Navvye/Olympiad-Problems-CSV huggingface dataset:
1999 Tournament Of Towns
$n$ diameters divide a disk into $2n$ equal sectors. $n$ of the sectors are coloured blue , and the other $n$ are coloured red (in arbitrary order) . Blue sectors are numbered from $1$ to $n$ in the anticlockwise direction, starting from an arbitrary blue sector, and red sectors are numbered from $1$ to $n$ in the clockwise direction, starting from an arbitrary red sector. Prove that there is a semi-disk containing sectors with all numbers from $1$ to $n$.



(V Proizvolov)
1999 Tournament Of Towns
The sides $AB$ and $AC$ are tangent at points $P$ and $Q$, respectively, to the incircle of a triangle $ABC. R$ and $S$ are the midpoints of the sides $AC$ and $BC$, respectively, and $T$ is the intersection point of the lines $PQ$ and $RS$. Prove that $T$ lies on the bisector of the angle $B$ of the triangle.



(M Evdokimov)
1999 Tournament Of Towns
A rook is allowed to move one cell either horizontally or vertically. After $64$ moves the rook visited all cells of the $8 \times 8$ chessboard and returned back to the initial cell. Prove that the number of moves in the vertical direction and the number of moves in the horizontal direction cannot be equal.



(A Shapovalov, R Sadykov)
1999 Tournament Of Towns
In a row are written $1999$ numbers such that except the first and the last , each is equal to the sum of its neighbours. If the first number is $1$, find the last number.



(V Senderov)
1999 Tournament Of Towns
Two people play a game on a $9 \times 9$ board. They move alternately. On each move, the first player draws a cross in an empty cell, and the second player draws a nought in an empty cell. When all $81$ cells are filled, the number $K$ of rows and columns in which there are more crosses and the number $H$ of rows and columns in which there are more noughts are counted. The score for the first player is the difference $B = K- H$. Find a value of $B$ such that the first player can guarantee a score of at least $B$, while the second player can hold the first player's score to at most B, regardless how the opponent plays.



(A Kanel)
1999 Tournament Of Towns
A convex polyhedron is floating in a sea. Can it happen that $90\%$ of its volume is below the water level, while more than half of its surface area is above the water level?



(A Shapovalov)
1999 Tournament Of Towns
Let all vertices of a convex quadrilateral $ABCD$ lie on the circumference of a circle with centre $O$. Let $F$ be the second point of intersection of the circumcircles of the triangles $ABO$ and $CDO$. Prove that the circle passing through the points $A, F$ and $D$ also passes through the point of intersection of the segments $AC$ and $BD$.



(A Zaslavskiy)
1999 Tournament Of Towns
Find all pairs $(x, y)$ of integers satisfying the following condition:

each of the numbers $x^3 + y$ and $x + y^3$ is divisible by $x^2 + y^2$ .



(S Zlobin)
1999 Tournament Of Towns
For every non-negative integer $i$, define the number $M(i)$ as follows:

write $i$ down as a binary number, so that we have a string of zeroes and ones, if the number of ones in this string is even, then set $M(i) = 0$, otherwise set $M(i) = 1$.  (The first terms of the sequence $M(i)$, $i = 0, 1, 2, ...$ are $0, 1, 1, 0, 1, 0, 0, 1,...$ )

(a) Consider the finite sequence $M(O), M(1), . . . , M(1000) $.

Prove that there are at least $320$ terms in this sequence which are equal to their neighbour on the right : $M(i) = M(i + 1 )$ .

(b) Consider the finite sequence $M(O), M(1), . . . , M(1000000)$ .

Prove that the number of terms $M(i)$ such that $M(i) = M(i +7)$ is at least $450000$.



(A Kanel)
1999 Tournament Of Towns
A right-angled triangle made of paper is folded along a straight line so that the vertex at the right angle coincides with one of the other vertices of the triangle and a quadrilateral is obtained .

