Navvye/Olympiad-Problems-CSV huggingface dataset:
2006 Tournament of Towns
An ant craws along a closed route along the edges of a dodecahedron, never going backwards.

Each edge of the route is passed exactly twice. Prove that one of the edges is passed both times in the same direction. (Dodecahedron has $12$ faces in the shape of pentagon, $30$ edges and $20$ vertices; each vertex emitting 3 edges). (8)
2006 Tournament of Towns
Two positive integers are written on the blackboard. Mary records in her notebook the square of the smaller number and replaces the larger number on the blackboard by the difference of the two numbers. With the new pair of numbers, she repeats the process, and continues until one of the numbers on the blackboard becomes zero. What will be the sum of the numbers in Mary's notebook at that point? (4)
2006 Tournament of Towns
A Knight always tells the truth. A Knave always lies. A Normal may either lie or tell the truth. You are allowed to ask questions that can be answered with ''yes" or ''no", such as ''is this person a Normal?"

(a) There are three people in front of you. One is a Knight, another one is a Knave, and the third one is a Normal. They all know the identities of one another. How can you too learn the identity of each? (1)

(b) There are four people in front of you. One is a Knight, another one is a Knave, and the other two are Normals. They all know the identities of one another. Prove that the Normals may agree in advance to answer your questions in such a way that you will not be able to learn the identity of any of the four people. (3)
2006 Tournament of Towns
(a) Prove that from $2007$ given positive integers, one of them can be chosen so the product of the remaining numbers is expressible in the form $a^2 - b^2$ for some positive integers $a$ and $b$. (2)

(b) One of $2007$ given positive integers is $2006$. Prove that if there is a unique number among them such that the product of the remaining numbers is expressible in the form $a^2 - b^2$ for some positive integers $a$ and $b$, then this unique number is $2006$. (2)
2006 Tournament of Towns
Given triangle $ABC, BC$ is extended beyond $B$ to the point $D$ such that $BD = BA$. The bisectors of the exterior angles at vertices $B$ and $C$ intersect at the point $M$. Prove that quadrilateral $ADMC$ is cyclic. (4)
2006 Tournament of Towns
A square is dissected into $n$ congruent non-convex polygons whose sides are parallel to the sides of the square, and no two of these polygons are parallel translates of each other. What is the maximum value of $n$? (4)
2006 Tournament of Towns
Two regular polygons, a $7$-gon and a $17$-gon are given. For each of them two circles are drawn, an inscribed circle and a circumscribed circle. It happened that rings containing the polygons have equal areas. Prove that sides of the polygons are equal. (3)
2006 Tournament of Towns
When Ann meets new people, she tries to find out who is acquainted with who. In order to memorize it she draws a circle in which each person is depicted by a chord; moreover, chords corresponding to acquainted persons intersect (possibly at the ends), while the chords corresponding to non-acquainted persons do not. Ann believes that such set of chords exists for any company. Is her judgement correct? (5)
2006 Tournament of Towns
A $3 \times 3$ square is filled with numbers: $a, b, c, d, e, f, g, h, i$ in the following way:

Given that the square is magic (sums of the numbers in each row, column and each of two diagonals are the same), show that

a) $2(a + c + g + i) = b + d + f + h + 4e$. (3)

b) $2(a^3 + c^3 + g^3 + i^3) = b^3 + d^3 + f^3 + h^3 + 4e^3$. (3)
2006 Tournament of Towns
A circle of radius $R$ is inscribed into an acute triangle. Three tangents to the circle split the triangle into three right angle triangles and a hexagon that has perimeter $Q$. Find the sum of diameters of circles inscribed into the three right triangles. (6)
2006 Tournament of Towns
Consider a square painting of size $1 \times 1$. A rectangular sheet of paper of area $2$ is called its “envelope” if one can wrap the painting with it without cutting the paper. (For instance, a $2 \times 1$ rectangle and a square with side $\sqrt2$ are envelopes.)

