Navvye/Olympiad-Problems-CSV huggingface dataset:
2004 Tournament Of Towns
P(x) and Q(x) are polynomials of positive degree such that for all x P(P(x))=Q(Q(x)) and P(P(P(x)))=Q(Q(Q(x))). Does this necessarily mean that P(x)=Q(x)?
2004 Tournament Of Towns
For which values of N is it possible to write numbers from 1 to N in some order so that for any group of two or more consecutive numbers, the arithmetic mean of these numbers is not whole?
2004 Tournament Of Towns
Functions f and g are defined on the whole real line and are mutually inverse: g(f(x))=x, f(g(y))=y for all x, y. It is known that f can be written as a sum of periodic and linear functions: f(x)=kx+h(x) for some number k and a periodic function h(x). Show that g can also be written as a sum of periodic and linear functions. (A functions h(x) is called periodic if there exists a non-zero number d such that h(x+d)=h(x) for any x.)
2004 Tournament Of Towns
Two persons are playing the following game. They have a pile of stones and take turns removing stones from it, with the first player taking the first turn. At each turn, the first player removes either 1 or 10 stones from the pile, and the second player removes either m or n stones. The player who can not make his move loses. It is known that for any number of stones in the pile, the first player can always win (regardless of the second player's moves). What are the possible values of m and n?
2004 Tournament Of Towns
A circle with the center $I$ is entirely inside of a circle with center $O$. Consider all possible chords $AB$ of the larger circle which are tangent to the smaller one. Find the locus of the centers of the circles circumscribed about the triangle $AIB$.
2004 Tournament Of Towns
Let n be a fixed prime number >3. A triangle is said to be admissible if the measure of each of its angles is of the form $\frac{m\cdot 180^{\circ}}{n}$ for some positive integer m.

We are given one admissible triangle. Every minute we cut one of the triangles we already have into two admissible triangles so that no two of the triangles we have after cutting are similar. After some time, it turns out that no more cuttings are possible. Prove that at this moment, the triangles we have contain all possible admissible triangles (we do not distinguish between triangles which have same sets of angles, i.e. similar triangles).
2004 Tournament Of Towns
Let AOB and COD be angles which can be identified by a rotation of the plane (such that rays OA and OC are identified). A circle is inscribed in each of these angles; these circles intersect at points E and F. Show that angles AOE and DOF are equal.
2005 Tournament of Towns
Anna and Boris move simultaneously towards each other, from points A and B respectively. Their speeds are constant, but not necessarily equal. Had Anna started 30 minutes earlier, they would have met 2 kilometers nearer to B. Had Boris started 30 minutes earlier instead, they would have met some distance nearer to A. Can this distance be uniquely determined?



(3 points)
2005 Tournament of Towns
Prove that one of the digits 1, 2 and 9 must appear in the base-ten expression of $n$ or $3n$ for any positive integer $n$.



(4 points)
2005 Tournament of Towns
There are eight identical Black Queens in the first row of a chessboard and eight identical White Queens in the last row. The Queens move one at a time, horizontally, vertically or diagonally by any number of squares as long as no other Queens are in the way. Black and White Queens move alternately. What is the minimal number of moves required for interchanging the Black and White Queens?



(5 points)
2005 Tournament of Towns
$M$ and $N$ are the midpoints of sides $BC$ and $AD$, respectively, of a square $ABCD$. $K$ is an arbitrary point on the extension of the diagonal $AC$ beyond $A$. The segment $KM$ intersects the side $AB$ at some point $L$. Prove that $\angle KNA = \angle LNA$.



(5 points)
2005 Tournament of Towns
In a certain big city, all the streets go in one of two perpendicular directions. During a drive in the city, a car does not pass through any place twice, and returns to the parking place along a street from which it started. If it has made 100 left turns, how many right turns must it have made?



(5 points)
2005 Tournament of Towns
On the graph of a polynomial with integral coefficients are two points with integral coordinates. Prove that if the distance between these two points is integral, then the segment connecting them is parallel to the $x$-axis.



(4 points)
2005 Tournament of Towns
The altitudes $AD$ and $BE$ of triangle $ABC$ meet at its orthocentre $H$. The midpoints of $AB$ and $CH$ are $X$ and $Y$, respectively. Prove that $XY$ is perpendicular to $DE$.



(5 points)
2005 Tournament of Towns
Baron Münchhausen’s watch works properly, but has no markings on its face. The hour, minute and second hands have distinct lengths, and they move uniformly. The Baron claims that since none of the mutual positions of the hands is repeats twice in the period between 8:00 and 19:59, he can use his watch to tell the time during the day. Is his assertion true?



