Navvye/Olympiad-Problems-CSV huggingface dataset:
1999 Baltic Way
Let $ABC$ be an isosceles triangle with $AB=AC$. Points $D$ and $E$ lie on the sides $AB$ and $AC$, respectively. The line passing through $B$ and parallel to $AC$ meets the line $DE$ at $F$. The line passing through $C$ and parallel to $AB$ meets the line $DE$ at $G$. Prove that

$$\frac{[DBCG]}{[FBCE]}=\frac{AD}{DE} $$
1999 Baltic Way
Let $ABC$ be a triangle with $\angle C=60^\circ$ and $AC<BC$. The point $D$ lies on the side $BC$ and satisfies $BD=AC$. The side $AC$ is extended to the point $E$ where $AC=CE$. Prove that $AB=DE$.
1999 Baltic Way
Find the smallest positive integer $k$ which is representable in the form $k=19^n-5^m$ for some positive integers $m$ and $n$.
1999 Baltic Way
Does there exist a finite sequence of integers $c_1,c_2,\ldots ,c_n$ such that all the numbers $a+c_1,a+c_2,\ldots ,a+c_n$ are primes for more than one but not infinitely many different integers $a$?
1999 Baltic Way
Let $m$ be a positive integer such that $m=2\pmod{4}$. Show that there exists at most one factorization $m=ab$ where $a$ and $b$ are positive integers satisfying

$$0<a-b<\sqrt{5+4\sqrt{4m+1}}$$
1999 Baltic Way
Prove that there exist infinitely many even positive integers $k$ such that for every prime $p$ the number $p^2+k$ is composite.
1999 Baltic Way
Let $a,b,c$ and $d$ be prime numbers such that $a>3b>6c>12d$ and $a^2-b^2+c^2-d^2=1749$. Determine all possible values of $a^2+b^2+c^2+d^2$ .
1998 Baltic Way
Find all functions $f$ of two variables, whose arguments $x,y$ and values $f(x,y)$ are positive integers, satisfying the following conditions (for all positive integers $x$ and $y$):

\begin{align*} f(x,x)& =x,\\ f(x,y)& =f(y,x),\\ (x+y)f(x,y)& =yf(x,x+y).\end{align*}
1998 Baltic Way
A triple $(a,b,c)$ of positive integers is called quasi-Pythagorean if there exists a triangle with lengths of the sides $a,b,c$ and the angle opposite to the side $c$ equal to $120^{\circ}$. Prove that if $(a,b,c)$ is a quasi-Pythagorean triple then $c$ has a prime divisor bigger than $5$.
1998 Baltic Way
Find all positive integer solutions to $2x^2+5y^2=11(xy-11)$.
1998 Baltic Way
Let $P$ be a polynomial with integer coefficients. Suppose that for $n=1,2,3,\ldots ,1998$ the number $P(n)$ is a three-digit positive integer. Prove that the polynomial $P$ has no integer roots.
1998 Baltic Way
Let $a$ be an odd digit and $b$ an even digit. Prove that for every positive integer $n$ there exists a positive integer, divisible by $2^n$, whose decimal representation contains no digits other than $a$ and $b$.
1998 Baltic Way
Let $P$ be a polynomial of degree $6$ and let $a,b$ be real numbers such that $0<a<b$. Suppose that $P(a)=P(-a),P(b)=P(-b),P'(0)=0$. Prove that $P(x)=P(-x)$ for all real $x$.
1998 Baltic Way
Let $\mathbb{R}$ be the set of all real numbers. Find all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ satisfying for all $x,y\in\mathbb{R}$ the equation $f(x)+f(y)=f(f(x)f(y))$.
1998 Baltic Way
Let $P_k(x)=1+x+x^2+\ldots +x^{k-1}$. Show that

$$ \sum_{k=1}^n \binom{n}{k} P_k(x)=2^{n-1} P_n \left( \frac{x+1}{2} \right) $$

for every real number $x$ and every positive integer $n$.
1998 Baltic Way
Let the numbers $\alpha ,\beta $ satisfy $0<\alpha <\beta <\frac{\pi}{2}$ and let $\gamma $ and $\delta $ be the numbers defined by the conditions:



$(\text{i})\ 0<\gamma<\frac{\pi}{2}$, and $\tan\gamma$ is the arithmetic mean of $\tan\alpha$ and $\tan\beta$;



$(\text{ii})\ 0<\delta<\frac{\pi}{2}$, and $\frac{1}{\cos\delta}$ is the arithmetic mean of $\frac{1}{\cos\alpha}$ and $\frac{1}{\cos\beta}$.