(a) What is the ratio into which the diagonals of this quadrilateral divide each other?

(b) This quadrilateral is cut along its longest diagonal. Find the area of the smallest piece of paper thus obtained if the area of the original triangle is $1$ .



(A Shapovalov)
1999 Tournament Of Towns
Let $d = a^{1999} + b^{1999} + c^{1999}$ , where $a, b$ and $c$ are integers such that $a + b + c = 0$.

(a) May it happen that $d = 2$?

(b) May it happen that $d$ is prime?



(V Senderov)
1999 Tournament Of Towns
There are $n$ straight lines in the plane such that each intersects exactly $1999$ of the others . Find all posssible values of $n$.



(R Zhenodarov)
1999 Tournament Of Towns
Every $24$ hours , the minute hand of an ordinary clock completes $24$ revolutions while the hour hand completes $2$. Every $24$ hours , the minute hand of an Italian clock completes $24$ revolutions while the hour hand completes only $1$ . The minute hand of each clock is longer than the hour hand, and "zero hour" is located at the top of the clock's face. How many positions of the two hands can occur on an Italian clock within a $24$-hour period that are possible on an ordinary one?



(Folklore)
1999 Tournament Of Towns
Is it possible to divide a $6 \times 6$ chessboard into $18$ rectangles, each either $1 \times 2$ or $2 \times 1$, and to draw exactly one diagonal on each rectangle such that no two of these diagonals have a common endpoint?



(A Shapovalov)
1999 Tournament Of Towns
The incentre of a triangle is joined by three segments to the three vertices of the triangle, thereby dividing it into three smaller triangles. If one of these three triangles is similar to the original triangle, find its angles.



(A Shapovalov)
1999 Tournament Of Towns
Prove that there exist infinitely many odd positive integers $n$ for which the number $2^n + n$ is composite.



(V Senderov)
1999 Tournament Of Towns
Is it possible to divide the integers from $1$ to $100$ inclusive into $50$ pairs such that for $1\le  k\le 50$, the difference between the two numbers in the $k$-th pair is $k$?



(V Proizvolov)
1999 Tournament Of Towns
Is it possible to divide a $8 \times 8$ chessboard into $32$ rectangles, each either $1 \times 2$ or $2 \times 1$, and to draw exactly one diagonal on each rectangle such that no two of these diagonals have a common endpoint?



(A Shapovalov)
1999 Tournament Of Towns
For what values o f $n$ is it possible to place the integers from $1$ to $n$ inclusive on a circle (not necessarily in order) so that the sum of any two successive integers in the circle is divisible by the next one in the clockwise order?



(A Shapovalov)
1999 Tournament Of Towns
On a rectangular piece of paper there are

(a) several marked points all on one straight line,

(b) three marked points (not necessarily on a straight line).

We are allowed to fold the paper several times along a straight line not containing marked points and then puncture the folded paper with a needle. Show that this can be done so that after the paper has been unfolded all the marked points are punctured and there are no extra holes.



(A Shapovalov)
1999 Tournament Of Towns
Tireless Thomas and Jeremy construct a sequence. At the beginning there is one positive integer in the sequence. Then they successively write new numbers in the sequence in the following way: Thomas obtains the next number by adding to the previous number one of its (decimal) digits, while Jeremy obtains the next number by subtracting from the previous number one of its digits. Prove that there is a number in this sequence which will be repeated at least $100$ times.



(A Shapovalov)
1999 Tournament Of Towns
Points $K, L$ on sides $AC, CB$ respectively of a triangle $ABC$ are the points of contact of the excircles with the corresponding sides . Prove that the straight line through the midpoints of $KL$ and $AB$

(a) divides the perimeter of triangle $ABC$ in half,

(b) is parallel to the bisector of angle $ACB$.



( L Emelianov)
1999 Tournament Of Towns
(a) The numbers $1, 2,... , 100$ are divided into two groups so that the sum of all numbers in one group is equal to that in the other. Prove that one can remove two numbers from each group so that the sums of all numbers in each group are still the same.

