Navvye/Olympiad-Problems-CSV huggingface dataset:
2001 Tournament Of Towns
(a) One black and one white pawn are placed on a chessboard. You may move the pawns in turn to the neighbouring empty squares of the chessboard using vertical and horizontal moves. Can you arrange the moves so that every possible position of the two pawns will appear on the chessboard exactly once?

(b) Same question, but you don’t have to move the pawns in turn.
2001 Tournament Of Towns
Let $AH_A$, $BH_B$ and $CH_C$ be the altitudes of triangle $\triangle ABC$. Prove that the triangle whose vertices are the intersection points of the altitudes of triangles $\triangle AH_BH_C$, $\triangle BH_AH_C$ and $\triangle CH_AH_B$ is equal to triangle $\triangle H_AH_BH_C$.
2001 Tournament Of Towns
Alex thinks of a two-digit integer (any integer between $10$ and $99$). Greg is trying to guess it. If the number Greg names is correct, or if one of its digits is equal to the corresponding digit of Alex’s number and the other digit differs by one from the corresponding digit of Alex’s number, then Alex says “hot”; otherwise, he says “cold”. (For example, if Alex’s number was $65$, then by naming any of $64, 65, 66, 55$ or $75$ Greg will be answered “hot”, otherwise he will be answered “cold”.)





(a) Prove that there is no strategy which guarantees that Greg will guess Alex’s number in no more than 18 attempts.

(b) Find a strategy for Greg to find out Alex’s number (regardless of what the chosen number was) using no more than $24$ attempts.

(c) Is there a $22$ attempt winning strategy for Greg?
2001 Tournament Of Towns
A bus that moves along a 100 km route is equipped with a computer, which predicts how much more time is needed to arrive at its final destination. This prediction is made on the assumption that the average speed of the bus in the remaining part of the route is the same as that in the part already covered. Forty minutes after the departure of the bus, the computer predicts that the remaining travelling time will be 1 hour. And this predicted time remains the same for the next 5 hours. Could this possibly occur? If so, how many kilometers did the bus cover when these 5 hours passed? (Average speed is the number of kilometers covered divided by the time it took to cover them.)
2001 Tournament Of Towns
The decimal expression of the natural number $a$ consists of $n$ digits, while that of $a^3$ consists of $m$ digits. Can $n + m$ be equal to 2001?
2001 Tournament Of Towns
Points $X$ and $Y$ are chosen on the sides $AB$ and $BC$ of the triangle $\triangle ABC$. The segments $AY$ and $CX$ intersect at the point $Z$. Given that $AY = YC$ and $AB = ZC$, prove that the points $B$, $X$, $Z$, and $Y$ lie on the same circle.
2001 Tournament Of Towns
Two persons play a game on a board divided into $3\times 100$ squares. They move in turn: the first places tiles of size $1\times2$ lengthwise (along the long axis of the board), the second, in the perpendicular direction. The loser is the one who cannot make a move. Which of the players can always win (no matter how his opponent plays), and what is the winning strategy?
2001 Tournament Of Towns
Nine points are drawn on the surface of a regular tetrahedron with an edge of $1$ cm.  Prove that among these points there are two located at a distance (in space) no greater than $0.5$ cm.
2001 Tournament Of Towns
Find at least one polynomial $P(x)$ of degree 2001 such that $P(x)+P(1- x)=1$ holds for all real numbers $x$.
2001 Tournament Of Towns
At the end of the school year it became clear that for any arbitrarily chosen group of no less than 5 students, 80% of the marks “F” received by this group were given to no more than 20% of the students in the group.  Prove that at least 3/4 of all “F” marks were given to the same student.
2001 Tournament Of Towns
Let $AH_A$, $BH_B$ and $CH_C$ be the altitudes of triangle $\triangle ABC$. Prove that the triangle whose vertices are the intersection points of the altitudes of $\triangle AH_BH_C$, $\triangle BH_AH_C$ and $\triangle CH_AH_B$ is congruent to $\triangle H_AH_BH_C$.
2001 Tournament Of Towns
There are two matrices $A$ and $B$ of size $m\times n$ each filled only by “0”s and “1”s. It is given that along any row or column its elements do not decrease (from left to right and from top to bottom).It is also given that the numbers of “1”s in both matrices are equal and for any $k = 1, . . .$ , $m$ the sum of the elements in the top $k$ rows of the matrix $A$ is no less than that of the matrix $B$. Prove for any $l = 1, . . . $, $n$ the sum of the elements in left $l$ columns of the matrix $A$ is no greater than that of the matrix $B$.
2001 Tournament Of Towns
In a chess tournament, every participant played with each other exactly once, receiving $1$ point for a win, $1/2$ for a draw and $0$ for a loss.





