Navvye/Olympiad-Problems-CSV huggingface dataset:
2003 Baltic Way
Prove that any real solution of $x^3+px+q=0$, where $p,q$ are real numbers, satisfies the inequality $4qx\le p^2$.
2003 Baltic Way
Let $x$, $y$ and $z$ be positive real numbers such that $xyz = 1$. Prove that

$$\left(1+x\right)\left(1+y\right)\left(1+z\right)\geq 2\left(1+\sqrt[3]{\frac{x}{z}}+\sqrt[3]{\frac{y}{x}}+\sqrt[3]{\frac{z}{y}}\right).$$
2003 Baltic Way
Let $a,b,c$ be positive real numbers. Prove that

$$ \frac{2a}{a^{2}+bc}+\frac{2b}{b^{2}+ca}+\frac{2c}{c^{2}+ab}\leq\frac{a}{bc}+\frac{b}{ca}+\frac{c}{ab} $$
2003 Baltic Way
The sequence $(a_n)$ is defined by $a_1=\sqrt{2}$, $a_2=2$, and $a_{n+1}=a_na_{n-1}^2$ for $n\ge 2$. Prove that for every $n\ge 1$

$$(1+a_1)(1+a_2)\cdots (1+a_n)<(2+\sqrt{2})a_1a_2\cdots a_n. $$
2003 Baltic Way
Let $n\ge 2$ and $d\ge 1$ be integers with $d\mid n$, and let $x_1,x_2,\ldots x_n$ be real numbers such that $x_1+x_2+\cdots + x_n=0$. Show that there are at least $\binom{n-1}{d-1}$ choices of $d$ indices $1\le i_1<i_2<\cdots <i_d\le n $ such that $x_{i_{1}}+x_{i_{2}}+\cdots +x_{i_{d}}\ge 0$.
2003 Baltic Way
A subset of $X$ of $\{1,2,3, \ldots 10000 \}$ has the following property: If $a,b$ are distinct elements of $X$, then $ab\not\in X$. What is the maximal number of elements in $X$?
2003 Baltic Way
There are $2003$ pieces of candy on a table. Two players alternately make moves. A move consists of eating one candy or half of the candies on the table (the “lesser half” if there are an odd number of candies). At least one candy must be eaten at each move. The loser is the one who eats the last candy. Which player has a winning strategy?
2003 Baltic Way
It is known that $n$ is a positive integer and $n \le 144$. Ten questions of the type “Is $n$ smaller than $a$?” are allowed. Answers are given with a delay: for $i = 1, \ldots , 9$, the $i$-th question is answered only after the $(i + 1)$-th question is asked. The answer to the tenth question is given immediately.

Find a strategy for identifying $n$.
2003 Baltic Way
A lattice point in the plane is a point with integral coordinates. The centroid of four points $(x_i,y_i )$, $i = 1, 2, 3, 4$, is the point  $\left(\frac{x_1 +x_2 +x_3 +x_4}{4},\frac{y_1 +y_2 +y_3 +y_4 }{4}\right)$.