(5 points)
2005 Tournament of Towns
A $10 \times 12$ paper rectangle is folded along the grid lines several times, forming a thick $1 \times 1$ square. How many pieces of paper can one possibly get by cutting this square along the segment connecting



(a) the midpoints of a pair of opposite sides;     (2 points)

(b) the midpoints of a pair of adjacent sides?    (4 points)
2005 Tournament of Towns
In a rectangular box are a number of rectangular blocks, not necessarily identical to one another. Each block has one of its dimensions reduced. Is it always possible to pack these blocks in a smaller rectangular box, with the sides of the blocks parallel to the sides of the box?



(6 points)
2005 Tournament of Towns
John and James wish to divide $25$ coins, of denominations $1, 2, 3, \ldots , 25$ kopeks. In each move, one of them chooses a coin, and the other player decides who must take this coin. John makes the initial choice of a coin, and in subsequent moves, the choice is made by the player having more kopeks at the time. In the event that there is a tie, the choice is made by the same player in the preceding move. After all the coins have been taken, the player with more kokeps wins. Which player has a winning strategy?



(6 points)
2005 Tournament of Towns
The squares of a chessboard are numbered in the following way. The upper left corner is numbered 1. The two squares on the next diagonal from top-right to bottom-left are numbered 2 and 3. The three squares on the next diagonal are numbered 4, 5 and 6, and so on. The two squares on the second-to-last diagonal are numbered 62 and 63, and the lower right corner is numbered 64. Peter puts eight pebbles on the squares of the chessboard in such a way that there is exactly one pebble in each column and each row. Then he moves each pebble to a square with a number greater than that of the original square. Can it happen that there is still exactly one pebble in each column and each row?



(8 points)
2005 Tournament of Towns
The graphs of four functions of the form $y = x^2 + ax + b$, where a and b are real coefficients, are plotted on the coordinate plane. These graphs have exactly four points of intersection, and at each one of them, exactly two graphs intersect. Prove that the sum of the largest and the smallest $x$-coordinates of the points of intersection is equal to the sum of the other two.



(3 points)
2005 Tournament of Towns
The base-ten expressions of all the positive integers are written on an infinite ribbon without spacing: $1234567891011\ldots$. Then the ribbon is cut up into strips seven digits long. Prove that any seven digit integer will:



(a) appear on at least one of the strips;    (3 points)

(b) appear on an infinite number of strips.    (1 point)
2005 Tournament of Towns
$M$ and $N$ are the midpoints of sides $BC$ and $AD$, respectively, of a square $ABCD$. $K$ is an arbitrary point on the extension of the diagonal $AC$ beyond $A$. The segment $KM$ intersects the side $AB$ at some point $L$. Prove that $\angle KNA = \angle LNA$.



(4 points)
2005 Tournament of Towns
In a certain big city, all the streets go in one of two perpendicular directions. During a drive in the city, a car does not pass through any place twice, and returns to the parking place along a street from which it started. If it has made $100$ left turns, how many right turns must it have made?



(4 points)
2005 Tournament of Towns
The sum of several positive numbers is equal to $10$, and the sum of their squares is greater than $20$. Prove that the sum of the cubes of these numbers is greater than $40$.



(5 points)
2005 Tournament of Towns
A circle $\omega_1$ with centre $O_1$ passes through the centre $O_2$ of a second circle $\omega_2$. The tangent lines to $\omega_2$ from a point $C$ on $\omega_1$ intersect $\omega_1$ again at points $A$ and $B$ respectively. Prove that $AB$ is perpendicular to $O_1O_2$.



(5 points)
2005 Tournament of Towns
John and James wish to divide $25$ coins, of denominations $1, 2, 3, \ldots , 25$ kopeks. In each move, one of them chooses a coin, and the other player decides who must take this coin. John makes the initial choice of a coin, and in subsequent moves, the choice is made by the player having more kopeks at the time. In the event that there is a tie, the choice is made by the same player in the preceding move. After all the coins have been taken, the player with more kopeks wins. Which player has a winning strategy?



(5 points)
2005 Tournament of Towns
For any function $f(x)$, define $f^1(x) = f(x)$ and $f^n (x) = f(f^{n-1}(x))$ for any integer $n \ge 2$.  Does there exist a quadratic polynomial $f(x)$ such that the equation $f^n(x) = 0$ has exactly $2^n$ distinct real roots for every positive integer $n$?



(6 points)
2005 Tournament of Towns
Prove that if a regular icosahedron and a regular dodecahedron have a common circumsphere, then they have a common insphere.