(a) Is it possible that for every player $P$, the sum of points of the players who were beaten by P is greater than the sum of points of the players who beat $P$?

(b) Is it possible that for every player $P$, the first sum is less than the second one?
2001 Tournament Of Towns
Prove that there exist $2001$ convex polyhedra such that any three of them do not have any common points but any two of them touch each other (i.e., have at least one common boundary point but no common inner points).
2001 Tournament Of Towns
Several boxes are arranged in a circle. Each box may be empty or may contain one or several chips. A move consists of taking all the chips from some box and distributing them one by one into subsequent boxes clockwise starting from the next box in the clockwise direction.



(a) Suppose that on each move (except for the first one) one must take the chips from the box where the last chip was placed on the previous move. Prove that after several moves the initial distribution of the chips among the boxes will reappear.

(b) Now, suppose that in each move one can take the chips from any box. Is it true

that for every initial distribution of the chips you can get any possible distribution?
2001 Tournament Of Towns
In the quadrilateral $ABCD$, $AD$ is parallel to $BC$. $K$ is a point on $AB$. Draw the line through $A$ parallel to $KC$ and the line through $B$ parallel to $KD$.



Prove that these two lines intersect at some point on $CD$.
2001 Tournament Of Towns
Clara computed the product of the first $n$ positive integers, and Valerie computed the product of the first $m$ even positive integers, where $m\ge2$.  They got the same answer.  Prove that one of them had made a mistake.
2001 Tournament Of Towns
Kolya is told that two of his four coins are fake. He knows that all real coins have the same weight, all fake coins have the same weight, and the weight of a real coin is greater than that of a fake coin.  Can Kolya decide whether he indeed has exactly two fake coins by using a balance twice?
2001 Tournament Of Towns
On an east-west shipping lane are ten ships sailing individually.  The first five from the west are sailing eastwards while the other five ships are sailing westwards.  They sail at the same constant speed at all times.  Whenever two ships meet, each turns around and sails in the opposite direction.  When all ships have returned to port, how many meetings of two ships have taken place?
2001 Tournament Of Towns
On the plane is a set of at least four points.  If any one point from this set is removed, the resulting set has an axis of symmetry.  Is it necessarily true that the whole set has an axis of symmetry?
2001 Tournament Of Towns
Do there exist postive integers $a_1<a_2<\cdots<a_{100}$ such that for $2\le k\le100$ the greatest common divisor of  $a_{k-1}$ and $a_k$ is greater than the greatest common divisor of $a_k$ and $a_{k+1}$?
2001 Tournament Of Towns
Let $n\ge3$ be an integer.  A circle is divided into $2n$ arcs by $2n$ points.  Each arc has one of three possible lengths, and no two adjacent arcs have the same lengths.  The $2n$ points are colored alternately red and blue.  Prove that the $n$-gon with red vertices and the $n$-gon with blue vertices have the same perimeter and the same area.
2001 Tournament Of Towns
Let $n\ge3$ be an integer.  Each row in an $(n-2)\times n$ array consists of the numbers 1,2,...,$n$ in some order, and the numbers in each column are all different.  Prove that this array can be expanded into an $n\times n$ array such that each row and each column consists of the numbers 1,2,...,$n$.
2001 Tournament Of Towns
Let $n\ge2$ be an integer.  A regular $(2n+1)-gon$ is divided in to $2n-1$ triangles by diagonals which do not meet except at the vertices.  Prove that at least three of these triangles are isosceles.
2001 Tournament Of Towns
Alex places a rook on any square of an empty $8\times8$ chessboard.  Then he places additional rooks one rook at a time, each  attacking an odd number of rooks which are already on the board.  A rook attacks to the left, to the right, above and below, and only the first rook in each direction.  What is the maximum number of rooks Alex can place on the chessboard?
2001 Tournament Of Towns
Several numbers are written in a row. In each move, Robert chooses any two adjacent numbers in which the one on the left is greater than the one on the right, doubles each of them and then switches them around.  Prove that Robert can make only a finite number of moves.
2001 Tournament Of Towns
It is given that $2^{333}$ is a 101-digit number whose first digit is 1.  How many of the numbers $2^k$, $1\le k\le 332$ have first digit 4?
2001 Tournament Of Towns
An altitude of a pentagon is the perpendicular drop from a vertex to the opposite side.  A median of a pentagon is the line joining a vertex to the midpoint of the opposite side.  If the five altitudes and the five medians all have the same length,