Let $n$ be the largest natural number for which there are $n$ distinct lattice points in the plane such that the centroid of any four of them is not a lattice point. Prove that $n = 12$.
2003 Baltic Way
Is it possible to select $1000$ points in the plane so that $6000$ pairwise distances between them are equal?
2003 Baltic Way
Points $M$ and $N$ are taken on the sides $BC$ and $CD$ respectively of a square $ABCD$ so that $\angle MAN=45^{\circ}$. Prove that the circumcentre of $\triangle AMN$ lies on $AC$.
2003 Baltic Way
In a rectangle $ABCD$ be a rectangle and $BC = 2AB$, let $E$ be the midpoint of $BC$ and $P$ an arbitrary inner point of $AD$. Let $F$ and $G$ be the feet of perpendiculars drawn correspondingly from $A$ to $BP$ and from $D$ to $CP$. Prove that the points $E,F,P,G$ are concyclic.
2003 Baltic Way
Equilateral triangles $AMB,BNC,CKA$ are constructed on the exterior of a triangle $ABC$. The perpendiculars from the midpoints of $MN, NK, KM$ to the respective lines $CA, AB, BC$ are constructed. Prove that these three perpendiculars pass through a single point.
2003 Baltic Way
The diagonals of a cyclic convex quadrilateral $ABCD$ intersect at $P$. A circle through $P$ touches the side $CD$ at its midpoint $M$ and intersects the segments $BD$ and $AC$ again at the points $Q$ and $R$ respectively. Let $S$ be the point on segment $BD$ such that $BS = DQ$. The line through $S$ parallel to $AB$ intersects $AC$ at $T$. Prove that $AT = RC$.
2003 Baltic Way
Find all pairs of positive integers $(a,b)$ such that $a-b$ is a prime number and $ab$ is a perfect square.
2003 Baltic Way
All the positive divisors of a positive integer $n$ are stored into an increasing array. Mary is writing a programme which decides for an arbitrarily chosen divisor $d > 1$ whether it is a prime. Let $n$ have $k$ divisors not greater than $d$. Mary claims that it suffices to check divisibility of $d$ by the first $\left\lceil\frac{k}{2}\right\rceil$ divisors of $n$: $d$ is prime if and only if none of them but $1$ divides $d$.

Is Mary right?
2003 Baltic Way
Every integer is to be coloured blue, green, red, or yellow. Can this be done in such a way that if $a, b, c, d$ are not all $0$ and have the same colour, then $3a-2b \neq 2c-3d$?



[Mod edit: Question fixed]
2003 Baltic Way
Let $a$ and $b$ be positive integers. Show that if $a^3+b^3$ is the square of an integer, then $a + b$ is not a product of two different prime numbers.
2003 Baltic Way
Suppose that the sum of all positive divisors of a natural number $n$, $n$ excluded, plus the number of these divisors is equal to $n$. Prove that $n = 2m^2$ for some integer $m$.
2002 Baltic Way
Solve the system of simultaneous equations

$$\begin{cases}a^3+3ab^2+3ac^2-6abc=1\\ b^3+3ba^2+3bc^2-6abc=1\\c^3+3ca^2+3cb^2-6abc=1\end{cases}$$

in real numbers.
2002 Baltic Way
Let $a,b,c,d$ be real numbers such that

$$a+b+c+d=-2$$

$$ab+ac+ad+bc+bd+cd=0$$

Prove that at least one of the numbers $a,b,c,d$ is not greater than $-1$.
2002 Baltic Way
Find all sequences $0\le a_0\le a_1\le a_2\le \ldots$ of real numbers such that

$$a_{m^2+n^2}=a_m^2+a_n^2 $$

for all integers $m,n\ge 0$.
2002 Baltic Way
Let $n$ be a positive integer. Prove that

$$\sum_{i=1}^nx_i(1-x_i)^2\le\left(1-\frac{1}{n}\right)^2 $$

for all nonnegative real numbers $x_1,x_2,\ldots ,x_n$ such that $x_1+x_2+\ldots x_n=1$.
2002 Baltic Way
Find all pairs $(a,b)$ of positive rational numbers such that

