Navvye/Olympiad-Problems-CSV huggingface dataset:
2002 Tournament Of Towns
Two circles $\Gamma_1,\Gamma_2$ intersect at $A,B$. Through $B$ a straight line $\ell$ is drawn and $\ell\cap \Gamma_1=K,\ell\cap\Gamma_2=M\;(K,M\neq B)$. We are given $\ell_1\parallel AM$ is tangent to $\Gamma_1$ at $Q$. $QA\cap \Gamma_2=R\;(\neq A)$ and further $\ell_2$ is tangent to $\Gamma_2$ at $R$.

Prove that:



$\ell_2\parallel AK$

$\ell,\ell_1,\ell_2$ have a common point.
2002 Tournament Of Towns
Define a sequence $\{a_n\}_{n\ge 1}$ such that $a_1=1,a_2=2$ and $a_{n+1}$ is the smallest positive integer $m$ such that $m$ hasn't yet occurred in the sequence and also $\text{gcd}(m,a_n)\neq 1$. Show all positive integers occur in the sequence.
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 $7\times 7$ lattice.
2003 Tournament Of Towns
$2003$ dollars are placed into $N$ purses, and the purses are placed into $M$ pockets. It is known that $N$ is greater than the number of dollars in any pocket. Is it true that there is a purse with less than $M$ dollars in it?
2003 Tournament Of Towns
Two players in turns color the sides of an $n$-gon. The first player colors any side that has $0$ or $2$ common vertices with already colored sides. The second player colors any side that has exactly $1$ common vertex with already colored sides. The player who cannot move, loses. For which $n$ the second player has a winning strategy?
2003 Tournament Of Towns
Points $K$ and $L$ are chosen on the sides $AB$ and $BC$ of the isosceles $\triangle ABC$ ($AB = BC$) so that $AK +LC = KL$. A line parallel to $BC$ is drawn through midpoint $M$ of the segment $KL$, intersecting side $AC$ at point $N$. Find the value of $\angle KNL$.
2003 Tournament Of Towns
Each term of a sequence of positive integers is obtained from the previous term by adding to it its largest digit. What is the maximal number of successive odd terms in such a sequence?
2003 Tournament Of Towns
Is it possible to tile $2003 \times 2003$ board by $1 \times 2$ dominoes placed horizontally and $1 \times 3$ rectangles placed vertically?
2003 Tournament Of Towns
Johnny writes down quadratic equation

$$ax^2 + bx + c = 0.$$

with positive integer coefficients $a, b, c$. Then Pete changes one, two, or none “$+$” signs to “$-$”. Johnny wins, if both roots of the (changed) equation are integers. Otherwise (if there are no real roots or at least one of them is not an integer), Pete wins. Can Johnny choose the coefficients in such a way that he will always win?
2003 Tournament Of Towns
Triangle $ABC$ is given. Prove that $\frac{R}{r} > \frac{a}{h}$, where $R$ is the radius of the circumscribed circle, $r$ is the radius of the inscribed circle, $a$ is the length of the longest side, $h$ is the length of the shortest altitude.
2003 Tournament Of Towns
In a tournament, each of $15$ teams played with each other exactly once. Let us call the game “odd” if the total number of games previously played by both competing teams was odd.



(a) Prove that there was at least one “odd” game.



(b) Could it happen that there was exactly one “odd” game?
2003 Tournament Of Towns
A chocolate bar in the shape of an equilateral triangle with side of the length $n$, consists of triangular chips with sides of the length $1$, parallel to sides of the bar. Two players take turns eating up the chocolate. Each player breaks off a triangular piece (along one of the lines), eats it up and passes leftovers to the other player (as long as bar contains more than one chip, the player is not allowed to eat it completely).