prove that the pentagon is regular.
2001 Tournament Of Towns
There exists a block of 1000 consecutive positive integers containing no prime numbers, namely, $1001!+2,1001!+3,...,1001!+1001$.  Does there exist a block of 1000 consecutive positive intgers containing exactly five prime numbers?
2001 Tournament Of Towns
On top of a thin square cake are triangular chocolate chips which are mutually disjoint.  Is it possible to cut the cake into convex polygonal pieces each containing exactly one chip?
2001 Tournament Of Towns
The only pieces on an $8\times8$ chessboard are three rooks.  Each moves along a row or a column without running to or jumping over another rook.  The white rook starts at the bottom left corner, the black rook starts in the square directly above the white rook, and the red rook starts in the square directly to the right of the white rook.  The white rook is to finish at the top right corner, the black rook in the square directly to the left of the white rook, and the red rook in the square directly below the white rook.  At all times, each rook must be either in the same row or the same column as another rook.  Is it possible to get the rooks to their destinations?
2001 Tournament Of Towns
On the plane is a triangle with red vertices and a triangle with blue vertices.  $O$ is a point inside both triangles such that the distance from $O$ to any red vertex is less than the distance from $O$ to any blue vertex.  Can the three red vertices and the three blue vertices all lie on the same circle?
2001 Tournament Of Towns
Do there exist positive integers $a_1<a_2<\ldots<a_{100}$ such that for $2\le k\le100$, the least common multiple of $a_{k-1}$ and $a_k$ is greater than the least common multiple of $a_k$ and $a_{k+1}$?
2001 Tournament Of Towns
An $8\times8$ array consists of the numbers $1,2,...,64$.  Consecutive numbers are adjacent along a row or a column.  What is the minimum value of the sum of the numbers along the diagonal?
2001 Tournament Of Towns
Let $F_1$ be an arbitrary convex quadrilateral.  For $k\ge2$, $F_k$ is obtained by cutting $F_{k-1}$ into two pieces along one of its diagonals, flipping one piece over, and the glueing them back together along the same diagonal.  What is the maximum number of non-congruent quadrilaterals in the sequence $\{F_k\}$?
2001 Tournament Of Towns
Let $a$ and $d$ be positive integers. For any positive integer $n$, the number $a+nd$ contains a block of consecutive digits which constitute the number $n$.  Prove that $d$ is a power of 10.
2001 Tournament Of Towns
In a row are 23 boxes such that for $1\le k \le 23$, there is a box containing exactly $k$ balls.  In one move, we can double the number of balls in any box by taking balls from another box which has more.  Is it always possible to end up with exactly $k$ balls in the $k$-th box for $1\le k\le 23$?
2001 Tournament Of Towns
The vertices of a triangle have coordinates $(x_1,y_1)$, $(x_2,y_2)$ and $(x_3,y_3)$.  For any integers $h$ and $k$, not both 0, both triangles whose vertices have coordinates $(x_1+h,y_1+k),(x_2+h,y_2+k)$ and $(x_3+h,y_3+k)$ has no common interior points with the original triangle.



(a)  Is it possible for the area of this triangle to be greater than $\tfrac{1}{2}$?



(b)  What is the maximum area of this triangle?
2002 Tournament Of Towns
There are many $a\times b$ rectangular cardboard pieces ($a,b\in\mathbb{N}$ such that $a<b$). It is given that by putting such pieces together without overlapping one can make $49\times 51$ rectangle, and $99\times 101$ rectangle. Can one uniquely determine $a,b$ from this?
2002 Tournament Of Towns
Can any triangle be cut into four convex figures: a triangle, a quadrilateral, a pentagon, a hexagon?
2002 Tournament Of Towns
Show that if the last digit of the number $x^2+xy+y^2$ is $0$ (where $x,y\in\mathbb{N}$ ) then last two digits are zero.
2002 Tournament Of Towns
Quadrilateral $ABCD$ is circumscribed about a circle $\Gamma$ and $K,L,M,N$ are points of tangency of sides $AB,BC,CD,DA$ with $\Gamma$ respectively. Let $S\equiv KM\cap LN$. If quadrilateral $SKBL$ is cyclic then show that $SNDM$ is also cyclic.
2002 Tournament Of Towns




There are $128$ coins of two different weights, $64$ each. How can one always find two coins of different weights by performing no more than $7$ weightings on a regular balance?