$$\sqrt{a}+\sqrt{b}=\sqrt{2+\sqrt{3}}. $$
2002 Baltic Way
The following solitaire game is played on an $m\times n$ rectangular board, $m,n\ge 2$, divided into unit squares. First, a rook is placed on some square. At each move, the rook can be moved an arbitrary number of squares horizontally or vertically, with the extra condition that each move has to be made in the $90^{\circ}$ clockwise direction compared to the previous one (e.g. after a move to the left, the next one has to be done upwards, the next one to the right etc). For which values of $m$ and $n$ is it possible that the rook visits every square of the board exactly once and returns to the first square? (The rook is considered to visit only those squares it stops on, and not the ones it steps over.)
2002 Baltic Way
We draw $n$ convex quadrilaterals in the plane. They divide the plane into regions (one of the regions is infinite). Determine the maximal possible number of these regions.
2002 Baltic Way
Let $P$ be a set of $n\ge 3$ points in the plane, no three of which are on a line. How many possibilities are there to choose a set $T$ of $\binom{n-1}{2}$ triangles, whose vertices are all in $P$, such that each triangle in $T$ has a side that is not a side of any other triangle in $T$?
2002 Baltic Way
Two magicians show the following trick. The first magician goes out of the room. The second magician takes a deck of $100$ cards labelled by numbers $1,2,\ldots ,100$ and asks three spectators to choose in turn one card each. The second magician sees what card each spectator has taken. Then he adds one more card from the rest of the deck. Spectators shuffle these $4$ cards, call the first magician and give him these $4$ cards. The first magician looks at the $4$ cards and “guesses” what card was chosen by the first spectator, what card by the second and what card by the third. Prove that the magicians can perform this trick.
2002 Baltic Way
Let $N$ be a positive integer. Two persons play the following game. The first player writes a list of positive integers not greater than $25$, not necessarily different, such that their sum is at least $200$. The second player wins if he can select some of these numbers so that their sum $S$ satisfies the condition $200-N\le S\le 200+N$. What is the smallest value of $N$ for which the second player has a winning strategy?
2002 Baltic Way
Let $n$ be a positive integer. Consider $n$ points in the plane such that no three of them are collinear and no two of the distances between them are equal. One by one, we connect each point to the two points nearest to it by line segments (if there are already other line segments drawn to this point, we do not erase these). Prove that there is no point from which line segments will be drawn to more than $11$ points.
2002 Baltic Way
A set $S$ of four distinct points is given in the plane. It is known that for any point $X\in S$ the remaining points can be denoted by $Y,Z$ and $W$ so that

$|XY|=|XZ|+|XW|$

Prove that all four points lie on a line.
2002 Baltic Way
Let $ABC$ be an acute triangle with $\angle BAC>\angle BCA$, and let $D$ be a point on side $AC$ such that $|AB|=|BD|$. Furthermore, let $F$ be a point on the circumcircle of triangle $ABC$ such that line $FD$ is perpendicular to side $BC$ and points $F,B$ lie on different sides of line $AC$. Prove that line $FB$ is perpendicular to side $AC$ .
2002 Baltic Way
Let $L,M$ and $N$ be points on sides $AC,AB$ and $BC$ of triangle $ABC$, respectively, such that $BL$ is the bisector of angle $ABC$ and segments $AN,BL$ and $CM$ have a common point. Prove that if $\angle ALB=\angle MNB$ then $\angle LNM=90^{\circ}$.
2002 Baltic Way
A spider and a fly are sitting on a cube. The fly wants to maximize the shortest path to the spider along the surface of the cube. Is it necessarily best for the fly to be at the point opposite to the spider?

(“Opposite” means “symmetric with respect to the centre of the cube”.)
2002 Baltic Way
Find all nonnegative integers $m$ such that

$$a_m=(2^{2m+1})^2+1 $$

is divisible by at most two different primes.
2002 Baltic Way
Show that the sequence

$$\binom{2002}{2002},\binom{2003}{2002},\binom{2004}{2002},\ldots $$

considred modulo $2002$, is periodic.
2002 Baltic Way
Find all integers $n>1$ such that any prime divisor of $n^6-1$ is a divisor of $(n^3-1)(n^2-1)$.
2002 Baltic Way
Does there exist an infinite non-constant arithmetic progression, each term of which is of the form $a^b$, where $a$ and $b$ are positive integers with $b\ge 2$?
2001 Baltic Way
A set of $8$ problems was prepared for an examination. Each student was given $3$ of them. No two students received more than one common problem. What is the largest possible number of students?
2001 Baltic Way
Let $n\ge 2$ be a positive integer. Find whether there exist $n$ pairwise nonintersecting nonempty subsets of $\{1, 2, 3, \ldots \}$ such that each positive integer can be expressed in a unique way as a sum of at most $n$ integers, all from different subsets.
2001 Baltic Way
The numbers $1, 2, \ldots 49$ are placed in a $7\times 7$ array, and the sum of the numbers in each row and in each column is computed. Some of these $14$ sums are odd while others are even. Let $A$ denote the sum of all the odd sums and $B$ the sum of all even sums. Is it possible that the numbers were placed in the array in such a way that $A = B$?
2001 Baltic Way
Let $p$ and $q$ be two different primes. Prove that