a)  Show that there exist other envelopes. (4)

b) Show that there exist infinitely many envelopes. (3)
2006 Tournament of Towns
Let $1 + 1/2 + 1/3 +... + 1/n = a_n/b_n$, where $a_n$ and $b_n$ are relatively prime. Show that there exist infinitely many positive integers $n$, such that $b_{n+1} < b_n$. (8)
2006 Tournament of Towns
A Magician has a deck of $52$ cards. Spectators want to know the order of cards in the deck(without specifying face-up or face-down). They are allowed to ask the questions “How many cards are there between such-and-such card and such-and-such card?” One of the spectators knows the card order. Find the minimal number of questions he needs to ask to be sure that the other spectators can learn the card order. (9)
2006 Tournament of Towns
Three positive integers $x$ and $y$ are written on the blackboard. Mary records in her notebook the product of any two of them and reduces the third number on the blackboard by $1$. With the new trio of numbers, she repeats the process, and continues until one of the numbers on the blackboard becomes zero. What will be the sum of the numbers in Mary's notebook at that point? (4)
2006 Tournament of Towns
The incircle of the quadrilateral $ABCD$ touches $AB,BC, CD$ and $DA$ at $E, F,G$ and $H$ respectively. Prove that the line joining the incentres of triangles $HAE$ and $FCG$ is perpendicular to the line joining the incentres of triangles $EBF$ and $GDH$. (4)
2006 Tournament of Towns
Each of the numbers $1, 2, 3,... , 2006^2$ is placed at random into a cell of a $2006\times 2006$ board. Prove that there exist two cells which share a common side or a common vertex such that the sum of the numbers in them is divisible by $4$. (4)
2006 Tournament of Towns
Every term of an infinite geometric progression is also a term of a given infinite arithmetic progression. Prove that the common ratio of the geometric progression is an integer. (4)
2006 Tournament of Towns
Can a regular octahedron be inscribed in a cube in such a way that all vertices of the octahedron are on cube's edges? (4)
2006 Tournament of Towns
Suppose $ABC$ is an acute triangle. Points $A_1, B_1$ and $C_1$ are chosen on sides $BC, AC$ and $AB$ respectively so that the rays $A_1A, B_1B$ and $C_1C$ are bisectors of triangle $A_1B_1C_1$. Prove that $AA_1, BB_1$ and $CC_1$ are altitudes of triangle $ABC$. (6)
2006 Tournament of Towns
The $n$-th digit of number $a = 0.12457...$ equals the first digit of the integer part of the number $n\sqrt2$. Prove that $a$ is irrational number. (6)
2006 Tournament of Towns
Is it possible to split a prism into disjoint set of pyramids so that each pyramid has its base on one base of the prism, while its vertex on another base of the prism ? (6)
2006 Tournament of Towns
Let us say that a deck of $52$ cards is arranged in a “regular” way if the ace of spades is on the very top of the deck and any two adjacent cards are either of the same value or of the same suit (top and bottom cards regarded adjacent as well). Prove that the number of ways to arrange a deck in regular way is

a) divisible by $12!$ (3)

b) divisible by $13!$ (5)
2006 Tournament of Towns
Positive numbers $x_1,..., x_k$ satisfy the following inequalities:

$x_1^2+...+ x_k^2 <\frac{x_1+...+x_k}{2}$ and $x_1+...+x_k < \frac{x_1^3+...+ x_k^3}{2}$.

a) Show that $k > 50$, (3)

b) Give an example of such numbers for some value of $k$ (3)

c) Find minimum $k$, for which such an example exists. (3)
2007 Tournament Of Towns
The sides of a convex pentagon are extended on both sides to form five triangles. If these triangles are congruent to one another, does it follow that the pentagon is regular?
2007 Tournament Of Towns
Two $2007$-digit numbers are given. It is possible to delete $7$ digits from each of them to obtain the same $2000$-digit number. Prove that it is also possible to insert $7$ digits into the given numbers so as to obtain the same $2014$-digit number.
2007 Tournament Of Towns
What is the least number of rooks that can be placed on a standard $8 \times 8$ chessboard so that all the white squares are attacked? (A rook also attacks the square it is on, in addition to every other square in the same row or column.)
2007 Tournament Of Towns
Three nonzero real numbers are given. If they are written in any order as coefficients of a quadratic trinomial, then each of these trinomials has a real root. Does it follow that each of these trinomials has a positive root?
2007 Tournament Of Towns
A triangular pie has the same shape as its box, except that they are mirror images of each other. We wish to cut the pie in two pieces which can t together in the box without turning either piece over. How can this be done if





(a) one angle of the triangle is three times as big as another;