(7 points)
2005 Tournament of Towns
A lazy rook can only move from a square to a vertical or a horizontal neighbour. It follows a path which visits each square of an $8 \times 8$ chessboard exactly once. Prove that the number of such paths starting at a corner square is greater than the number of such paths starting at a diagonal neighbour of a corner square.



(7 points)
2005 Tournament of Towns
Every two of $200$ points in space are connected by a segment, no two intersecting each other. Each segment is painted in one colour, and the total number of colours is $k$. Peter wants to paint each of the $200$ points in one of the colours used to paint the segments, so that no segment connects two points both in the same colour as the segment itself. Can Peter always do this if



(a) k = 7; (4 points)

(b) k = 10? (4 points)
2005 Tournament of Towns
In triangle $ABC$, points $M_1, M_2$ and $M_3$ are midpoints of sides $AB$, $BC$ and $AC$, respectively, while points $H_1, H_2$ and $H_3$ are bases of altitudes drawn from $C$, $A$ and $B$, respectively. Prove that one can construct a triangle from segments $H_1M_2, H_2M_3$ and $H_3M_1$.



(3 points)
2005 Tournament of Towns
A number is written in each corner of the cube. On each step, each number is replaced with the average of three numbers in the three adjacent corners (all the numbers are replaced simultaneously). After ten such steps, every number returns to its initial value. Must all numbers have been originally equal?



(3 points)
2005 Tournament of Towns
A segment of unit length is cut into eleven smaller segments, each with length of no more than $a$. For what values of $a$, can one guarantee that any three segments form a triangle?



(4 points)
2005 Tournament of Towns
A chess piece moves as follows: it can jump 8 or 9 squares either vertically or horizontally. It is not allowed to visit the same square twice. At most, how many squares can this piece visit on a $15 \times 15$ board (it can start from any square)?



(4 points)
2005 Tournament of Towns
Among 6 coins one is counterfeit (its weight differs from that real one and neither weights is known). Using scales that show the total weight of coins placed on the cup, find the counterfeit coin in 3 weighings.



(5 points)
2005 Tournament of Towns
A palindrome is a positive integer which reads in the same way in both directions (for example, $1$, $343$ and $2002$ are palindromes, while $2005$ is not). Is it possible to find $2005$ pairs in the form of $(n, n + 110)$ where both numbers are palindromes?



(3 points)
2005 Tournament of Towns
The extensions of sides $AB$ and $CD$ of a convex quadrilateral $ABCD$ intersect at $K$. It is known that $AD = BC$. Let $M$ and $N$ be the midpoints of sides $AB$ and $CD$. Prove that the triangle $MNK$ is obtuse.



(5 points)
2005 Tournament of Towns
Originally, every square of $8 \times 8$ chessboard contains a rook. One by one, rooks which attack an odd number of others are removed. Find the maximal number of rooks that can be removed. (A rook attacks another rook if they are on the same row or column and there are no other rooks between them.)



(6 points)
2005 Tournament of Towns
Two ants crawl along the perimeter of a polygonal table, so that the distance between them is always $10$ cm. Each side of the table is more than $1$ meter long. At the initial moment both ants are on the same side of the table.



(a) (2 points) Suppose that the table is a convex polygon. Is it always true that both ants can visit each point on the perimeter?



(b) (4 points) Is it always true (this time without assumption of convexity) that each point on the perimeter can be visited by at least one ant?
2005 Tournament of Towns
Find the largest positive integer $N$ such that the equation $99x + 100y + 101z = N$ has an unique solution in the positive integers $x, y, z$.



(7 points)
2005 Tournament of Towns
Karlsson-on-the-Roof has $1000$ jars of jam. The jars are not necessarily identical; each contains no more than $\dfrac{1}{100}$-th of the total amount of the jam. Every morning, Karlsson chooses any $100$ jars and eats the same amount of the jam from each of them. Prove that Karlsson can eat all the jam.



(8 points)
2005 Tournament of Towns
Can two perfect cubes fit between two consecutive perfect squares? In other words, do there exist positive integers $a$, $b$, $n$ such that $n^2 < a^3 < b^3 < (n + 1)^2$?



(3 points)
2005 Tournament of Towns
A segment of length $\sqrt2 + \sqrt3 + \sqrt5$ is drawn. Is it possible to draw a segment of unit length using a compass and a straightedge?



(3 points)
2005 Tournament of Towns
Among 6 coins one is counterfeit (its weight differs from that real one and neither weights is known). Using scales that show the total weight of coins placed on the cup, find the counterfeit coin in 3 weighings.