$$\left\lfloor\frac{p}{q}\right\rfloor+\left\lfloor\frac{2p}{q}\right\rfloor+\left\lfloor\frac{3p}{q}\right\rfloor+\ldots +\left\lfloor\frac{(q-1)p}{q}\right\rfloor=\frac{1}{2}(p-1)(q-1) $$
2001 Baltic Way
Let $2001$ given points on a circle be coloured either red or green. In one step all points are recoloured simultaneously in the following way: If both direct neighbours of a point $P$ have the same colour as $P$, then the colour of $P$ remains unchanged, otherwise $P$ obtains the other colour. Starting with the first colouring $F_1$, we obtain the colourings $F_2,F_3 ,\ldots .$ after several recolouring steps. Prove that there is a number $n_0\le 1000$ such that $F_{n_0}=F_{n_0 +2}$. Is the assertion also true if $1000$ is replaced by $999$?
2001 Baltic Way
The points $A, B, C, D, E$ lie on the circle $c$ in this order and satisfy $AB\parallel EC$ and $AC\parallel ED$. The line tangent to the circle $c$ at $E$ meets the line $AB$ at $P$. The lines $BD$ and $EC$ meet at $Q$. Prove that $|AC|=|PQ|$.
2001 Baltic Way
Given a parallelogram $ABCD$. A circle passing through $A$ meets the line segments $AB, AC$ and $AD$ at inner points $M,K,N$, respectively. Prove that

$$|AB|\cdot |AM | + |AD|\cdot |AN|=|AK|\cdot |AC|$$
2001 Baltic Way
Let $ABCD$ be a convex quadrilateral, and let $N$ be the midpoint of $BC$. Suppose further that $\angle AND=135^{\circ}$.

Prove that $|AB|+|CD|+\frac{1}{\sqrt{2}}\cdot |BC|\ge |AD|.$
2001 Baltic Way
Given a rhombus $ABCD$, find the locus of the points $P$ lying inside the rhombus and satisfying $\angle APD+\angle BPC=180^{\circ}$.
2001 Baltic Way
In a triangle $ABC$, the bisector of $\angle BAC$ meets the side $BC$ at the point $D$. Knowing that $|BD|\cdot |CD|=|AD|^2$ and $\angle ADB=45^{\circ}$, determine the angles of triangle $ABC$.
2001 Baltic Way
The real-valued function $f$ is defined for all positive integers. For any integers $a>1, b>1$ with $d=\gcd (a, b)$, we have

$$f(ab)=f(d)\left(f\left(\frac{a}{d}\right)+f\left(\frac{b}{d}\right)\right) $$

Determine all possible values of $f(2001)$.
2001 Baltic Way
Let $a_1, a_2,\ldots , a_n$ be positive real numbers such that $\sum_{i=1}^na_i^3=3$ and $\sum_{i=1}^na_i^5=5$. Prove that $\sum_{i=1}^na_i>\frac{3}{2}$.
2001 Baltic Way
Let $a_0, a_1, a_2,\ldots $ be a sequence of real numbers satisfying $a_0=1$ and $a_n=a_{\lfloor 7n/9\rfloor}+a_{\lfloor n/9\rfloor}$ for $n=1, 2,\ldots $