Prove that $\gamma <\delta $.
1998 Baltic Way
Let $n\ge 4$ be an even integer. A regular $n$-gon and a regular $(n-1)$-gon are inscribed into the unit circle. For each vertex of the $n$-gon consider the distance from this vertex to the nearest vertex of the $(n-1)$-gon, measured along the circumference. Let $S$ be the sum of these $n$ distances. Prove that $S$ depends only on $n$, and not on the relative position of the two polygons.
1998 Baltic Way
If $a,b,c$ be the lengths of the sides of a triangle. Let $R$ denote its circumradius. Prove that

$$ R\ge \frac{a^2+b^2}{2\sqrt{2a^2+2b^2-c^2}}$$

When does equality hold?
1998 Baltic Way
In a triangle $ABC$, $\angle BAC =90^{\circ}$. Point $D$ lies on the side $BC$ and satisfies $\angle BDA=2\angle BAD$. Prove that

$$\frac{2}{AD}=\frac{1}{BD}+\frac{1}{CD} $$
1998 Baltic Way
In convex pentagon $ABCDE$, the sides $AE,BC$ are parallel and $\angle ADE=\angle BDC$. The diagonals $AC$ and $BE$ intersect at $P$. Prove that $\angle EAD=\angle BDP$ and $\angle CBD=\angle ADP$.
1998 Baltic Way
Given triangle $ABC$ with $AB<AC$. The line passing through $B$ and parallel to $AC$ meets the external angle bisector of $\angle BAC$ at $D$. The line passing through $C$ and parallel to $AB$ meets this bisector at $E$. Point $F$ lies on the side $AC$ and satisfies the equality $FC=AB$. Prove that $DF=FE$.
1998 Baltic Way
Given acute triangle $ABC$. Point $D$ is the foot of the perpendicular from $A$ to $BC$. Point $E$ lies on the segment $AD$ and satisfies the equation

$$\frac{AE}{ED}=\frac{CD}{DB}$$

Point $F$ is the foot of the perpendicular from $D$ to $BE$. Prove that $\angle AFC=90^{\circ}$.
1998 Baltic Way
Is it possible to cover a $13\times 13$ chessboard with forty-two pieces of dimensions $4\times 1$ such that only the central square of the chessboard remains uncovered?
1998 Baltic Way
Let $n$ and $k$ be positive integers. There are $nk$ objects (of the same size) and $k$ boxes, each of which can hold $n$ objects. Each object is coloured in one of $k$ different colours. Show that the objects can be packed in the boxes so that each box holds objects of at most two colours.
1998 Baltic Way
Determine all positive integers $n$ for which there exists a set $S$ with the following properties:

(i) $S$ consists of $n$ positive integers, all smaller than $2^{n-1}$;