(4 points)
2005 Tournament of Towns
On all three sides of a right triangle $ABC$ external squares are constructed; their centers denoted by $D$, $E$, $F$. Show that the ratio of the area of triangle $DEF$ to the area of triangle $ABC$ is:



a) (2 points) greater than $1$;

b) (2 points) at least $2$.
2005 Tournament of Towns
A cube lies on the plane. After being rolled a few times (over its edges), it is brought back to its initial location with the same face up. Could the top face have been rotated by 90 degrees?



(5 points)
2005 Tournament of Towns
For which $n \ge 2$ can one find a sequence of distinct positive integers $a_1, a_2, \ldots , a_n$ so that the sum

$$\frac{a_1}{a_2} + \frac{a_2}{a_3} + \ldots +\frac{a_n}{a_1}$$

is an integer?



(3 points)
2005 Tournament of Towns
Two ants crawl along the perimeter of a polygonal table, so that the distance between them is always 10 cm. Each side of the table is more than 1 meter long. At the initial moment both ants are on the same side of the table.



(a) (2 points) Suppose that the table is a convex polygon. Is it always true that both ants can visit each point on the perimeter?

(b) (3 points) Is it always true (this time without assumption of convexity) that each point on the perimeter can be visited by at least one ant?
2005 Tournament of Towns
Originally, every square of $8 \times 8$ chessboard contains a rook. One by one, rooks which attack an odd number of others are removed. Find the maximal number of rooks that can be removed. (A rook attacks another rook if they are on the same row or column and there are no other rooks between them.)



(5 points)
2005 Tournament of Towns
Several positive numbers each not exceeding 1 are written on the circle. Prove that one can divide the circle into three arcs so that the sums of numbers on any two arcs differ by no more than 1. (If there are no numbers on an arc, the sum is equal to zero.)



(6 points)
2005 Tournament of Towns
In triangle $ABC$ bisectors $AA_1, BB_1$ and $CC_1$ are drawn. Given $\angle A : \angle B : \angle C = 4 : 2 : 1$, prove that $A_1B_1 = A_1C_1$.



(7 points)
2005 Tournament of Towns
Two operations are allowed:



(i) to write two copies of number $1$;

(ii) to replace any two identical numbers $n$ by $(n + 1)$ and $(n - 1)$.



Find the minimal number of operations that required to produce the number $2005$ (at the beginning there are no numbers).



(8 points)
2006 Tournament of Towns
Let $\angle A$ in a triangle $ABC$ be $60^\circ$. Let point $N$ be the intersection of $AC$ and perpendicular bisector to the side $AB$ while point $M$ be the intersection of $AB$ and perpendicular bisector to the side $AC$. Prove that $CB = MN$.



(3 points)
2006 Tournament of Towns
A $n \times n$ table is filled with the numbers as follows: the first column is filled with $1$’s, the second column with $2$’s, and so on. Then, the numbers on the main diagonal (from top-left to bottom-right) are erased. Prove that the total sums of the numbers on both sides of the main diagonal differ in exactly two times.



(3 points)
2006 Tournament of Towns
Let $a$ be some positive number. Find the number of integer solutions $x$ of inequality $2 < xa < 3$ given that inequality $1 < xa < 2$ has exactly $3$ integer solutions. Consider all possible cases.



(4 points)
2006 Tournament of Towns
Anna, Ben and Chris sit at the round table passing and eating nuts. At first only Anna has the nuts that she divides equally between Ben and Chris, eating a leftover (if there is any). Then Ben does the same with his pile. Then Chris does the same with his pile. The process repeats itself: each of the children divides his/her pile of nuts equally between his/her neighbours eating the leftovers if there are any. Initially, the number of nuts is large enough (more than 3). Prove that

a) at least one nut is eaten; (3 points)

b) all nuts cannot be eaten. (3 points)
2006 Tournament of Towns
Pete has $n^3$ white cubes of the size $1\times 1\times 1$. He wants to construct a $n\times n\times n$ cube with all its faces being completely white. Find the minimal number of the faces of small cubes that Basil must paint (in black colour) in order to prevent Pete from fulfilling his task. Consider the cases:

a)  $n = 2$; (2 points)

b)  $n = 3$. (4 points)
2006 Tournament of Towns
There is a billiard table in shape of rectangle $2 \times 1$, with pockets at its corners and at midpoints of its two largest sizes. Find the minimal number of balls one has to place on the table interior so that any pocket is on a straight line with some two balls. (Assume that pockets and balls are points).



(4 points)
2006 Tournament of Towns
Prove that one can find 100 distinct pairs of integers such that every digit of each number is no less than 6 and the product of the numbers in each pair is also a number with all its digits being no less than 6.