(b) one angle of the triangle is obtuse and is twice as big as one of the acute angles?
2007 Tournament Of Towns
Let $n$ be a positive integer. In order to find the integer closest to $\sqrt n$, Mary finds $a^2$, the closest perfect square to $n$. She thinks that a is then the number she is looking for. Is she always correct?
2007 Tournament Of Towns
$K, L, M$ and $N$ are points on sides $AB, BC, CD$ and $DA$, respectively, of the unit square $ABCD$ such that $KM$ is parallel to $BC$ and $LN$ is parallel to $AB$. The perimeter of triangle $KLB$ is equal to $1$. What is the area of triangle $MND$?
2007 Tournament Of Towns
Anna's number is obtained by writing down $20$ consecutive positive integers, one after another in arbitrary order. Bob's number is obtained in the same way, but with $21$ consecutive positive integers. Can they obtain the same number?
2007 Tournament Of Towns
Several diagonals (possibly intersecting each other) are drawn in a convex $n$-gon in such a way that no three diagonals intersect in one point. If the $n$-gon is cut into triangles, what is the maximum possible number of these triangles?
2007 Tournament Of Towns
Find all (finite) increasing arithmetic progressions, consisting only of prime numbers, such that the number of terms is larger than the common difference.
2007 Tournament Of Towns
In the quadrilateral $ABCD$, $AB = BC = CD$ and $\angle BMC = 90^\circ$, where $M$ is the midpoint of $AD$. Determine the acute angle between the lines $AC$ and $BD$.
2007 Tournament Of Towns
Nancy shuffles a deck of $52$ cards and spreads the cards out in a circle face up, leaving one spot empty. Andy, who is in another room and does not see the cards, names a card. If this card is adjacent to the empty spot, Nancy moves the card to the empty spot, without telling Andy; otherwise nothing happens. Then Andy names another card and so on, as many times as he likes, until he says "stop."





(a) Can Andy guarantee that after he says "stop," no card is in its initial spot?

(b) Can Andy guarantee that after he says "stop," the Queen of Spades is not adjacent to

the empty spot?
2007 Tournament Of Towns
A $9 \times 9$ chessboard with the standard checkered pattern has white squares at its four corners. What is the least number of rooks that can be placed on this board so that all the white squares are attacked? (A rook also attacks the square it is on, in addition to every other square in the same row or column.)
2007 Tournament Of Towns
The polynomial $x^3 + px^2 + qx + r$ has three roots in the interval $(0,2)$. Prove that $-2 <p + q + r < 0$.
2007 Tournament Of Towns
$B$ is a point on the line which is tangent to a circle at the point $A$. The line segment $AB$ is rotated about the centre of the circle through some angle to the line segment $A'B'$. Prove that the line $AA'$ passes through the midpoint of $BB'$.
2007 Tournament Of Towns
A binary sequence is constructed as follows. If the sum of the digits of the positive integer $k$ is even, the $k$-th term of the sequence is $0$. Otherwise, it is $1$. Prove that this sequence is not periodic.
2007 Tournament Of Towns
$A,B,C$ and $D$ are points on the parabola $y = x^2$ such that $AB$ and $CD$ intersect on the $y$-axis. Determine the $x$-coordinate of $D$ in terms of the $x$-coordinates of $A,B$ and $C$, which are $a, b$ and $c$ respectively.
2007 Tournament Of Towns
A convex figure $F$ is such that any equilateral triangle with side $1$ has a parallel translation that takes all its vertices to the boundary of $F$. Is $F$ necessarily a circle?
2007 Tournament Of Towns
Let $f(x)$ be a polynomial of nonzero degree. Can it happen that for any real number $a$, an even number of real numbers satisfy the equation $f(x) = a$?
2007 Tournament Of Towns
From a regular octahedron with edge $1$, cut off a pyramid about each vertex. The base of each pyramid is a square with edge $\frac 13$. Can copies of the polyhedron so obtained, whose faces are either regular hexagons or squares, be used to tile space?
2007 Tournament Of Towns
Let $a_0$ be an irrational number such that $0 < a_0 < \frac 12$ . Define $a_n = \min \{2a_{n-1},1 - 2a_{n-1}\}$ for $n \geq 1$.





(a) Prove that $a_n < \frac{3}{16}$ for some $n$.