(ii) for any two distinct subsets $A$ and $B$ of $S$, the sum of the elements of $A$ is different from the sum of the elements of $B$.
1998 Baltic Way
Consider a ping-pong match between two teams, each consisting of $1000$ players. Each player played against each player of the other team exactly once (there are no draws in ping-pong). Prove that there exist ten players, all from the same team, such that every member of the other team has lost his game against at least one of those ten players.
1998 Baltic Way
We say that some positive integer $m$ covers the number $1998$, if $1,9,9,8$ appear in this order as digits of $m$. (For instance $1998$ is covered by $2\textbf{1}59\textbf{9}36\textbf{98}$ but not by $213326798$.) Let $k(n)$ be the number of positive integers that cover $1998$ and have exactly $n$ digits ($n\ge 5$), all different from $0$. What is the remainder of $k(n)$ on division by $8$?
1997 Baltic Way
Determine all functions $f$ from the real numbers to the real numbers, different from the zero function, such that $f(x)f(y)=f(x-y)$ for all real numbers $x$ and $y$.
1997 Baltic Way
Given a sequence $a_1,a_2,a_3,\ldots $ of positive integers in which every positive integer occurs exactly once. Prove that there exist integers $\ell $ and $m,\ 1<\ell <m$, such that $a_1+a_m=2a_{\ell}$.
1997 Baltic Way
Let $x_1=1$ and $x_{n+1} =x_n+\left\lfloor \frac{x_n}{n}\right\rfloor +2$, for $n=1,2,3,\ldots $ where $x$ denotes the largest integer not greater than $x$. Determine $x_{1997}$.
1997 Baltic Way
Prove that the arithmetic mean $a$ of $x_1,\ldots ,x_n$ satisfies

$$ (x_1-a)^2+\ldots +(x_n-a)^2\le \frac{1}{2}(|x_1-a|+\ldots +|x_n-a|)^2$$
1997 Baltic Way
In a sequence $u_0,u_1,\ldots $ of positive integers, $u_0$ is arbitrary, and for any non-negative integer $n$,

$$ u_{n+1}=\begin{cases}\frac{1}{2}u_n & \text{for even }u_n \\ a+u_n & \text{for odd }u_n \end{cases} $$

where $a$ is a fixed odd positive integer. Prove that the sequence is periodic from a certain step.
1997 Baltic Way
Find all triples $(a,b,c)$ of non-negative integers satisfying $a\ge b\ge c$ and

$$1\cdot a^3+9\cdot b^2+9\cdot c+7=1997 $$
1997 Baltic Way
Let $P$ and $Q$ be polynomials with integer coefficients. Suppose that the integers $a$ and $a+1997$ are roots of $P$, and that $Q(1998)=2000$. Prove that the equation $Q(P(x))=1$ has no integer solutions.
1997 Baltic Way
If we add $1996$ to $1997$, we first add the unit digits $6$ and $7$. Obtaining $13$, we write down $3$ and “carry” $1$ to the next column. Thus we make a carry. Continuing, we see that we are to make three carries in total.



Does there exist a positive integer $k$ such that adding $1996\cdot k$ to $1997\cdot k$ no carry arises during the whole calculation?
1997 Baltic Way
The worlds in the Worlds’ Sphere are numbered $1,2,3,\ldots $ and connected so that for any integer $n\ge 1$, Gandalf the Wizard can move in both directions between any worlds with numbers $n,2n$ and $3n+1$. Starting his travel from an arbitrary world, can Gandalf reach every other world?
1997 Baltic Way
Prove that in every sequence of $79$ consecutive positive integers written in the decimal system, there is a positive integer whose sum of digits is divisible by $13$.
1997 Baltic Way
On two parallel lines, the distinct points $A_1,A_2,A_3,\ldots $ respectively $B_1,B_2,B_3,\ldots $ are marked in such a way that $|A_iA_{i+1}|=1$ and $|B_iB_{i+1}|=2$ for $i=1,2,\ldots $. Provided that $A_1A_2B_1=\alpha$, find the infinite sum $\angle A_1B_1A_2+\angle A_2B_2A_3+\angle A_3B_3A_4+\ldots $
1997 Baltic Way
Two circles $\mathcal{C}_1$ and $\mathcal{C}_2$ intersect in $P$ and $Q$. A line through $P$ intersects $\mathcal{C}_1$ and $\mathcal{C}_2$ again at $A$ and $B$, respectively, and $X$ is the midpoint of $AB$. The line through $Q$ and $X$ intersects $C_1$ and $C_2$ again at $Y$ and $Z$, respectively. Prove that $X$ is the midpoint of $YZ$.
1997 Baltic Way
Five distinct points $A,B,C,D$ and $E$ lie on a line with $|AB|=|BC|=|CD|=|DE|$. The point $F$ lies outside the line. Let $G$ be the circumcentre of the triangle $ADF$ and $H$ the circumcentre of the triangle $BEF$. Show that the lines $GH$ and $FC$ are perpendicular.
1997 Baltic Way
In the triangle $ABC$, $AC^2$ is the arithmetic mean of $BC^2$ and $AB^2$. Show that $\cot^2B\ge \cot A\cdot\cot C$.
1997 Baltic Way
In the acute triangle $ABC$, the bisectors of $A,B$ and $C$ intersect the circumcircle again at $A_1,B_1$ and $C_1$, respectively. Let $M$ be the point of intersection of $AB$ and $B_1C_1$, and let $N$ be the point of intersection of $BC$ and $A_1B_1$. Prove that $MN$ passes through the incentre of $\triangle ABC$.
1997 Baltic Way
On a $5\times 5$ chessboard, two players play the following game: The first player places a knight on some square. Then the players alternately move the knight according to the rules of chess, starting with the second player. It is not allowed to move the knight to a square that was visited previously. The player who cannot move loses. Which of the two players has a winning strategy?
1997 Baltic Way
A rectangle can be divided into $n$ equal squares. The same rectangle can also be divided into $n+76$ equal squares. Find $n$.
1997 Baltic Way
a) Prove the existence of two infinite sets $A$ and $B$, not necessarily disjoint, of non-negative integers such that each non-negative integer $n$ is uniquely representable in the form $n=a+b$ with $a\in A,b\in B$.

