graph of the parabola $y = x^2$ is drawn. Three distinct points $A$, $B$, and $C$ are marked on the graph with $A$ lying between $B$ and $C$. Point $N$ is marked on $BC$ so that $AN$ is parallel to the y-axis. Let $K_1$ and $K_2$ are the areas of triangles $ABN$ and $ACN$, respectively. Express $AN$ in terms of $K_1$ and $K_2$.
2001 Belarusian National Olympiad
Prove for postitive $a$ and natural $n$

$$a^n+\frac{1}{a^n}-2 \geq n^2(a+\frac{1}{a}-2)$$
2001 Belarusian National Olympiad
Three distinct points $A$, $B$, and $N$ are marked on the line $l$, with $B$ lying between $A$ and $N$. For an arbitrary angle $\alpha \in (0,\frac{\pi}{2})$, points $C$ and $D$ are marked in the plane on the same side of $l$ such that $N$, $C$, and $D$ are collinear; $\angle NAD = \angle NBC = \alpha$; and $A$, $B$, $C$, and $D$ are concyclic. Find the locus of the intersection points of the diagonals of $ABCD$ as $\alpha$ varies between $0$ and $\frac{\pi}{2}$.
2001 Belarusian National Olympiad
The problem committee of a mathematical olympiad prepares some variants of the contest. Each variant contains $4$ problems, chosen from a shortlist of $n$ problems, and any two variants have at most one problem in common.

(a) If $n = 14$, determine the largest possible number of variants the problem committee can prepare.

(b) Find the smallest value of n such that it is possible to prepare ten variants of the contest.
2001 Belarusian National Olympiad
In the increasing sequence of positive integers $a_1$, $a_2$,. . . , the number $a_k$ is said to be funny if it can be represented as the sum of some other terms (not necessarily distinct) of the sequence.

(a) Prove that all but finitely terms of the sequence are funny.

(b) Does the result in (a) always hold if the terms of the sequence can be any positive rational numbers?
2001 Belarusian National Olympiad
Let $n$ be a positive integer. Each square of a $(2n-1) \times (2n - 1)$ square board contains an arrow, either pointing up, down,to the left, or to the right. A beetle sits in one of the cells. Each year it creeps from one square in the direction of the arrow in that square, either reaching another square or leaving the board. Each time the beetle moves, the arrow in the square it leaves turns $\frac{\pi}{2}$ clockwise. Prove that the beetle leaves the board in at most $2^{3n-1}(n-1)!-4$ years after it first moves.
2001 Belarusian National Olympiad
The convex quadrilateral $ABCD$ is inscribed in the circle $S_1$. Let $O$ be the intersection of $AC$ and $BD$. Circle $S_2$ passes through $D$ and $ O$, intersecting $AD$ and $CD$ at $ M$ and $ N$, respectively. Lines $OM$ and $AB$ intersect at $R$, lines $ON$ and $BC$ intersect at $T$, and $R$ and $T$ lie on the same side of line $BD$ as $ A$.

Prove that $O$, $R$,$T$, and $B$ are concyclic.
2001 Belarusian National Olympiad
There are $n$ aborigines on an island. Any two of them are either friends or enemies. One day, the chieftain orders that all

citizens (including himself) make and wear a necklace with zero or more stones so that:

(i) given a pair of friends, there exists a color such that each has a stone of that color;

(ii) given a pair of enemies,there does not exist a color such that each a stone of that color.



(a) Prove that the aborigines can carry out the chieftain’s order.

(b) What is the minimum number of colors of stones required for the

aborigines to carry out the chieftain’s order?
2004 Belarusian National Olympiad
A connected graph with at least one vertex of an odd degree is given. Show that one can color the edges of the graph red and blue in such a way that, for each vertex, the absolute difference between the numbers of red and blue edges at that vertex does not exceed 1.
2004 Belarusian National Olympiad
Let $C$ be a semicircle with diameter $AB$. Circles $S$, $S_1$, $S_2$ with radii $r$, $r_1$, $r_2$, respectively, are tangent to $C$ and the segment $AB$, and moreover $S_1$ and $S_2$ are externally tangent to $S$. Prove that $\frac{1}{\sqrt{r_1}}+\frac{1}{\sqrt{r_2}}=\frac{2\sqrt{2}}{\sqrt{r}}$
2004 Belarusian National Olympiad
The cells of an $n\times n$ table ($n\ge 3$) are painted black and white in the chess-like manner. Per move one can choose any $2\times 2$ square and reverse the color of the cells inside it. Find all $n$ for which one can obtain a table with all cells of the same color after finitely many such moves.
2004 Belarusian National Olympiad
For a positive integer $A = \overline{a_n ...a_1a_0}$ with nonzero digits which are not all the same ($n \ge 0$), the numbers $A_k = \overline{a_{n-k}...a_1a_0a_n ...a_{n-k+1}}$ are obtained for $k = 1,2,...,n$ by cyclic permutations of its digits. Find all $A$ for which each of the $A_k$ is divisible by $A$.
2004 Belarusian National Olympiad
Suppose that $A$ and $B$ are sets of real numbers such that

$$A\subset B+\alpha \mathbb{Z}\quad \text{and}\quad B\subset A+\alpha\mathbb{Z}\quad \text{for all}\quad \alpha>0$$(where $X+\alpha\mathbb=\{x+\alpha n|x\in\mathbb{X}, n\in\mathbb{Z}\}$