(b) The numbers $1, 2 , ... , n$ are divided into two groups so that the sum of all numbers in one group is equal to that in the other . Is it true that for every such$ n > 4$ one can remove two numbers from each group so that the sums of all numbers in each group are still the same?



(A Shapovalov)
1999 Tournament Of Towns
On a large chessboard $2n$ of its $1 \times 1$ squares  have been marked such thar the rook (which moves only horizontally or vertically) can visit all the marked squares without jumpin over any unmarked ones. Prove that the figure consisting of all the marked squares can be cut into rectangles.



(A Shapovalov)
1999 Tournament Of Towns
Prove that any convex polyhedron with $10n$ faces, has at least $n$ faces with the same number of sides.



(A Kanel)
2000 Tournament Of Towns
Can the product of $2$ consecutive natural numbers equal the product of $2$ consecutive even natural numbers?

(natural means positive integers)
2000 Tournament Of Towns
In a quadrilateral $ABCD$ of area $1$, the parallel sides $BC$ and $AD$ are in the ratio $1 :2$ . $K$ is the midpoint of the diagonal $AC$ and $L$ is the point of intersection of the line $DK$ and the side $AB$. Determine the area of the quadrilateral $BCKL$ .



(M G Sonkin)
2000 Tournament Of Towns
The base of a prism is an $n$-gon. We wish to colour its $2n$ vertices in three colours in such a way that every vertex is connected by edges to vertices of all three colours.

(a) Prove that if $n$ is divisible by $3$, then the task is possible.

{b) Prove that if the task is possible, then $n$ is divisible by $3$.



(A Shapovalov)
2000 Tournament Of Towns
Can one place positive integers at all vertices of a cube in such a way that for every pair of numbers connected by an edge, one will be divisible by the other , and there are no other pairs of numbers with this property?



(A Shapovalov)
2000 Tournament Of Towns
Determine all real numbers that satisfy the equation $(x+1)^{21}+(x+1)^{20}(x-1)+(x+1)^{19}(x-1)^2+...+(x-1)^{21}=0$



(RM Kuznec)
2000 Tournament Of Towns
Two parallel sides of a quadrilateral have integer lengths. Prove that this quadrilateral can be cut into congruent triangles.



(A Shapovalov)
2000 Tournament Of Towns
$A$ is a fixed point inside a given circle. Determine the locus of points $C$ such that $ABCD$ is a rectangle with $B$ and $D$ on the circumference of the given circle.



(M Panov)
2000 Tournament Of Towns
Give and Take divide $100$ coins between themselves as follows. In each step, Give chooses a handful of coins from the heap, and Take decides who gets this handful. This is repeated until all coins have been taken, or one of them has $9$ handfuls. In the latter case, the other gets all the remaining coins. What is the largest number of coins that Give can be sure of getting no matter what Take does?



(A Shapovalov)
2000 Tournament Of Towns
What is the largest number of knights that can be put on a $5 \times 5$ chess board so that each knight attacks exactly two other knights?



(M Gorelov)
2000 Tournament Of Towns
The diagonals of a convex quadrilateral $ABCD$ meet at $P$. The sum of the areas of triangles $PAB$ and $PCD$ is equal to the sum of areas of triangles $PAD$ and $PCB$. Prove that $P$ is the midpoint of either $AC$ or $BD$.



(Folklore)
2000 Tournament Of Towns
Each of a pair of opposite faces of a unit cube is marked by a dot. Each of another pair of opposite faces is marked by two dots. Each of the remaining two faces is marked by three dots. Eight such cubes are used to construct a $2\times 2 \times  2$ cube. Count the total number of dots on each of its six faces. Can we obtain six consecutive numbers?