b) Prove that for each such pair $(A,B)$, either $A$ or $B$ contains only multiples of some integer $k>1$.
1997 Baltic Way
In a forest each of $n$ animals ($n\ge 3$) lives in its own cave, and there is exactly one separate path between any two of these caves. Before the election for King of the Forest some of the animals make an election campaign. Each campaign-making animal visits each of the other caves exactly once, uses only the paths for moving from cave to cave, never turns from one path to another between the caves and returns to its own cave in the end of its campaign. It is also known that no path between two caves is used by more than one campaign-making animal.



a) Prove that for any prime $n$, the maximum possible number of campaign-making animals is $\frac{n-1}{2}$.



b) Find the maximum number of campaign-making animals for $n=9$.
1997 Baltic Way
Twelve cards lie in a row. The cards are of three kinds: with both sides white, both sides black, or with a white and a black side. Initially, nine of the twelve cards have a black side up. The cards $1-6$ are turned, and subsequently four of the twelve cards have a black side up. Now cards $4-9$ are turned, and six cards have a black side up. Finally, the cards $1-3$ and $10-12$ are turned, after which five cards have a black side up. How many cards of each kind were there?
1996 Baltic Way
Let $\alpha$ be the angle between two lines containing the diagonals of a regular $1996$-gon, and let $\beta\not= 0$ be another such angle. Prove that $\frac{\alpha}{\beta}$ is a rational number.
1996 Baltic Way
In the figure below, you see three half-circles. The circle $C$ is tangent to two of the half-circles and to the line $PQ$ perpendicular to the diameter $AB$. The area of the shaded region is $39\pi$, and the area of the circle $C$ is $9\pi$. Find the length of the diameter $AB$.
1996 Baltic Way
Let $ABCD$ be a unit square and let $P$ and $Q$ be points in the plane such that $Q$ is the circumcentre of triangle $BPC$ and $D$ be the circumcentre of triangle $PQA$. Find all possible values of the length of segment $PQ$.
1996 Baltic Way
$ABCD$ is a trapezium where $AD\parallel BC$. $P$ is the point on the line $AB$ such that $\angle CPD$ is maximal. $Q$ is the point on the line $CD$ such that $\angle BQA$ is maximal. Given that $P$ lies on the segment $AB$, prove that $\angle CPD=\angle BQA$.
1996 Baltic Way
Let $ABCD$ be a cyclic convex quadrilateral and let $r_a,r_b,r_c,r_d$ be the radii of the circles inscribed in the triangles $BCD, ACD, ABD, ABC$, respectively. Prove that $r_a+r_c=r_b+r_d$.
1996 Baltic Way
A sequence of integers $a_1,a_2,\ldots $ is such that $a_1=1,a_2=2$ and for $n\ge 1$,

$$a_{n+2}=\left\{\begin{array}{cl}5a_{n+1}-3a_{n}, &\text{if}\ a_n\cdot a_{n+1}\ \text{is even},\\ a_{n+1}-a_{n}, &\text{if}\ a_n\cdot a_{n+1}\ \text{is odd},\end{array}\right. $$