There are $8$ coins of two different weights, $4$ each. How can one always find two coins of different weights by performing two weightings on a regular balance?
2002 Tournament Of Towns
Let $a,b,c$ be sides of a triangle. Show that $a^3+b^3+3abc>c^3$.
2002 Tournament Of Towns
A game is played on a $23\times 23$ board. The first player controls two white chips which start in the bottom left and top right corners. The second player controls two black ones which start in bottom right and top left corners. The players move alternately. In each move, a player moves one of the chips under control to a square which shares a side with the square the chip is currently in. The first player wins if he can bring the white chips to squares which share a side with each other. Can the second player prevent the first player from winning?
2002 Tournament Of Towns
Let $E$ and $F$ be the respective midpoints of $BC,CD$ of a convex quadrilateral $ABCD$. Segments $AE,AF,EF$ cut the quadrilateral into four triangles whose areas are four consecutive integers. Find the maximum possible area of $\Delta BAD$.
2002 Tournament Of Towns
There are $n$ lamps in a row. Some of which are on. Every minute all the lamps already on go off. Those which were off and were adjacent to exactly one lamp which was on will go on. For which $n$ one can find an initial configuration of lamps which were on, such that at least one lamp will be on at any time?
2002 Tournament Of Towns
An acute triangle was dissected by a straight cut into two pieces which are not necessarily triangles. Then one of the pieces were dissected by a straight cut into two pieces and so on. After a few dissections it turns out the pieces were all triangles. Is it possible they were all obtuse?
2002 Tournament Of Towns
In an infinite increasing sequence of positive integers, every term from the $2002^{\text{th}}$ term divides the sum of all preceding terms. Prove that every term starting from some term is equal to the sum of all preceding terms.
2002 Tournament Of Towns
Some domino pieces are placed in a chain according to standard rules. In each move, we may remove a sub-chain with equal numbers at its ends, turn the whole sub-chain around, and put it back in the same place. Prove that for every two legal chains formed from the same pieces and having the same numbers at their ends, we can transform one to another in a finite sequence of moves.
2002 Tournament Of Towns
$\Delta ABC$ and its mirror reflection $\Delta A^{\prime}B^{\prime}C^{\prime}$ is arbitrarily placed on the plane. Prove the midpoints of $AA^{\prime},BB^{\prime},CC^{\prime}$ are collinear.
2002 Tournament Of Towns
There are $6$ pieces of cheese of different weights. For any two pieces we can identify the heavier piece. Given that it is possible to divide them into two groups of equal weights with three pieces in each. Give the explicit way to find these groups by performing two weightings on a regular balance.
2002 Tournament Of Towns
In how many ways can we place the numbers from $1$ to $100$ in a $2\times 50$ rectangle (divided into $100$ unit squares) so that any two consecutive numbers are always placed in squares with a common side?
2002 Tournament Of Towns
Does there exist a regular triangular prism that can be covered (without overlapping) by different equilateral triangles? (One is allowed to bend the triangles around the edges of the prism.)
2002 Tournament Of Towns
In a triangle $ABC$ it is given $\tan A,\tan B,\tan C$ are integers. Find their values.
2002 Tournament Of Towns
Does there exist points $A,B$ on the curve $y=x^3$ and on $y=x^3+|x|+1$ respectively such that distance between $A,B$ is less than $\frac{1}{100}$ ?
2002 Tournament Of Towns
The spectators are seated in a row with no empty places. Each is in a seat which does not match the spectator's ticket. An usher can order two spectators in adjacent seats to trade places unless one of them is already seated correctly. Is it true that from any initial arrangement, the spectators can be brought to their correct seats?
2002 Tournament Of Towns
Let $AA_1,BB_1,CC_1$ be the altitudes of acute $\Delta ABC$. Let $O_a,O_b,O_c$ be the incentres of $\Delta AB_1C_1,\Delta BC_1A_1,\Delta CA_1B_1$ respectively. Also let $T_a,T_b,T_c$ be the points of tangency of the incircle of $\Delta ABC$ with $BC,CA,AB$ respectively. Prove that $T_aO_cT_bO_aT_cO_b$ is an equilateral hexagon.
2002 Tournament Of Towns
The $52$ cards of a standard deck are placed in a $13\times 4$ array. If every two adjacent cards, vertically or horizontally, have the same suit or have the same value, prove that all $13$ cards of the same suit are in the same row.
2002 Tournament Of Towns
Do there exist irrational numbers $a,b$ both greater than $1$, such that $\lfloor{a^m}\rfloor\neq \lfloor{b^n}\rfloor$ for all $m,n\in\mathbb{N}$ ?
2002 Tournament Of Towns
In a convex $2002\text{-gon}$ several diagonals are drawn so that they do not intersect inside the polygon. As a result the polygon splits into $2000$ triangles. Isit possible that exactly $1000$ triangles have diagonals for all their three sides?
2002 Tournament Of Towns
John and Mary select a natural number each and tell that to Bill. Bill wrote their sum and product in two papers hid one paper and showed the other to John and Mary.