Prove that there exists a positive integer $k$ with $a_k<\frac{k}{2001!}$.
2001 Baltic Way
There are $2n$ cards. On each card some real number $x$, $(1\le x\le 2n)$, is written (there can be different numbers on different cards). Prove that the cards can be divided into two heaps with sums $s_1$ and $s_2$ so that $\frac{n}{n+1}\le\frac{s_1}{s_2}\le 1$.
2001 Baltic Way
Let $a_0,a_1,a_2,\ldots $ be a sequence of positive real numbers satisfying $i\cdot a_2\ge (i + 1)\cdot a_{i_1}a_{i+1}$ for $i=1, 2, \ldots $ Furthermore, let $x$ and $y$ be positive reals, and let $b_i=xa_i+ya_{i-1}$ for $i=1, 2, \ldots $

Prove that the inequality $i\cdot b_2\ge (i + 1)\cdot b_{i-1}b_{i+1}$ holds for all integers $i\ge 2$.
2001 Baltic Way
Let $f$ be a real-valued function defined on the positive integers satisfying the following condition: For all $n>1$ there exists a prime divisor $p$ of $n$ such that $f(n)=f\left(\frac{n}{p}\right)-f(p)$. Given that $f(2001)=1$, what is the value of $f(2002)$?
2001 Baltic Way
Let $n$ be a positive integer. Prove that at least $2^{n-1}+n$ numbers can be chosen from the set $\{1, 2, 3,\ldots ,2^n\}$ such that for any two different chosen numbers $x$ and $y$, $x+y$ is not a divisor of $x\cdot y$.
2001 Baltic Way
Let $a$ be an odd integer. Prove that $a^{2^m}+2^{2^m}$ and $a^{2^n}+2^{2^n}$ are relatively prime for all positive integers $n$ and $m$ with $n\not= m$.
2001 Baltic Way
What is the smallest positive odd integer having the same number of positive divisors as $360$?
2001 Baltic Way
From a sequence of integers $(a, b, c, d)$ each of the sequences

$$(c, d, a, b),\quad (b, a, d, c),\quad (a + nc, b + nd, c, d),\quad (a + nb, b, c + nd, d)$$