(A Shapovalov)
2000 Tournament Of Towns
Prove the inequality $$ 1^k+2^k+...+n^k \le \frac{n^{2k}-(n-1)^k}{n^k-(n-1)^k}$$(L Emelianov)
2000 Tournament Of Towns
(a) Does there exist an infinite sequence of real numbers such that the sum of every ten successive numbers is positive, while for every $n$ the sum of the first $10n + 1$ successive numbers is negative?

(b) Does there exist an infinite sequence of integers with the same properties?



(AK Tolpygo)
2000 Tournament Of Towns
Positive integers $m$ and $n$ have no common divisor greater than one. What is the largest possible value of the greatest common divisor of $m + 2000n$ and $n + 2000m$ ?



(S Zlobin)
2000 Tournament Of Towns
The chords $AC$ and $BD$ of a, circle with centre $O$ intersect at the point $K$. The circumcentres of triangles $AKB$ and $CKD$ are $M$ and $N$ respectively. Prove that $OM = KN$.



(A Zaslavsky )
2000 Tournament Of Towns
Peter plays a solitaire game with a deck of cards, some of which are face-up while the others are face-down. Peter loses if all the cards are face-down. As long as at least one card is face up, Peter must choose a stack of consecutive cards from the deck, so that the top and the bottom cards of the stack are face-up. They may be the same card. Then Peter turns the whole stack over and puts it back into the deck in exactly the same place as before. Prove that Peter always loses.



(A Shapovalov)
2000 Tournament Of Towns
Each vertex of a convex polygon has integer coordinates, and no side of this polygon is horizontal or vertical. Prove that the sum of the lengths of the segments of lines of the form $x = m$, $m$ an integer, that lie within the polygon is equal to the sum of the lengths of the segments of lines of the form $y = n$, $n$ an integer, that lie within the polygon.



(G Galperin)
2000 Tournament Of Towns
What is the largest number $N$ for which there exist $N$ consecutive positive integers such that the sum of the digits in the $k$-th integer is divisible by $k$ for $1 \le  k \le  N$ ?



(S Tokarev)
2000 Tournament Of Towns
In a chess tournament , every two participants play each other exactly once. A win is worth one point , a draw is worth half a point and a loss is worth zero points. Looking back at the end of the tournament, a game is called an upset if the total number of points obtained by the winner of that game is less than the total number of points obtained by the loser of that game.

(a) Prove that the number of upsets is always strictly less than three-quarters of the total number of games in the tournament.

(b) Prove that three-quarters cannot be replaced by a smaller number.



(S Tokarev)



PS. part (a) for Juniors, both parts for Seniors
2000 Tournament Of Towns
Each of the $16$ squares in a $4 \times 4$ table contains a number. For any square, the sum of the numbers in the squares sharing a common side with the chosen square is equal to $1$. Determine the sum of all $16$ numbers in the table.



(R Zhenodarov)
2000 Tournament Of Towns
$ABCD$ is parallelogram, $M$ is the midpoint of side $CD$ and $H$ is the foot of the perpendicular from $B$ to line $AM$. Prove that $BCH$ is an isosceles triangle.



(M Volchkevich)
2000 Tournament Of Towns
(a) On a blackboard are written $100$ different numbers. Prove that you can choose $8$ of them so that their average value is not equal to that of any $9$ of the numbers on the blackboard.

(b) On a blackboard are written $100$ integers. For any $8$ of them, you can find $9$ numbers on the blackboard so that the average value of the $8$ numbers is equal to that of the $9$. Prove that all the numbers on the blackboard are equal.



(A Shapovalov)
2000 Tournament Of Towns
Among a set of $32$ coins , all identical in appearance, $30$ are real and $2$ are fake. Any two real coins have the same weight . The fake coins have the same weight , which is different from the weight of a real coin. How can one divide the coins into two groups of equal total weight by using a balance at most $4$ times?



(A Shapovalov)
2000 Tournament Of Towns
Each $1 \times 1$ square of an $n \times n$ table contains a different number. The smallest number in each row is marked, and these marked numbers are in different columns. Then the smallest number in each column is marked, and these marked numbers are in different rows. Prove that the two sets of marked numbers are identical.