A player who has no move or leaves exactly one chip to the opponent, loses. For each $n$, find who has a winning strategy.
2003 Tournament Of Towns
What is the largest number of squares on $9 \times 9$ square board that can be cut along their both diagonals so that the board does not fall apart into several pieces?
2003 Tournament Of Towns
A trapezoid with bases $AD$ and $BC$ is circumscribed about a circle, $E$ is the intersection point of the diagonals. Prove that $\angle AED$ is not acute.
2003 Tournament Of Towns
$100$-gon made of $100$ sticks. Could it happen that it is not possible to construct a polygon from any lesser number of these sticks?
2003 Tournament Of Towns
Point $M$ is chosen in triangle $ABC$ so that the radii of the circumcircles of triangles $AMC, BMC$, and $BMA$ are no smaller than the radius of the circumcircle of $ABC$. Prove that all four radii are equal.
2003 Tournament Of Towns
In the sequence $00, 01, 02, 03,\ldots , 99$ the terms are rearranged so that each term is obtained from the previous one by increasing or decreasing one of its digits by $1$ (for example, $29$ can be followed by $19, 39$, or $28$, but not by $30$ or $20$). What is the maximal number of terms that could remain on their places?
2003 Tournament Of Towns
Prove that one can cut $a \times  b$ rectangle, $\frac{b}{2} < a < b$, into three pieces and rearrange them into a square (without overlaps and holes).
2003 Tournament Of Towns
A triangular pyramid $ABCD$ is given. Prove that $\frac Rr > \frac ah$, where $R$ is the radius of the circumscribed sphere, $r$ is the radius of the inscribed sphere, $a$ is the length of the longest edge, $h$ is the length of the shortest altitude (from a vertex to the opposite face).
2003 Tournament Of Towns
$P(x)$ is a polynomial with real coefficients such that $P(a_1) = 0, P(a_{i+1}) = a_i$ ($i = 1, 2,\ldots$) where $\{a_i\}_{i=1,2,\ldots}$ is an infinite sequence of distinct natural numbers. Determine the possible values of degree of $P(x)$.
2003 Tournament Of Towns
Can one cover a cube by three paper triangles (without overlapping)?
2003 Tournament Of Towns
A right triangle $ABC$ with hypotenuse $AB$ is inscribed in a circle. Let $K$ be the midpoint of the arc $BC$ not containing $A, N$ the midpoint of side $AC$, and $M$ a point of intersection of ray $KN$ with the circle. Let $E$ be a point of intersection of tangents to the circle at points $A$ and $C$. Prove that $\angle EMK = 90^\circ$.
2003 Tournament Of Towns
Prior to the game John selects an integer greater than $100$.



Then Mary calls out an integer $d$ greater than $1$. If John's integer is divisible by $d$, then Mary wins. Otherwise, John subtracts $d$ from his number and the game continues (with the new number). Mary is not allowed to call out any number twice. When John's number becomes negative, Mary loses. Does Mary have a winning strategy?
2003 Tournament Of Towns
The signs "$+$" or "$-$" are placed in all cells of a $4 \times 4$ square table. It is allowed to change a sign of any cell altogether with signs of all its adjacent cells (i.e. cells having a common side with it). Find the number of different tables that could be obtained by iterating this procedure.
2003 Tournament Of Towns
A square is triangulated in such way that no three vertices are collinear. For every vertex (including vertices of the square) the number of sides issuing from it is counted. Can it happen that all these numbers are even?
2003 Tournament Of Towns
There is $3 \times 4 \times 5$ - box with its faces divided into $1 \times 1$ - squares. Is it possible to place numbers in these squares so that the sum of numbers in every stripe of squares (one square wide) circling the box, equals $120$?
2003 Tournament Of Towns
In $7$-gon $A_1A_2A_3A_4A_5A_6A_7$ diagonals $A_1A_3, A_2A_4, A_3A_5, A_4A_6, A_5A_7, A_6A_1$ and $A_7A_2$ are congruent to each other and diagonals $A_1A_4, A_2A_5, A_3A_6, A_4A_7, A_5A_1, A_6A_2$ and $A_7A_3$ are also congruent to each other. Is the polygon necessarily regular?
2003 Tournament Of Towns
For any integer $n+1,\ldots, 2n$ ($n$ is a natural number) consider its greatest odd divisor. Prove that the sum of all these divisors equals $n^2.$
2003 Tournament Of Towns
There are $N$ points on the plane; no three of them belong to the same straight line. Every pair of points is connected by a segment. Some of these segments are colored in red and the rest of them in blue. The red segments form a closed broken line without self-intersections(each red segment having only common endpoints with its two neighbors and no other common points with the other segments), and so do the blue segments. Find all possible values of $N$ for which such a disposition of $N$ points and such a choice of red and blue segments are possible.
2003 Tournament Of Towns
$25$ checkers are placed on $25$ leftmost squares of $1 \times N$ board. Checker can either move to the empty adjacent square to its right or jump over adjacent right checker to the next square if it is empty. Moves to the left are not allowed. Find minimal $N$ such that all the checkers could be placed in the row of $25$ successive squares but in the reverse order.
2003 Tournament Of Towns
An increasing arithmetic progression consists of one hundred positive integers. Is it possible that every two of them are relatively prime?
2003 Tournament Of Towns
Smallville is populated by unmarried men and women, some of them are acquainted. Two city’s matchmakers are aware of all acquaintances. Once, one of matchmakers claimed: “I could arrange that every brunette man would marry a woman he was acquainted with”. The other matchmaker claimed “I could arrange that every blonde woman would marry a man she was acquainted with”. An amateur mathematician overheard their conversation and said “Then both arrangements could be done at the same time! ” Is he right?
2003 Tournament Of Towns
Find all positive integers $k$ such that there exist two positive integers $m$ and $n$ satisfying