(a) Does it follow that $A=B$

(b) The same question, with the assumption that $B$ is bounded
2004 Belarusian National Olympiad
At a mathematical olympiad, eight problems were given to 30 contestants. In order to take the difficulty of each problem into account, the jury decided to assign weights to the problems as follows: a problem is worth $n$ points if it was not solved by exactly $n$ contestants. For example, if a problem was solved by all contestants, then it is worth no points. (It is assumed that there are no partial marks for a problem.) Ivan got less points than any other contestant. Find the greatest score he can have.
2004 Belarusian National Olympiad
A cube $ABCDA_1B_1C_1D_1$ is given. Find the locus of points $E$ on the face $A_1B_1C_1D_1$ for which there exists a line intersecting the lines $AB$, $A_1D_1$, $B_1D$, and $EC$.
2004 Belarusian National Olympiad
Tom Sawyer must whitewash a circular fence consisting of $N$ planks. He whitewashes the fence going clockwise and following the rule: He whitewashes the first plank, skips two planks, whitewashes one, skips three, and so on. Some planks may be whitewashed several times. Tom believes that all planks will be whitewashed sooner or later, but aunt Polly is sure that some planks will remain unwhitewashed forever. Prove that Tom is right if $N$ is a power of two, otherwise

aunt Polly is right.
2005 Belarusian National Olympiad
Prove for positive numbers:

$$(a^2+b+\frac{3}{4})(b^2+a+\frac{3}{4}) \geq (2a+\frac{1}{2})(2b+\frac{1}{2})$$
2005 Belarusian National Olympiad
A line parallel to the side $AC$ of a triangle $ABC$ with $\angle C = 90$ intersects side $AB$ at $M$ and side $BC$ at $N$, so that $CN/BN = AC/BC = 2/1$. The segments $CM$ and $AN$ meet at $O$. Let $K$ be a point on the segment $ON$ such that $MO+OK = KN$. The bisector of $\angle ABC$ meets the line through $K$ perpendicular to $AN$ at point $T$.

Determine $\angle MTB$.
2005 Belarusian National Olympiad
Solve in positive integers $a>b$:

$$(a-b)^{ab}=a^bb^a$$
2005 Belarusian National Olympiad
An $n \times n$ table is called good if one can paint its cells with three colors so that, for any two different rows and two different columns, the four cells at their intersections are not all of the same color.

a)Show, that exists good $9 \times 9$ good table.

b)Prove, that fif $n \times n$  table is good, then  $n<11$
2005 Belarusian National Olympiad
For $0<a,b,c,d<\frac{\pi}{2}$ is true that $$\cos 2a+\cos 2b+ \cos 2c+ \cos 2d= 4 (\sin a \sin b \sin c \sin d -\cos a \cos b \cos c \cos d)$$Find all possible values of $a+b+c+d$
2005 Belarusian National Olympiad
$f(n+f(n))=f(n)$ for every $n \in \mathbb{N}$.

a)Prove, that if $f(n)$ is finite, then $f$ is periodic.

b) Give example nonperiodic function.



PS. $0 \not \in \mathbb{N}$
2005 Belarusian National Olympiad
The deputies in a parliament were split into $10$ fractions. According to regulations, no fraction may consist of less than five people, and no two fractions may have the same number of members. After the vacation, the fractions disintegrated and several new fractions arose instead. Besides, some deputies became independent. It turned out that no two deputies that were in the same fraction before the vacation entered the same fraction after the vacation.

Find the smallest possible number of independent deputies after the vacation.
2005 Belarusian National Olympiad
Does there exist a convex pentagon such that for any of its inner angles, the angle bisector contains one of the diagonals?
2007 Belarusian National Olympiad
Find all polynomials with degree $\leq n$ and nonnegative coefficients, such that $P(x)P(\frac{1}{x}) \leq P(1)^2$ for every positive $x$
2007 Belarusian National Olympiad
Circles $S_1$ and $S_2$ with centers $O_1$ and $O_2$, respectively, pass through the centers of each other. Let $A$ be one of their intersection points. Two points $M_1$ and $M_2$ begin to move simultaneously starting from $A$. Point $M_1$ moves along $S_1$ and point $M_2$ moves along $S_2$. Both points move in clockwise direction and have the same linear velocity $v$.



(a) Prove that all triangles $AM_1M_2$ are equilateral.

(b) Determine the trajectory of the movement of the center of the triangle $AM_1M_2$ and find its linear velocity.
2007 Belarusian National Olympiad
Given a $2n \times 2m$ table $(m,n \in \mathbb{N})$ with one of two signs ”+” or ”-” in each of its cells. A union of all the cells of some row and some column is called a cross. The cell on the intersectin of this row and this column is called the center of the cross. The following procedure we call a transformation of the table: we mark all cells which contain ”−” and then, in turn, we replace the signs in all cells of the crosses which centers are marked by the opposite signs. (It is easy to see that the order of the choice of the crosses doesn’t matter.) We call a table attainable if it can be obtained from some table applying such transformations one time.