(V Klepcyn)
2000 Tournament Of Towns
In triangle $ABC, AB = AC$. A line is drawn through $A$ parallel to $BC$. Outside triangle $ABC$, a circle is drawn tangent to this line, to the line $BC$, to $AB$ and to the incircle of $ABC$. If the radius of this circle is $1$ , determine the inradius of $ABC$.



(RK Gordin)
2000 Tournament Of Towns
The least common multiple of positive integers $a, b, c$ and $d$ is equal to $a + b + c + d$. Prove that $abcd$ is divisible by at least one of $3$ and $5$.



( V Senderov)
2000 Tournament Of Towns
In how many ways can $31$ squares be marked on an $8 \times 8$ chessboard so that no two of the marked squares have a common side?



(R Zhenodarov)
2000 Tournament Of Towns
A weight of $11111$ grams is placed in the left pan of a balance. Weights are added one at a time, the first weighing $1$ gram, and each subsequent one weighing twice as much as the preceding one. Each weight may be added to either pan. After a while, equilibrium is achieved. Is the $16$ gram weight placed in the left pan or the right pan?



( AV Kalinin)
2000 Tournament Of Towns
In the spring round of the Tournament of Towns this year, $6$ problems were posed in the Senior A-Level paper. In a certain country, each problem was solved by exactly $1000$ participants, but no two participants solved all $6$ problems between them. What is the smallest possible number of participants from this country in the spring round Senior A-Level paper?



(R Zhenodarov)
2000 Tournament Of Towns
A student has $100$ cards on which the integers $1$ to $100$ are printed, as well as a sufficiently large number of cards on which the symbols $+$ and $=$ are printed. What is the maximal number of correct equalities the student can construct, if each card is used at most once?



(R Zhenodarov)
2000 Tournament Of Towns
Triangle $ABC$ is inscribed in a circle. Chords $AM$ and $AN$ intersect side $BC$ at points $K$ and $L$ respectively. Prove that if a circle passes through all of the points $K, L, M$ and $N$, then $ABC$ is an isosceles triangle.



(V Zhgun)
2000 Tournament Of Towns
Positive integers $a, b, c, d$ satisfy the inequality $ad - bc > 1$. Prove that at least one of the numbers $a, b, c, d$ is not divisible by $ad - bc$.



(A Spivak)
2000 Tournament Of Towns
In each lateral face of a pentagonal prism at least one of the four angles is equal to $f$. Find all possible values of $f$.



(A Shapovalov)
2000 Tournament Of Towns
Among a set of $2N$ coins, all identical in appearance, $2N - 2$ are real and $2$ are fake. Any two real coins have the same weight . The fake coins have the same weight, which is different from the weight of a real coin. How can one divide the coins into two groups of equal total weight by using a balance at most $4$ times, if

(a) $N = 16$,

( b ) $N = 11$ ?



(A Shapovalov)
2000 Tournament Of Towns
What is the largest integer $n$ such that one can find $n$ points on the surface of a cube, not all lying on one face and being the vertices of a regular $n$-gon?



(A Shapovalov)
2000 Tournament Of Towns
In a triangle $ABC, AB = c, BC = a, CA = b$, and $a < b < c$. Points $B'$ and $A'$ are chosen on the rays $BC$ and $AC$ respectively so that $BB'= AA'= c$. Points $C''$ and $B''$ are chosen on the rays $CA$ and $BA$ so that $CC'' = BB'' = a$. Find the ratio of the segment $A'B'$ to the segment $C'' B''$.



(R Zhenodarov)
2000 Tournament Of Towns
Let $a_1 , a_2 , ..., a_n$ be non-zero integers that satisfy the equation

$$a_1 +\frac{1}{a_2+\frac{1}{a_3+ ... \frac{1}{a_n+\frac{1}{x}}  }  }  = x$$for all values of $x$ for which the lefthand side of the equation makes sense.