John looked at the number (which was $2002$ ) and declared he couldn't determine Mary's number. Knowing this Mary also said she couldn't determine John's number as well.

What was Mary's Number?
2002 Tournament Of Towns




A test was conducted in class. It is known that at least $\frac{2}{3}$ of the problems were hard. Each such problems were not solved by at least $\frac{2}{3}$ of the students. It is also known that at least $\frac{2}{3}$ of the students passed the test. Each such student solved at least $\frac{2}{3}$ of the suggested problems. Is this possible?

Previous problem with $\frac{2}{3}$ replaced by $\frac{3}{4}$.

Previous problem with $\frac{2}{3}$ replaced by $\frac{7}{10}$.
2002 Tournament Of Towns
$2002$ cards with numbers $1,2,\ldots ,2002$ written on them are put on a table face up. Two players $A,B$ take turns to pick up a card until all are gone. $A$ goes first. The player who gets the last digit of the sum of his cards larger than his opponent wins.

Who has a winning strategy and how should one play to win?
2002 Tournament Of Towns
An angle and a point $A$ inside it is given. Is it possible to draw through $A$ three straight lines so that on either side of the angle one of three points of intersection of these lines be the midpoint of two other points of intersection with that side?
2002 Tournament Of Towns
There are $2002$ employees in a bank. All the employees came to celebrate the bank's jubilee and were seated around one round table. It is known that the difference in salaries of any two adjacent employees is $2$ or $3$ dollars. Find the maximal difference in salaries of two employees, if it is known all salaries are different.
2002 Tournament Of Towns
All the species of plants existing in Russia are catalogued (numbered by integers from $2$ to $2000$ ; one after another, without omissions or repetitions). For any pair of species the gcd of their catalogue numbers was calculated and recorded but the catalogue numbers themselves were lost. Is it possible to restore the catalogue numbers from the data in hand?
2002 Tournament Of Towns
The vertices of a $50\text{-gon}$ divide a circumference into $50$ arcs, whose lengths are $1,2,\ldots 50$ in some order. It is known that any two opposite arcs (corresponding to opposite sides) differ by $25$. Prove that the polygon has two parallel sides.
2002 Tournament Of Towns
Point $P$ is chosen in the plane of triangle $ABC$ such that $\angle{ABP}$ is congruent to $\angle{ACP}$ and $\angle{CBP}$ is congruent to $\angle{CAP}$. Show $P$ is the orthocentre.
2002 Tournament Of Towns
A convex $N\text{-gon}$ is divided by diagonals into triangles so that no two diagonals intersect inside the polygon. The triangles are painted in black and white so that any two triangles are painted in black and white so that any two triangles with a common side are painted in different colors. For each $N$ find the maximal difference between the numbers of black and white triangles.
2002 Tournament Of Towns
There's a large pile of cards. On each card a number from $1,2,\ldots n$ is written. It is known that sum of all numbers on all of the cards is equal to $k\cdot n!$ for some $k$. Prove that it is possible to arrange cards into $k$ stacks so that sum of numbers written on the cards in each stack is equal to $n!$.
2002 Tournament Of Towns




A power grid with the shape of a $3\times 3$ lattice with $16$ nodes (vertices of the lattice) joined by wires (along the sides of squares. It may have happened that some of the wires have burned out. In one test technician can choose any two nodes and check if electrical current circulates between them (i.e there is a chain of intact wires joining the chosen nodes) . Technicial knows that current will circulate from any node to another node. What is the least number of tests required to demonstrate this?

Previous problem for the grid of $5\times 5$ lattice.
2002 Tournament Of Towns
Several straight lines such that no two are parallel, cut the plane into several regions. A point $A$ is marked inside of one region. Prove that a point, separated from $A$ by each of these lines, exists if and only if $A$ belongs to an unbounded region.
2002 Tournament Of Towns
$x,y,z\in\left(0,\frac{\pi}{2}\right)$ are given. Prove that:

$$ \frac{x\cos x+y\cos y+z\cos z}{x+y+z}\le \frac{\cos x+\cos y+\cos z}{3} $$
2002 Tournament Of Towns
An infinite sequence of natural number $\{x_n\}_{n\ge 1}$ is such that $x_{n+1}$ is obtained by adding one of the non-zero digits of $x_n$ to itself. Show this sequence contains an even number.
2002 Tournament Of Towns
A cube is cut by a plane such that the cross section is a pentagon. Show there is a side of the pentagon of length $\ell$ such that the inequality holds:

$$ |\ell-1|>\frac{1}{5} $$