This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

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Found problems: 28

2019 Tournament Of Towns, 1
Tags: combinatorics , Sum , consecutive , Tournament of Towns , ToT
Consider a sequence of positive integers with total sum $20$ such that no number and no sum of a set of consecutive numbers is equal to $3$. Is it possible for such a sequence to contain more than $10$ numbers? (Alexandr Shapovalov)

2024/2025 TOURNAMENT OF TOWNS, P2
Tags: ToT , geometry
There are $100$ lines in the plane, such that no two are parallel and no three are concurrent. Consider the quadrilaterals such that all their sides lie on these lines (including the quadrilaterals whose interior is crossed by some of these lines). Is it true that the number of convex quadrilaterals equals the number of non-convex ones?

2024/2025 TOURNAMENT OF TOWNS, P2
Tags: combinatorics , ToT
In a $2025 \times 2025$ table, several cells are marked. At each move, Cyril can get to know the number of marked cells in any checkered square inside the initial table, with side less than $2025$. What is the minimal number of moves, which allows to determine the total number of marked cells for sure? (5 marks)

2024/2025 TOURNAMENT OF TOWNS, P1
Tags: ToT , number theory , floor function
On the blackboard, there are numbers $1, 2, \dots , 100$. At each move, Bob erases arbitrary two numbers $a$ and $b$, where $a \ge b > 0$, and writes the single number $\lfloor{a/b}\rfloor$. After $99$ such moves the blackboard will contain a single number. What is its maximum possible value? (Reminder that $\lfloor{x}\rfloor$ is the maximum integer not exceeding $x$.)

2024/2025 TOURNAMENT OF TOWNS, P1
Tags: ToT , number theory
The teacher has chosen two different figures from $\{1, 2, 3, \dots, 9\}$. Nick intends to find a seven-digit number divisible by $7$ such that its decimal representation contains no figures besides these two. Is this possible for each teacher’s choice? (4 marks)

2020/2021 Tournament of Towns, P2
Tags: geometry , algebra , polynomial , Vieta , Tournament of Towns , ToT , Baron Munchausen
Baron Munchausen presented a new theorem: if a polynomial $x^{n} - ax^{n-1} + bx^{n-2}+ \dots$ has $n$ positive integer roots then there exist $a$ lines in the plane such that they have exactly $b$ intersection points. Is the baron’s theorem true?

2024/2025 TOURNAMENT OF TOWNS, P5
Tags: ToT , algebra
Given $15$ coins of the same appearance. It is known that one of them weighs $1$g, two coins weigh $2$g each, three coins weigh $3$g each, four coins weigh $4$g each, and five coins weigh $5$g each. There are inscriptions on the coins, indicating their weight. It is allowed to perform two weighings on a balance without additional weights. Find a way to check that there are no wrong inscriptions. (It is not required to check which inscriptions are wrong and which ones are correct.) (8 marks)

2019 Tournament Of Towns, 2
Tags: Line , game , game strategy , combinatorics , Tournament of Towns , ToT , Kvant
$2019$ point grasshoppers sit on a line. At each move one of the grasshoppers jumps over another one and lands at the point the same distance away from it. Jumping only to the right, the grasshoppers are able to position themselves so that some two of them are exactly $1$ mm apart. Prove that the grasshoppers can achieve the same, jumping only to the left and starting from the initial position. (Sergey Dorichenko)

2024/2025 TOURNAMENT OF TOWNS, P1
Tags: ToT , number theory
Find the minimum positive integer such that some four of its natural divisors sum up to $2025$.

2016 Tournament Of Towns, 6
Tags: geometry , ToT
Q. An automatic cleaner of the disc shape has passed along a plain floor. For each point of its circular boundary there exists a straight line that has contained this point all the time. Is it necessarily true that the center of the disc stayed on some straight line all the time? ($9$ marks)

2024/2025 TOURNAMENT OF TOWNS, P4
Tags: ToT , combinatorics
Several jugs (not necessarily of the same size) with juices are placed along a circle. It is allowed to transfuse any part of juice (maybe nothing or the total content) from any jug to the neighboring one on the right, so that the latter one is not overflowed and the sugariness of its content becomes equal to $10\%$. It is known that at the initial moment such transfusion is possible from each jug. Prove that it is possible to perform several transfusions in some order, at most one transfusion from each jug, such that the sugariness of the content of each non-empty jug will become equal to $10\%$. (Sugariness is the percent of sugar in a jug, by weight. Sugar is always uniformly distributed in a jug.)