for arbitrary integer $n$ can be obtained by one step. Is it possible to obtain $(3, 4, 5, 7)$ from $(1, 2, 3, 4)$ through a sequence of such steps?
2000 Baltic Way
Let $ K$ be a point inside the triangle $ ABC$. Let $ M$ and $ N$ be points such that $ M$ and $ K$ are on opposite sides of the line $ AB$, and $ N$ and $ K$ are on opposite sides of the line $ BC$. Assume that $ \angle MAB = \angle MBA = \angle NBC = \angle NCB = \angle KAC = \angle KCA$. Show that $ MBNK$ is a parallelogram.
2000 Baltic Way
Given an isosceles triangle $ ABC$ with $ \angle A = 90^{\circ}$. Let $ M$ be the midpoint of $ AB$. The line passing through $ A$ and perpendicular to $ CM$ intersects the side $ BC$ at $ P$. Prove that $ \angle AMC = \angle BMP$.
2000 Baltic Way
Given a triangle $ ABC$ with $ \angle A = 90^{\circ}$ and $ AB \neq AC$. The points $ D$, $ E$, $ F$ lie on the sides $ BC$, $ CA$, $ AB$, respectively, in such a way that $ AFDE$ is a square. Prove that the line $ BC$, the line $ FE$ and the line tangent at the point $ A$ to the circumcircle of the triangle $ ABC$ intersect in one point.
2000 Baltic Way
Given a triangle $ ABC$ with $ \angle A = 120^{\circ}$. The points $ K$ and $ L$ lie on the sides $ AB$ and $ AC$, respectively. Let $ BKP$ and $ CLQ$ be equilateral triangles constructed outside the triangle $ ABC$. Prove that $ PQ \ge\frac{\sqrt 3}{2}\left(AB + AC\right)$.
2000 Baltic Way
Let $ ABC$ be a triangle such that $$ \frac{BC}{AB - BC}=\frac{AB + BC}{AC}$$ Determine the ratio $ \angle A : \angle C$.
2000 Baltic Way
Fredek runs a private hotel. He claims that whenever $ n \ge 3$ guests visit the hotel, it is possible to select two guests that have equally many acquaintances among the other guests, and that also have a common acquaintance or a common unknown among the guests. For which values of $ n$ is Fredek right? (Acquaintance is a symmetric relation.)
2000 Baltic Way
In a $ 40 \times 50$ array of control buttons, each button has two states: on and off. By touching a button, its state and the states of all buttons in the same row and in the same column are switched. Prove that the array of control buttons may be altered from the all-off state to the all-on state by touching buttons successively, and determine the least number of touches needed to do so.
2000 Baltic Way
Fourteen friends met at a party. One of them, Fredek, wanted to go to bed early. He said goodbye to 10 of his friends, forgot about the remaining 3, and went to bed. After a while he returned to the party, said goodbye to 10 of his friends (not necessarily the same as before), and went to bed. Later Fredek came back a number of times, each time saying goodbye to exactly 10 of his friends, and then went back to bed. As soon as he had said goodbye to each of his friends at least once, he did not come back again. In the morning Fredek realized that he had said goodbye a different number of times to each of his thirteen friends! What is the smallest possible number of times that Fredek returned to the party?
2000 Baltic Way
There is a frog jumping on a $ 2k \times 2k$ chessboard, composed of unit squares. The frog's jumps are $ \sqrt{1 + k^2}$ long and they carry the frog from the center of a square to the center of another square. Some $ m$ squares of the board are marked with an $ \times$, and all the squares into which the frog can jump from an $ \times$'d square (whether they carry an $ \times$ or not) are marked with an $ \circ$. There are $ n$ $ \circ$'d squares. Prove that $ n \ge m$.
2000 Baltic Way
Two positive integers are written on the blackboard. Initially, one of them is $2000$ and the other is smaller than $2000$. If the arithmetic mean $ m$ of the two numbers on the blackboard is an integer, the following operation is allowed: one of the two numbers is erased and replaced by $ m$. Prove that this operation cannot be performed more than ten times. Give an example where the operation is performed ten times.
2000 Baltic Way
A sequence of positive integers $a_1,a_2,\ldots $ is such that for each $m$ and $n$ the following holds: if $m$ is a divisor of $n$ and $m<n$, then $a_m$ is a divisor of $a_n$ and $a_m<a_n$. Find the least possible value of $a_{2000}$.
2000 Baltic Way
Let $x_1,x_2,\ldots x_n$ be positive integers such that no one of them is an initial fragment of any other (for example, $12$ is an initial fragment of $\underline{12},\underline{12}5$ and $\underline{12}405$). Prove that

$$\frac{1}{x_1}+\frac{1}{x_2}+\ldots+\frac{1}{x_n}<3. $$
2000 Baltic Way
Let $a_1,a_2 ,\ldots, a_n$ be an arithmetic progression of integers such that $i|a_i$ for $i=1, 2,\ldots ,n-1$ and $n\nmid a_n$. Prove that $n$ is a prime power.
2000 Baltic Way
Find all positive integers $n$ such that $n$ is equal to $100$ times the number of positive divisors of $n$.
2000 Baltic Way
Let $n$ be a positive integer not divisible by $2$ or $3$. Prove that for all integers $k$, the number $(k+1)^n-k^n-1$ is divisible by $k^2+k+1$.
2000 Baltic Way
Prove that for all positive real numbers $a,b,c$ we have

$$\sqrt{a^2-ab+b^2}+\sqrt{b^2-bc+c^2}\ge\sqrt{a^2+ac+c^2} $$
2000 Baltic Way
Find all real solutions to the following system of equations:

$$\begin{cases}  x+y+z+t=5\\xy+yz+zt+tx=4\\xyz+yzt+ztx+txy=3\\xyzt=-1\end{cases}$$
2000 Baltic Way
Determine all positive real numbers $x$ and $y$ satisfying the equation