(b) Can it happen that $a_n > \frac{7}{40}$ for all $n$?
2007 Tournament Of Towns
$T$ is a point on the plane of triangle $ABC$ such that $\angle ATB = \angle  BTC = \angle  CTA = 120^\circ$. Prove that the lines symmetric to $AT, BT$ and $CT$ with respect to $BC, CA$ and $AB$, respectively, are concurrent.
2007 Tournament Of Towns
Black and white checkers are placed on an $8 \times 8$ chessboard, with at most one checker on each cell. What is the maximum number of checkers that can be placed such that each row and each column contains twice as many white checkers as black ones?
2007 Tournament Of Towns
Initially, the number $1$ and a non-integral number $x$ are written on a blackboard. In each step, we can choose two numbers on the blackboard, not necessarily different, and write their sum or their difference on the blackboard. We can also choose a non-zero number of the blackboard and write its reciprocal on the blackboard. Is it possible to write $x^2$ on the blackboard in a finite number of moves?
2007 Tournament Of Towns
$D$ is the midpoint of the side $BC$ of triangle $ABC$. $E$ and $F$ are points on $CA$ and $AB$ respectively, such that $BE$ is perpendicular to $CA$ and $CF$ is perpendicular to $AB$. If $DEF$ is an equilateral triangle, does it follow that $ABC$ is also equilateral?
2007 Tournament Of Towns
Each cell of a $29 \times 29$ table contains one of the integers $1, 2, 3, \ldots , 29$, and each of these integers appears $29$ times. The sum of all the numbers above the main diagonal is equal to three times the sum of all the numbers below this diagonal. Determine the number in the central cell of the table.
2007 Tournament Of Towns
The audience chooses two of five cards, numbered from $1$ to $5$ respectively. The assistant of a magician chooses two of the remaining three cards, and asks a member of the audience to take them to the magician, who is in another room. The two cards are presented to the magician in arbitrary order. By an arrangement with the assistant beforehand, the magician is able to deduce which two cards the audience has chosen only from the two cards he receives. Explain how this may be done.
2007 Tournament Of Towns
Let $ABCD$ be a rhombus. Let $K$ be a point on the line $CD$, other than $C$ or $D$, such that $AD = BK$. Let $P$ be the point of intersection of $BD$ with the perpendicular bisector of $BC$. Prove that $A, K$ and $P$ are collinear.
2007 Tournament Of Towns
(a) Each of Peter and Basil thinks of three positive integers. For each pair of his numbers, Peter writes down the greatest common divisor of the two numbers. For each pair of his numbers, Basil writes down the least common multiple of the two numbers. If both Peter and Basil write down the same three numbers, prove that these three numbers are equal to each other.



(b) Can the analogous result be proved if each of Peter and Basil thinks of four positive integers instead?
2007 Tournament Of Towns
Michael is at the centre of a circle of radius $100$ metres. Each minute, he will announce the direction in which he will be moving. Catherine can leave it as is, or change it to the opposite direction. Then Michael moves exactly $1$ metre in the direction determined by Catherine. Does Michael have a strategy which guarantees that he can get out of the circle, even though Catherine will try to stop him?
2007 Tournament Of Towns
Two players take turns entering a symbol in an empty cell of a $1 \times n$ chessboard, where $n$ is an integer greater than $1$. Aaron always enters the symbol $X$ and Betty always enters the symbol $O$. Two identical symbols may not occupy adjacent cells. A player without a move loses the game. If Aaron goes first, which player has a winning strategy?
2007 Tournament Of Towns
Attached to each of a number of objects is a tag which states the correct mass of the object. The tags have fallen off and have been replaced on the objects at random. We wish to determine if by chance all tags are in fact correct. We may use exactly once a horizontal lever which is supported at its middle. The objects can be hung from the lever at any point on either side of the support. The lever either stays horizontal or tilts to one side. Is this task always possible?
2007 Tournament Of Towns
The audience arranges $n$ coins in a row. The sequence of heads and tails is chosen arbitrarily. The audience also chooses a number between $1$ and $n$ inclusive. Then the assistant turns one of the coins over, and the magician is brought in to examine the resulting sequence. By an agreement with the assistant beforehand, the magician tries to determine the number chosen by the audience.





(a) Prove that if this is possible for some $n$, then it is also possible for $2n$.

(b) Determine all $n$ for which this is possible.
2007 Tournament Of Towns
For each letter in the English alphabet, William assigns an English word which contains that letter. His first document consists only of the word assigned to the letter $A$. In each subsequent document, he replaces each letter of the preceding document by its assigned word. The fortieth document begins with “Till whatsoever star that guides my moving.” Prove that this sentence reappears later in this document.
2007 Tournament Of Towns
(from The Good Soldier Svejk) Senior military doctor Bautze exposed $abccc$ malingerers among $aabbb$ draftees who claimed not to be fit for the military service. He managed to expose all but one draftees. (He would for sure expose this one too, if the lucky guy was not taken by a stroke at the very moment when the doctor yelled at him "Turn around !. . . ") How many malingerers were exposed by the vigilant doctor?



Each digit substitutes a letter. The same digits substitute the same letters, while distinct digits substitute distinct letters.



(1 point)
2007 Tournament Of Towns
Let us call a triangle “almost right angle triangle” if one of its angles differs from $90^\circ$ by no more than $15^\circ$. Let us call a triangle “almost isosceles triangle” if two of its angles differs from each other by no more than $15^\circ$. Is it true that that any acute triangle is either “almost right angle triangle” or “almost isosceles triangle”?