Prove that $a_n\not= 0$ for all $n$.
1996 Baltic Way
Consider the sequence: $x_1=19,x_2=95,x_{n+2}=\text{lcm} (x_{n+1},x_n)+x_n$, for $n>1$, where $\text{lcm} (a,b)$ means the least common multiple of $a$ and $b$. Find the greatest common divisor of $x_{1995}$ and $x_{1996}$.
1996 Baltic Way
Let $n$ and $k$ be integers, $1\le k\le n$. Find an integer $b$ and a set $A$ of $n$ integers satisfying the following conditions:

(i) No product of $k-1$ distinct elements of $A$ is divisible by $b$.

(ii) Every product of $k$ distinct elements of $A$ is divisible by $b$.

(iii) For all distinct $a,a'$ in $A$, $a$ does not divide $a'$.
1996 Baltic Way
Denote by $d(n)$ the number of distinct positive divisors of a positive integer $n$ (including $1$ and $n$). Let $a>1$ and $n>0$ be integers such that $a^n+1$ is a prime. Prove that $d(a^n-1)\ge n$.
1996 Baltic Way
Real numbers $x_1,x_2,\ldots ,x_{1996}$ have the following property: For any polynomial $W$ of degree $2$ at least three of the numbers $W(x_1),W(x_2),\ldots ,W(x_{1996})$ are equal. Prove that at least three of the numbers $x_1,x_2,\ldots ,x_{1996}$ are equal.
1996 Baltic Way
Let $S$ be a set of integers containing the numbers $0$ and $1996$. Suppose further that any integer root of any non-zero polynomial with coefficients in $S$ also belongs to $S$. Prove that $-2$ belongs to $S$.
1996 Baltic Way
Consider the functions $f$ defined on the set of integers such that

$$f(x)=f(x^2+x+1)$$

for all integer $x$. Find $(a)$ all even functions, $(b)$ all odd functions of this kind.
1996 Baltic Way
The graph of the function $f(x)=x^n+a_{n-1}x_{n-1}+\ldots +a_1x+a_0$ (where $n>1$) intersects the line $y=b$ at the points $B_1,B_2,\ldots ,B_n$ (from left to right), and the line $y=c\ (c\not= b)$ at the points $C_1,C_2,\ldots ,C_n$ (from left to right). Let $P$ be a point on the line $y=c$, to the right to the point $C_n$. Find the sum

$$\cot (\angle B_1C_1P)+\ldots +\cot (\angle B_nC_nP) $$
1996 Baltic Way
For which positive real numbers $a,b$ does the inequality

$$x_1x_2+x_2x_3+\ldots x_{n-1}x_n+x_nx_1\ge x_1^ax_2^bx_3^a+  x_2^ax_3^bx_4^a+\ldots +x_n^ax_1^bx_2^a$$

hold for all integers $n>2$ and positive real numbers $x_1,\ldots ,x_n$?
1996 Baltic Way
On an infinite checkerboard two players alternately mark one unmarked cell. One of them uses $\times$, the other $\circ$. The first who fills a $2\times 2$ square with his symbols wins. Can the player who starts always win?
1996 Baltic Way
Using each of the eight digits $1,3,4,5,6,7,8$ and $9$ exactly once, a three-digit number $A$, two two-digit numbers $B$ and $C$, $B<C$, and a one digit number $D$ are formed. The numbers are such that $A+D=B+C=143$. In how many ways can this be done?
1996 Baltic Way
The jury of an Olympiad has $30$ members in the beginning. Each member of the jury thinks that some of his colleagues are competent, while all the others are not, and these opinions do not change. At the beginning of every session a voting takes place, and those members who are not competent in the opinion of more than one half of the voters are excluded from the jury for the rest of the olympiad. Prove that after at most $15$ sessions there will be no more exclusions. (Note that nobody votes about his own competence.)
1996 Baltic Way
Four heaps contain $38,45,61$ and $70$ matches respectively. Two players take turn choosing any two of the heaps and take some non-zero number of matches from one heap and some non-zero number of matches from the other heap. The player who cannot make a move, loses. Which one of the players has a winning strategy ?
1996 Baltic Way
Is it possible to partition all positive integers into disjoint sets $A$ and $B$ such that