Find the number of all attainable tables.
2007 Belarusian National Olympiad
Each point of a circle is painted in one of the $ N$ colors ($N \geq 2$). Prove that there exists an inscribed trapezoid such that all its vertices are painted the same color.
2007 Belarusian National Olympiad
Let $O$ be the intersection point of the diagonals of the convex quadrilateral $ABCD$, $AO = CO$. Points $P$ and $Q$ are marked on the segments $AO$ and $CO$, respectively, such that $PO = OQ$. Let $N$ and $K$ be the intersection points of the sides $AB$, $CD$, and the lines $DP$ and $BQ$ respectively.

Prove that the points $N$, $O$, and $K$ are colinear.
2007 Belarusian National Olympiad
Let $a$ be the sum and $b$ the product of the real roots of the equation $x^4-x^3-1=0$



Prove that $b < -\frac{11}{10}$ and $a > \frac{6}{11}$.
2007 Belarusian National Olympiad
Find solution in positive integers : $$n^5+n^4=7^m-1$$
2007 Belarusian National Olympiad
Let $(m,n)$ be a pair of positive integers.

(a) Prove that the set of all positive integers can be partitioned into four pairwise disjoint nonempty subsets such that none of them has two numbers with absolute value of their difference equal to either $m$, $n$, or $m+n$.

(b) Find all pairs $(m,n)$ such that the set of all positive integers can not be partitioned into three pairwise disjoint nonempty subsets satisfying the above condition.
2018 Belarusian National Olympiad
Prove that the set of all divisors of a positive integer which is not a perfect square can be divided into pairs so that in each pair one number is divisible by another.
2018 Belarusian National Olympiad
For every integer $n\geqslant2$ prove the inequality

$$

\frac{1}{2!}+\frac{2}{3!}+\ldots+\frac{2^{n-2}}{n!}\leqslant\frac{3}{2},

$$where $k!=1\cdot2\cdot\ldots\cdot k$.
2018 Belarusian National Olympiad
The bisector of angle $CAB$ of triangle $ABC$ intersects the side $CB$ at $L$. The point $D$ is the foot of the perpendicular from $C$ to $AL$ and the point $E$ is the foot of perpendicular from $L$ to $AB$. The lines $CB$ and $DE$ meet at $F$.

Prove that $AF$ is an altitude of triangle $ABC$.
2018 Belarusian National Olympiad
Three $n\times n$ squares form the figure $\Phi$ on the checkered plane as shown on the picture. (Neighboring squares are tpuching along the segment of length $n-1$.)

Find all $n > 1$ for which the figure $\Phi$ can be covered with tiles $1\times 3$ and  $3\times 1$ without overlapping.
2018 Belarusian National Olympiad
The quadrilateral $ABCD$ is inscribed in the parabola $y=x^2$. It is known that angle $BAD=90$, the dioganal $AC$ is parallel to the axis $Ox$ and $AC$ is the bisector of the angle BAD.

Find the area of the quadrilateral $ABCD$ if the length of the dioganal $BD$ is equal to $p$.
2018 Belarusian National Olympiad
For all positive integers $m$ and $n$ prove the inequality

$$

|n\sqrt{n^2+1}-m|\geqslant \sqrt{2}-1.

$$
2018 Belarusian National Olympiad
A point $O$ is choosen inside a triangle $ABC$ so that the length of segments $OA$, $OB$ and $OC$ are equal to $15$,$12$ and $20$, respectively. It is known that the feet of the perpendiculars from $O$ to the sides of the triangle $ABC$ are the vertices of an equilateral triangle.

Find the value of the angle $BAC$.
2018 Belarusian National Olympiad
A positive integer $n$ is fixed. Numbers $0$ and $1$ are placed in all cells (exactly one number in any cell) of a $k \times n$ table ($k$ is a number of the rows in the table, $n$ is the number of the columns in it). We call a table nice if the following property is fulfilled: for any partition of the set of the rows of the table into two nonempty subsets $R$1 and $R$2 there exists a nonempty set $S$ of the columns such that on the intersection of any row from $R$1 with the columns from $S$ there are even number of $1's$ while on the intersection of any row from $R$2 with the columns from $S$ there are odd number of $1's$.

Find the greatest number of $k$ such that there exists at least one nice $k \times n$ table.
2018 Belarusian National Olympiad
The extension of the median $AM$ of the triangle $ABC$ intersects its circumcircle at $D$. The circumcircle of triangle $CMD$ intersects the line $AC$ at $C$ and $E$.The circumcircle of triangle $AME$ intersects the line $AB$ at $A$ and $F$. Prove that $CF$ is the altitude of triangle $ABC$.
2018 Belarusian National Olympiad
Determine, whether there exist a function $f$ defined on the set of all positive real numbers and taking positive values such that $f(x+y)\geqslant yf(x)+f(f(x))$ for all positive x and y?