(2 points)
2007 Tournament Of Towns
A triangle with sides $a, b, c$ is folded along a line $\ell$ so that a vertex $C$ is on side $c$. Find the segments on which point $C$ divides $c$, given that the angles adjacent to $\ell$ are equal.



(2 points)
2007 Tournament Of Towns
From the first 64 positive integers are chosen two subsets with 16 numbers in each. The first subset contains only odd numbers while the second one contains only even numbers. Total sums of both subsets are the same. Prove that among all the chosen numbers there are two whose sum equals 65.



(3 points)
2007 Tournament Of Towns
Two players in turns color the squares of a $4 \times 4$ grid, one square at the time. Player loses if after his move a square of $2\times2$ is colored completely. Which of the players has the winning strategy, First or Second?



(4 points)
2007 Tournament Of Towns
Pictures are taken of $100$ adults and $100$ children, with one adult and one child in each, the adult being the taller of the two. Each picture is reduced to $\frac 1k$ of its original size, where $k$ is a positive integer which may vary from picture to picture. Prove that it is possible to have the reduced image of each adult taller than the reduced image of every child.
2007 Tournament Of Towns
Initially, the number $1$ and two positive numbers $x$ and $y$ are written on a blackboard. In each step, we can choose two numbers on the blackboard, not necessarily different, and write their sum or their difference on the blackboard. We can also choose a non-zero number of the blackboard and write its reciprocal on the blackboard. Is it possible to write on the blackboard, in a finite number of moves, the number





a) $x^2$;

b) $xy$?
2007 Tournament Of Towns
Give a construction by straight-edge and compass of a point $C$ on a line $\ell$ parallel to a segment $AB$, such that the product $AC \cdot BC$ is minimum.
2007 Tournament Of Towns
The audience chooses two of twenty-nine cards, numbered from $1$ to $29$ respectively. The assistant of a magician chooses two of the remaining twenty-seven cards, and asks a member of the audience to take them to the magician, who is in another room. The two cards are presented to the magician in an arbitrary order. By an arrangement with the assistant beforehand, the magician is able to deduce which two cards the audience has chosen only from the two cards he receives. Explain how this may be done.
2007 Tournament Of Towns
A square of side length $1$ centimeter is cut into three convex polygons. Is it possible that the diameter of each of them does not exceed





a) $1$ centimeter;

b) $1.01$ centimeters;

c) $1.001$ centimeters?
2007 Tournament Of Towns
Let $K, L, M$ and $N$ be the midpoints of the sides $AB, BC, CD$ and $DA$ of a cyclic quadrilateral $ABCD$. Let $P$ be the point of intersection of $AC$ and $BD$. Prove that the circumradii of triangles $PKL, PLM, PMN$ and $PNK$ are equal to one another.
2007 Tournament Of Towns
Determine all finite increasing arithmetic progressions in which each term is the reciprocal of a positive integer and the sum of all the terms is $1$.
2007 Tournament Of Towns
Let $P$ and $Q$ be two convex polygons. Let $h$ be the length of the projection of $Q$ onto a line perpendicular to a side of $P$ which is of length $p$. Define $f(P,Q)$ to be the sum of the products $hp$ over all sides of $P$. Prove that $f(P,Q) = f(Q, P)$.
2007 Tournament Of Towns
There are $100$ boxes, each containing either a red cube or a blue cube. Alex has a sum of money initially, and places bets on the colour of the cube in each box in turn. The bet can be anywhere from $0$ up to everything he has at the time. After the bet has been placed, the box is opened. If Alex loses, his bet will be taken away. If he wins, he will get his bet back, plus a sum equal to the bet. Then he moves onto the next box, until he has bet on the last one, or until he runs out of money. What is the maximum factor by which he can guarantee to increase his amount of money, if he knows that the exact number of blue cubes is





(a) $1$;

(b) some integer $k$, $1 < k \leq 100$.
2007 Tournament Of Towns
A straight line is colored with two colors. Prove that there are three points $A, B, C$ of the same color such that $AB = BC$.



(1 point)
2007 Tournament Of Towns
A student did not notice multiplication sign between two three-digit numbers and wrote it as a six-digit number. Result was 7 times more that it should be. Find these numbers.



(2 points)
2007 Tournament Of Towns
Two players in turns color the squares of a $4 \times 4$ grid, one square at the time. Player loses if after his move a square of $2\times2$ is colored completely. Which of the players has the winning strategy, First or Second?



(3 points)