$$x+y+\frac{1}{x}+\frac{1}{y}+4=2\cdot (\sqrt{2x+1}+\sqrt{2y+1})$$
2000 Baltic Way
Let $t\ge\frac{1}{2}$ be a real number and $n$ a positive integer. Prove that

$$t^{2n}\ge (t-1)^{2n}+(2t-1)^n$$
2000 Baltic Way
For every positive integer $n$, let

$$x_n=\frac{(2n+1)(2n+3)\cdots (4n-1)(4n+1)}{(2n)(2n+2)\cdots (4n-2)(4n)}$$

Prove that $\frac{1}{4n}<x_n-\sqrt{2}<\frac{2}{n}$.
1999 Baltic Way
Determine all real numbers $a,b,c,d$ that satisfy the following equations

$$\begin{cases} abc + ab + bc + ca + a + b + c = 1\\  bcd + bc + cd + db + b + c + d = 9\\ cda + cd + da + ac + c + d + a = 9\\  dab + da + ab + bd + d + a + b = 9\end{cases}$$
1999 Baltic Way
Determine all positive integers $n$ with the property that the third root of $n$ is obtained by removing its last three decimal digits.
1999 Baltic Way
Determine all positive integers $n\ge 3$ such that the inequality

$$a_1a_2+a_2a_3+\ldots a_{n-1}a_n\le 0$$

holds for all real numbers $a_1,a_2,\ldots , a_n$ which satisfy $a_1+a_2+\ldots +a_n=0$.
1999 Baltic Way
For all positive real numbers $x$ and $y$ let

$$f(x,y)=\min\left( x,\frac{y}{x^2+y^2}\right) $$

Show that there exist $x_0$ and $y_0$ such that $f(x, y)\le f(x_0, y_0)$ for all positive $x$ and $y$, and find $f(x_0,y_0)$.
1999 Baltic Way
The point $(a,b)$ lies on the circle $x^2+y^2=1$. The tangent to the circle at this point meets the parabola $y=x^2+1$ at exactly one point. Find all such points $(a,b)$.
1999 Baltic Way
What is the least number of moves it takes a knight to get from one corner of an $n\times n$ chessboard, where $n\ge 4$, to the diagonally opposite corner?
1999 Baltic Way
Two squares on an $8\times 8$ chessboard are called adjacent if they have a common edge or common corner. Is it possible for a king to begin in some square and visit all squares exactly once in such a way that all moves except the first are made into squares adjacent to an even number of squares already visited?
1999 Baltic Way
We are given $1999$ coins. No two coins have the same weight. A machine is provided which allows us with one operation to determine, for any three coins, which one has the middle weight. Prove that the coin that is the $1000$th by weight can be determined using no more than $1000000$ operations and that this is the only coin whose position by weight can be determined using this machine.
1999 Baltic Way
A cube with edge length $3$ is divided into $27$ unit cubes. The numbers $1, 2,\ldots ,27$ are distributed arbitrarily over the unit cubes, with one number in each cube. We form the $27$ possible row sums (there are nine such sums of three integers for each of the three directions parallel with the edges of the cube). At most how many of the $27$ row sums can be odd?
1999 Baltic Way
May the points of a disc of radius $1$ (including its circumference) be partitioned into three subsets in such a way that no subset contains two points separated by a distance $1$?
1999 Baltic Way
Prove that for any four points in the plane, no three of which are collinear, there exists a circle such that three of the four points are on the circumference and the fourth point is either on the circumference or inside the circle.
1999 Baltic Way
In a triangle $ABC$ it is given that $2AB=AC+BC$. Prove that the incentre of $\triangle ABC$, the circumcentre of $\triangle ABC$, and the midpoints of $AC$ and $BC$ are concyclic.
1999 Baltic Way
The bisectors of the angles $A$ and $B$ of the triangle $ABC$ meet the sides $BC$ and $CA$ at the points $D$ and $E$, respectively. Assuming that $AE+BD=AB$, determine the angle $C$.