(a) Prove that $n$ is even.

(b) What is the smallest n for which such numbers $a_1 , a_2 , ..., a_n$ exist?



(M Skopenko)
2000 Tournament Of Towns
Each of the cells of an $m \times n$ table is coloured either black or white. For each cell, the total number of the cells which are in the same row or in the same column and of the same colour as this cell is strictly less than the total number of the cells which are in the same row or in the same column and of the other colour as this cell. Prove that in each row and in each column the number of white cells is the same as the number of black ones.



(A Shapovalov)
2000 Tournament Of Towns
a) Several black squares of side $1$ cm are nailed to a white plane with a nail of thickness $0 . 1$ cm so that they form a black polygon. Can it happen that the perimeter of this polygon is $1$ km long? (The nail is not allowed to touch the boundary of any of the squares . )

(b) The same problem as in (a) but with a nail of thickness $0$ (a point ) .

(c) Several black squares of side $1$ cm lie on a white plane so that they form a black polygon (possibly having more than one piece and/ or having holes) . Can it happen that the ratio of its perimeter (in centimetres) to its area (in square centimetres) is more than $100000$?



(Hungarian Folklore)
2001 Tournament Of Towns
The natural number $n$ can be replaced by $ab$ if $a + b = n$, where $a$ and $b$ are natural numbers. Can the number $2001$ be obtained from $22$ after a sequence of such replacements?
2001 Tournament Of Towns
One of the midlines of a triangle is longer than one of its medians. Prove that the triangle has an obtuse angle.
2001 Tournament Of Towns
Twenty kilograms of cheese are on sale in a grocery store. Several customers are lined up to buy this cheese. After a while, having sold the demanded portion of cheese to the next customer, the salesgirl calculates the average weight of the portions of cheese already sold and declares the number of customers for whom there is exactly enough cheese if each customer will buy a portion of cheese of weight exactly equal to the average weight of the previous purchases. Could it happen that the salesgirl can declare, after each of the first $10$ customers has made their purchase, that there just enough cheese for the next $10$ customers? If so, how much cheese will be left in the store after the first $10$ customers have made their purchases? (The average weight of a series of purchases is the total weight of the cheese sold divided by the number of purchases.)
2001 Tournament Of Towns
a. There are $5$ identical paper triangles on the table. Each can be moved in any direction parallel to itself (i.e., without rotating it). Is it true that then any one of them can be covered by the $4$ others?



b. There are $5$ identical equilateral paper triangles on the table. Each can be moved in any direction parallel to itself. Prove that any one of them can be covered by the $4$ others in this way.
2001 Tournament Of Towns
On a square board divided into $15 \times 15$ little squares there are $15$ rooks that do not attack each other. Then each rook makes one move like that of a knight. Prove that after this is done a pair of rooks will necessarily attack each other.
2001 Tournament Of Towns
In a certain country $10\%$ of the employees get $90\%$ of the total salary paid in this country. Supposing that the country is divided in several regions, is it possible that in every region the total salary of any 10% of the employees is no greater than $11\%$ of the total salary paid in this region?
2001 Tournament Of Towns
In three piles there are $51, 49$, and $5$ stones, respectively. You can combine any two piles into one pile or divide a pile consisting of an even number of stones into two equal piles. Is it possible to get $105$ piles with one stone in each?
2001 Tournament Of Towns
Point $A$ lies inside an angle with vertex $M$. A ray issuing from point $A$ is reflected in one side of the angle at point $B$, then in the other side at point $C$ and then returns back to point $A$ (the ordinary rule of reflection holds). Prove that the center of the circle circumscribed about triangle $\triangle BCM$ lies on line $AM$.
2001 Tournament Of Towns
Several non-intersecting diagonals divide a convex polygon into triangles. At each vertex of the polygon the number of triangles adjacent to it is written. Is it possible to reconstruct all the diagonals using these numbers if the diagonals are erased?