(i) no three numbers of $A$ form an arithmetic progression,

(ii) no infinite non-constant arithmetic progression can be formed by numbers of $B$?
1995 Baltic Way
Find all triples $(x,y,z)$ of positive integers satisfying the system of equations

$$\begin{cases} x^2=2(y+z)\\ x^6=y^6+z^6+31(y^2+z^2)\end{cases}$$
1995 Baltic Way
Let $a$ and $k$ be positive integers such that $a^2+k$ divides $(a-1)a(a+1)$. Prove that $k\ge a$.
1995 Baltic Way
The positive integers $a,b,c$ are pairwise relatively prime, $a$ and $c$ are odd and the numbers satisfy the equation $a^2+b^2=c^2$. Prove that $b+c$ is the square of an integer.
1995 Baltic Way
Josh is older than Fred. Josh notices that if he switches the two digits of his age (an integer), he gets Fred’s age. Moreover, the difference between the squares of their ages is a square of an integer. How old are Josh and Fred?
1995 Baltic Way
Let $a<b<c$ be three positive integers. Prove that among any $2c$ consecutive positive integers there exist three different numbers $x,y,z$ such that $abc$ divides $xyz$.
1995 Baltic Way
Prove that for positive $a,b,c,d$

$$\frac{a+c}{a+b}+\frac{b+d}{b+c}+\frac{c+a}{c+d}+\frac{d+b}{d+a}\ge 4$$
1995 Baltic Way
Prove that $\sin^318^{\circ}+\sin^218^{\circ}=\frac18$.
1995 Baltic Way
The real numbers $a,b$ and $c$ satisfy the inequalities $|a|\ge |b+c|,|b|\ge |c+a|$ and $|c|\ge |a+b|$. Prove that $a+b+c=0$.
1995 Baltic Way
Prove that

$$\frac{1995}{2}-\frac{1994}{3}+\frac{1993}{4}-\ldots -\frac{2}{1995}+\frac{1}{1996}=\frac{1}{999}+\frac{3}{1000}+\ldots +\frac{1995}{1996}$$
1995 Baltic Way
Find all real-valued functions $f$ defined on the set of all non-zero real numbers such that:



(i) $f(1)=1$,

(ii) $f\left(\frac{1}{x+y}\right)=f\left(\frac{1}{x}\right)+f\left(\frac{1}{y}\right)$ for all non-zero $x,y,x+y$,