$$m(m + k) = n(n + 1).$$
2003 Tournament Of Towns
Several squares on a $15 \times 15$ chessboard are marked so that a bishop placed on any square of the board attacks at least two of marked squares. Find the minimal number of marked squares.
2003 Tournament Of Towns
A point $O$ lies inside of the square $ABCD$. Prove that the difference between the sum of angles $OAB, OBC, OCD , ODA$ and $180^{\circ}$ does not exceed $45^{\circ}$.
2003 Tournament Of Towns
An ant crawls on the outer surface of the box in a shape of rectangular parallelepiped. From ant’s point of view, the distance between two points on a surface is defined by the length of the shortest path ant need to crawl to reach one point from the other. Is it true that if ant is at vertex then from ant’s point of view the opposite vertex be the most distant point on the surface?
2003 Tournament Of Towns
Two players in turn play a game. First Player has cards with numbers $2, 4, \ldots, 2000$ while Second Player has cards with numbers $1, 3, \ldots, 2001$. In each his turn, a player chooses one of his cards and puts it on a table; the opponent sees it and puts his card next to the first one. Player, who put the card with a larger number, scores 1 point. Then both cards are discarded. First Player starts. After $1000$ turns the game is over; First Player has used all his cards and Second Player used all but one. What are the maximal scores, that players could guarantee for themselves, no matter how the opponent would play?
2003 Tournament Of Towns
What least possible number of unit squares $(1\times1)$ must be drawn in order to get a picture of $25 \times 25$-square divided into $625$ of unit squares?
2003 Tournament Of Towns
A salesman and a customer altogether have $1999$ rubles in coins and bills of $1, 5, 10, 50, 100, 500 , 1000$ rubles. The customer has enough money to buy a Cat in the Bag which costs the integer number of rubles. Prove that the customer can buy the Cat and get the correct change.
2003 Tournament Of Towns
Each side of $1 \times 1$ square is a hypothenuse of an exterior right triangle. Let $A, B, C, D$ be the vertices of the right angles and $O_1, O_2, O_3, O_4$ be the centers of the incircles of these triangles. Prove that

$a)$ The area of quadrilateral $ABCD$ does not exceed $2$;

$b)$ The area of quadrilateral $O_1O_2O_3O_4$ does not exceed $1$.
2003 Tournament Of Towns
A paper tetrahedron is cut along some of so that it can be developed onto the plane. Could it happen that this development cannot be placed on the plane in one layer?
2003 Tournament Of Towns
Prove that every positive integer can be represented in the form

$$3^{u_1} \ldots 2^{v_1} + 3^{u_2} \ldots 2^{v_2} + \ldots + 3^{u_k} \ldots 2^{v_k}$$

with integers $u_1, u_2, \ldots , u_k, v_1, \ldots, v_k$ such that $u_1 > u_2 >\ldots > u_k\ge 0$ and $0 \le v_1 < v_2 <\ldots < v_k$.
2003 Tournament Of Towns
In a triangle $ABC$, let $H$ be the point of intersection of altitudes, $I$ the center of incircle, $O$ the center of circumcircle, $K$ the point where the incircle touches $BC$. Given that $IO$ is parallel to $BC$, prove that $AO$ is parallel to $HK$.
2003 Tournament Of Towns
Let $O$ be the center of insphere of a tetrahedron $ABCD$. The sum of areas of faces $ABC$ and $ABD$ equals the sum of areas of faces $CDA$ and $CDB$. Prove that $O$ and midpoints of $BC, AD, AC$ and $BD$ belong to the same plane.
2003 Tournament Of Towns
A $m \times n$ table is filled with signs $"+"$ and $"-"$. A table is called irreducible if one cannot reduce it to the table filled with $"+"$, applying the following operations (as many times as one wishes).

$a)$ It is allowed to flip all the signs in a row or in a column. Prove that an irreducible table contains an irreducible $2\times 2$ sub table.