(4 points)
2006 Tournament of Towns
On sides $AB$ and $BC$ of an acute triangle $ABC$ two congruent rectangles $ABMN$ and $LBCK$ are constructed (outside of the triangle), so that $AB = LB$. Prove that straight lines $AL, CM$ and $NK$ intersect at the same point.



(5 points)
2006 Tournament of Towns
Is there exist some positive integer $n$, such that the first decimal of $2^n$ (from left to the right) is $5$ while the first decimal of $5^n$ is $2$?



(5 points)
2006 Tournament of Towns
Numbers $0, 1$ and $2$ are placed in a table $2005 \times 2006$ so that total sums of the numbers in each row and in each column are factors of $3$. Find the maximal possible number of $1$'s that can be placed in the table.



(6 points)
2006 Tournament of Towns
Let us call a pentagon curved, if all its sides are arcs of some circles. Are there exist a curved pentagon $P$ and a point $A$ on its boundary so that any straight line passing through $A$ divides perimeter of $P$ into two parts of the same length?



(7 points)
2006 Tournament of Towns
Anna and Boris have the same copy of $5\times5$ table filled with $25$ distinct numbers. After choosing the maximal number in the table, Anna erases the row and the column that contain this number. Then she continue the same operations with a smaller table till it is possible.Boris basically does the same; however, each time choosing the minimal number in a table. Can it happen that the total sum of the numbers chosen by Boris

a) is greater than the total sum of the numbers chosen by Anna? (6 points)

b)  is greater than the total sum of any $5$ numbers of initial table given that no two of the numbers are in the same row or in the same column? (2 points)
2006 Tournament of Towns
All vertices of a convex polyhedron with 100 edges are cut off by some planes. The planes do not intersect either inside or on the surface of the polyhedron. For this new polyhedron find

a) the number of vertices; (1 point)

b) the number of edges. (2 points)
2006 Tournament of Towns
Do there exist functions $p(x)$ and $q(x)$, such that $p(x)$ is an even function while $p(q(x))$ is an odd function (different from 0)?



(3 points)
2006 Tournament of Towns
Let $a$ be some positive number. Find the number of integer solutions $x$ of inequality $100 < xa < 1000$ given that inequality $10 < xa < 100$ has exactly $5$ integer solutions. Consider all possible cases.



(4 points)
2006 Tournament of Towns
Quadrilateral $ABCD$ is a cyclic, $AB = AD$. Points $M$ and $N$ are chosen on sides $BC$ and $CD$ respectfully so that $\angle MAN =1/2 (\angle BAD)$. Prove that $MN = BM + ND$.



(5 points)
2006 Tournament of Towns
Pete has $n^3$ white cubes of the size $1\times1\times1$. He wants to construct a $n\times n\times n$ cube with all

its faces being completely white. Find the minimal number of the faces of small cubes that Basil must paint (in black colour) in order to prevent Pete from fulfilling his task. Consider the cases:

a) $n = 3$; (3 points)

b) $n = 1000$. (3 points)
2006 Tournament of Towns
Prove that one can always mark $50$ points inside of any convex $100$-gon, so that each its vertix is on a straight line connecting some two marked points. (4)
2006 Tournament of Towns
Are there exist some positive integers $n$ and $k$, such that the first decimals of $2^n$ (from left to the right) represent the number $5^k$ while the first decimals of $5^n$ represent the number $2^k$ ? (5)
2006 Tournament of Towns
Consider a polynomial $P(x) = x^4+x^3-3x^2+x+2$. Prove that at least one of the coefficients of $(P(x))^k$, ($k$ is any positive integer) is negative. (5)
2006 Tournament of Towns
In triangle $ABC$ let $X$ be some fixed point on bisector $AA'$ while point $B'$  be intersection of  $BX$ and $AC$ and point $C'$ be intersection of $CX$ and $AB$. Let point $P$ be intersection of segments $A'B'$ and $CC'$ while point $Q$  be intersection of segments $A'C'$ and $BB'$. Prove τhat $\angle PAC = \angle QAB$.
2006 Tournament of Towns
Prove that one can find infinite number of distinct pairs of integers such that every digit of each number is no less than $7$ and the product of two numbers in each pair is also a number with all its digits being no less than $7$. (6)
2006 Tournament of Towns
On a circumference at some points sit $12$ grasshoppers. The points divide the circumference into $12$ arcs. By a signal each grasshopper jumps from its point to the midpoint of its arc (in clockwise direction). In such way new arcs are created. The process repeats for a number of times. Can it happen that at least one of the grasshoppers returns to its initial point after

a) $12$ jumps? (4)

a) $13$ jumps? (3)