(iii) $(x+y)\cdot f(x+y)=xy\cdot f(x)\cdot f(y)$ for all non-zero $x,y,x+y$.
1995 Baltic Way
In how many ways can the set of integers $\{1,2,\ldots ,1995\}$ be partitioned into three non-empty sets so that none of these sets contains any pair of consecutive integers?
1995 Baltic Way
Assume we have $95$ boxes and $19$ balls distributed in these boxes in an arbitrary manner. We take $6$ new balls at a time and place them in $6$ of the boxes, one ball in each of the six. Can we, by repeating this process a suitable number of times, achieve a situation in which each of the $95$ boxes contains an equal number of balls?
1995 Baltic Way
Consider the following two person game. A number of pebbles are situated on the table. Two players make their moves alternately. A move consists of taking off the table $x$ pebbles where $x$ is the square of any positive integer. The player who is unable to make a move loses. Prove that there are infinitely many initial situations in which the second player can win no matter how his opponent plays.
1995 Baltic Way
There are $n$ fleas on an infinite sheet of triangulated paper. Initially the fleas are in different small triangles, all of which are inside some equilateral triangle consisting of $n^2$ small triangles. Once a second each flea jumps from its original triangle to one of the three small triangles having a common vertex but no common side with it. For which natural numbers $n$ does there exist an initial configuration such that after a finite number of jumps all the $n$ fleas can meet in a single small triangle?
1995 Baltic Way
A polygon with $2n+1$ vertices is given. Show that it is possible to assign numbers $1,2,\ldots ,4n+2$ to the vertices and midpoints of the sides of the polygon so that for each side the sum of the three numbers assigned to it is the same.
1995 Baltic Way
In the triangle $ABC$, let $\ell$ be the bisector of the external angle at $C$. The line through the midpoint $O$ of $AB$ parallel to $\ell$ meets $AC$ at $E$. Determine $|CE|$, if $|AC|=7$ and $|CB|=4$.
1995 Baltic Way
Prove that there exists a number $\alpha$ such that for any triangle $ABC$ the inequality

$$ \max(h_A,h_B,h_C)\le \alpha\cdot\min(m_A,m_B,m_C)$$

where $h_A,h_B,h_C$ denote the lengths of the altitudes and $m_A,m_B,m_C$ denote the lengths of the medians. Find the smallest possible value of $\alpha$.
1995 Baltic Way
Let $M$ be the midpoint of the side $AC$ of a triangle $ABC$ and let $H$ be the foot of the altitude from $B$. Let $P$ and $Q$ be orthogonal projections of $A$ and $C$ on the bisector of the angle $B$. Prove that the four points $H,P,M$ and $Q$ lie on the same circle.
1995 Baltic Way
The following construction is used for training astronauts:

A circle $C_2$ of radius $2R$ rolls along the inside of another, fixed circle $C_1$ of radius $nR$, where $n$ is an integer greater than $2$. The astronaut is fastened to a third circle $C_3$ of radius $R$ which rolls along the inside of circle $C_2$ in such a way that the touching point of the circles $C_2$ and $C_3$ remains at maximum distance from the touching point of the circles $C_1$ and $C_2$ at all times. How many revolutions (relative to the ground) does the astronaut perform together with the circle $C_3$ while the circle $C_2$ completes one full lap around the inside of circle $C_1$?
1994 Baltic Way
Let $a\circ b=a+b-ab$. Find all triples $(x,y,z)$ of integers such that

$$(x\circ y)\circ z +(y\circ z)\circ x +(z\circ x)\circ y=0$$
1994 Baltic Way
Let $a_1,a_2,\ldots ,a_9$ be any non-negative numbers such that $a_1=a_9=0$ and at least one of the numbers is non-zero. Prove that for some $i$, $2\le i\le 8$, the inequality $a_{i-1}+a_{i+1}<2a_i$ holds. Will the statement remain true if we change the number $2$ in the last inequality to $1.9$?
1994 Baltic Way
Find the largest value of the expression

$$xy+x\sqrt{1-x^2}+y\sqrt{1-y^2}-\sqrt{(1-x^2)(1-y^2)}$$
1994 Baltic Way
Is there an integer $n$ such that $\sqrt{n-1}+\sqrt{n+1}$ is a rational number?
1994 Baltic Way
Let $p(x)$ be a polynomial with integer coefficients such that both equations $p(x)=1$ and $p(x)=3$ have integer solutions. Can the equation $p(x)=2$ have two different integer solutions?
1994 Baltic Way
Prove that any irreducible fraction $p/q$, where $p$ and $q$ are positive integers and $q$ is odd, is equal to a fraction $\frac{n}{2^k-1}$ for some positive integers $n$ and $k$.
1994 Baltic Way
Let $p>2$ be a prime number and

$$1+\frac{1}{2^3}+\frac{1}{3^3}+\ldots +\frac{1}{(p-1)^3}=\frac{m}{n}$$

where $m$ and $n$ are relatively prime. Show that $m$ is a multiple of $p$.