$b)$ It is allowed to flip all the signs in a row or in a column or on a diagonal (corner cells are diagonals of length $1$). Prove that an irreducible table contains an irreducible $4\times 4$ sub table.
2004 Tournament Of Towns
In triangle $ABC$ the bisector of angle $A$, the perpendicular to side $AB$ from its midpoint, and the altitude from vertex $B$, intersect in the same point. Prove that the bisector of angle $A$, the perpendicular to side $AC$ from its midpoint, and the altitude from vertex $C$ also intersect in the same point.
2004 Tournament Of Towns
Find all possible values of $n \ge 1$ for which there exist $n$ consecutive positive integers whose sum is a prime number.
2004 Tournament Of Towns
Bucket $A$ contains 3 litres of syrup. Bucket $B$ contains $n$ litres of water. Bucket $C$ is empty.

We can perform any combination of the following operations:

- Pour away the entire amount in bucket $X$,

- Pour the entire amount in bucket $X$ into bucket $Y$,

- Pour from bucket $X$ into bucket $Y$ until buckets $Y$ and $Z$ contain the same amount.



(a) How can we obtain 10 litres of 30% syrup if $n = 20$?

(b) Determine all possible values of $n$ for which the task in (a) is possible.
2004 Tournament Of Towns
A positive integer $a > 1$ is given (in decimal notation). We copy it twice and obtain a number $b = \overline{aa}$ which happens to be a multiple of $a^2$. Find all possible values of $b/a^2$.
2004 Tournament Of Towns
Two 10-digit integers are called neighbours if they diﬀer in exactly one digit (for example, integers $1234567890$ and $1234507890$ are neighbours). Find the maximal number of elements in the set of 10-digit integers with no two integers being neighbours.
2004 Tournament Of Towns
The sum of all terms of a finite arithmetical progression of integers is a power of two. Prove that the number of terms is also a power of two.
2004 Tournament Of Towns
What is the maximal number of checkers that can be placed on an $8\times 8$ checkerboard so that each checker stands on the middle one of three squares in a row diagonally, with exactly one of the other two squares occupied by another checker?
2004 Tournament Of Towns
Each day, the price of the shares of the corporation “Soap Bubble, Limited” either increases or decreases by $n$ percent, where $n$ is an integer such that $0 < n < 100$. The price is calculated with unlimited precision. Does there exist an $n$ for which the price can take the same value twice?
2004 Tournament Of Towns
Two circles intersect in points $A$ and $B$. Their common tangent nearer $B$ touches the circles at points $E$ and $F$, and intersects the extension of $AB$ at the point $M$. The point $K$ is chosen on the extention of $AM$ so that $KM = MA$. The line $KE$ intersects the circle containing $E$ again at the point $C$. The line $KF$ intersects the circle containing $F$ again at the point $D$. Prove that the points $A, C$ and $D$ are collinear.
2004 Tournament Of Towns
All sides of a polygonal billiard table are in one of two perpendicular directions. A tiny billiard ball rolls out of the vertex $A$ of an inner $90^o$ angle and moves inside the billiard table, bouncing off its sides according to the law “angle of reflection equals angle of incidence”. If the ball passes a vertex, it will drop in and srays there. Prove that the ball will never return to $A$.
2004 Tournament Of Towns
At the beginning of a two-player game, the number $2004!$ is written on the blackboard. The players move alternately. In each move, a positive integer smaller than the number on the blackboard and divisible by at most $20$ different prime numbers is chosen. This is subtracted from the number on the blackboard, which is erased and replaced by the difference. The winner is the player who obtains $0$. Does the player who goes first or the one who goes second have a guaranteed win, and how should that be achieved?
2004 Tournament Of Towns
Segments $AB, BC$ and $CD$ of the broken line $ABCD$ are equal and are tangent to a circle with centre at the point $O$. Prove that the point of contact of this circle with $BC$, the point $O$ and the intersection point of $AC$ and $BD$ are collinear.
2004 Tournament Of Towns
Perimeter of a convex quadrilateral is $2004$ and one of its diagonals is $1001$. Can another diagonal be $1$ ? $2$ ? $1001$ ?
2004 Tournament Of Towns
Arithmetical progression $a_1, a_2, a_3, a_4,...$  contains $a_1^2 , a_2^2$ and $a_3^2$ at some positions. Prove that all terms of this progression are integers.
2004 Tournament Of Towns
Two $10$-digit integers are called neighbours if they differ in exactly one digit (for example, integers $1234567890$ and $1234507890$ are neighbours). Find the maximal number of elements in the set of $10$-digit integers with no two integers being neighbours.
2004 Tournament Of Towns
The perpendicular projection of a triangular pyramid on some plane has the largest possible area. Prove that this plane is parallel to either a face or two opposite edges of the pyramid.
2004 Tournament Of Towns
The parabola $y = x^2$ intersects a circle at exactly two points $A$ and $B$. If their tangents at $A$ coincide, must their tangents at $B$ also coincide?
2004 Tournament Of Towns
The audience shuffles a deck of $36$ cards, containing $9$ cards in each of the suits spades, hearts, diamonds and clubs. A magician predicts the suit of the cards, one at a time, starting with the uppermost one in the face-down deck. The design on the back of each card is an arrow.

An assistant examines the deck without changing the order of the cards, and points the arrow on the back each card either towards or away from the magician, according to some system agreed upon in advance with the magician. Is there such a system which enables the magician to guarantee the correct prediction of the suit of at least

(a) $19$ cards;

(b) $20$ cards?
2004 Tournament Of Towns
Is it possible to arrange numbers from 1 to 2004 in some order so that the sum of any 10 consecutive numbers is divisble by 10?
2004 Tournament Of Towns
A box contains red, green, blue, and white balls, 111 balls in all. If you take out 100 balls without looking, then there will always be 4 balls of different colors among them. What is the smallest number of balls you must take out without looking to guarantee that among them there will always be balls of at least 3 different colors?
2004 Tournament Of Towns
We have a number of towns, with bus routes between some of them (each bus route goes from a town to another town without any stops). It is known that you can get from any town to any other by bus (possibly changing buses several times). Mr. Ivanov bought one ticket for each of the bus routes (a ticket allows single travel in either direction, but not returning on the same route). Mr. Petrov bought n tickets for each of the bus routes. Both Ivanov and Petrov started at town A. Ivanov used up all his tickets without buying any new ones and finished his travel at town B. Petrov, after using some of his tickets, got stuck at town X: he can not leave it without buying a new ticket. Prove that X is either A or B.
2004 Tournament Of Towns
We have a circle and a line which does not intersect the circle. Using only compass and straightedge, construct a square whose two adjacent vertices are on the circle, and two other vertices are on the given line (it is known that such a square exists).
2004 Tournament Of Towns
How many different ways are there to write 2004 as a sum of one or more positive integers which are all "aproximately equal" to each other? Two numbers are called aproximately equal if their difference is at most 1. The order of terms does not matter: two ways which only differ in the order of terms are not considered different.
2004 Tournament Of Towns
Let us call a triangle rational if each of its angles is a rational number when measured in degrees. Let us call a point inside triangle rational if joining it to the three vertices of the triangle we get three rational triangles. Show that any acute rational triangle contains at least three distinct rational points.
2004 Tournament Of Towns
The incircle of the triangle ABC touches the sides BC, AC, and AB at points A', B', and C', respectively. It is known that AA'=BB'=CC'. Does the triangle ABC have to be equilateral?



(I am very interested in ingenious solution of this problem, because I found an ugly one using Stewart's theorem and tons of algebra during the contest).
2004 Tournament Of Towns
What is the maximal number of knights that can be placed on the usual 8x8 chessboard so that each of then threatens at most 7 others?
2004 Tournament Of Towns
Vanya has chosen two positive numbers, x and y. He wrote the numbers x+y, x-y, x/y, and xy, and has shown these numbers to Petya. However, he didn't say which of the numbers was obtained from which operation. Show that Petya can uniquely recover x and y.
2004 Tournament Of Towns
Let K be a point on the side BC of the triangle ABC. The incircles of the triangles ABK and ACK touch BC at points M and N, respectively. Show that BM\cdot CN>KM \cdot KN.
2004 Tournament Of Towns
Two persons are dividing a piece of cheese. The first person cuts it into two pieces, then the second person cuts one of these pieces into two, then again the first person cuts one of the pieces into two, and so until they have 5 pieces. After that the first person chooses one of the pieces, then the second person chooses one of remaining pieces and so on until all pieces are taken. For each of the players, what is the maximal amount of cheese he can get for certain, regardless of the other's actions?
2004 Tournament Of Towns
Let A and B be two rectangles such that it is possible to get rectangle similar to A by putting together rectangles equal to B. Show that it is possible to get rectangle similar to B by putting together rectangles equal to A.
2004 Tournament Of Towns
Three circles pass through point X. Their intersection points (other than X) are denoted A, B, C. Let A' be the second point of intersection of line AX and the circle circumscribed around triangle BCX, and define similarly points B', C'. Prove that triangles ABC', AB'C, and A'BC are similar.
2004 Tournament Of Towns
A box contains red, blue, and white balls, $100$ balls in total. It is known that among any $26$ of them there are always $10$ balls of the same color. Find the minimal number $N$ such that among any $N$ balls there are always $30$ balls of the same color.