2020 Tournament Of Towns, 5
Tags: ToT
On the 8×8 chessboard there are two identical markers in the squares a1 and c3. Alice and Bob in turn make the following moves (the first move is Alice’s): a player picks any marker and moves it horizontally to the right or vertically upwards through any number of squares. The aim of each player is to get tothe square h8. Which player has a winning strategy no matter what does his opponent? (There may be only one marker on a square,the markers may not go through each other.) The 8x8 chessboard consists of columns lettered a to h from left to right and rows numbered 1-8 from bottom to top [img]https://cdn.artofproblemsolving.com/attachments/1/f/4c5548f606fda915e0a50a8cf886ff93e1f86d.png[/img]

2024/2025 TOURNAMENT OF TOWNS, P6
Tags: geometry , ToT
An equilateral triangle is dissected into white and black triangles. It is known that all white triangles are right-angled and mutually congruent, and all black triangles are isosceles and also mutually congruent. Is it necessarily true that a) all angles of white triangles are multiples of $30^{\circ}$; (4 marks) b) all angles of black triangles are multiples of $30^{\circ}$ ? (5 marks)

2024/2025 TOURNAMENT OF TOWNS, P4
Tags: number theory , ToT , geometry
Given $2N$ real numbers. It is known that if they are arbitrarily divided into two groups of $N$ numbers each then the products of the numbers of each group differ by $2$ at most. Is it necessarily true that if we arbitrarily place these numbers along a circle then there are two neighboring numbers that differ by $2$ at most, for a) $N=50$; (3 marks) b) $N=25$? (5 marks)

2024/2025 TOURNAMENT OF TOWNS, P5
Tags: algebra , ToT , number theory , greatest common divisor , polynomial
Given a polynomial with integer coefficients, which has at least one integer root. The greatest common divisor of all its integer roots equals $1$. Prove that if the leading coefficient of the polynomial equals $1$ then the greatest common divisor of the other coefficients also equals $1$.

2020 Tournament Of Towns, 1
Tags: ToT
The Quadrumland map is a 6 × 6 square where each square cell is either a kingdom or a disputed territory. There are 27 kingdoms and 9 disputed territories. Each disputed territory is claimed by those and only those kingdoms that are neighbouring with it (adjacent by an edge or a vertex). Is it possible that for each disputed territory the numbers of claims are different? You can discuss your solutions here

2024/2025 TOURNAMENT OF TOWNS, P4
Tags: ToT , geometry
There was a tub on the plane, with its upper base greater that the lower one. The tub was overturned. Prove that the area of its visible shade did decrease. (The tub is a frustum of a right circular cone: its bases are two discs in parallel planes, such that their centers lie on a line perpendicular to these planes. The visible shade is the total shade besides the shade under the tub. Consider the sun rays as parallel.)

2024/2025 TOURNAMENT OF TOWNS, P3
Tags: geometry , ToT
In a triangle $ABC$ with right angle $C$, the altitude $CH$ is drawn. An arbitrary circle passing through points $C$ and $H$ meets the segments $AC$, $CB$ and $BH$ for the second time at points $Q$, $P$ and $R$ respectively. Segments $HP$ and $CR$ meet at point $T$. What is greater: the area of triangle $CPT$ or the sum of areas of triangles $CQH$ and $HTR$? (5 marks)

2001 Tournament Of Towns, 5
Tags: combinatorics unsolved , combinatorics , ToT , Tournament of Towns
On a square board divided into $15 \times 15$ little squares there are $15$ rooks that do not attack each other. Then each rook makes one move like that of a knight. Prove that after this is done a pair of rooks will necessarily attack each other.

2018 Tournament Of Towns, 7.
Tags: number theory , ToT
You are in a strange land and you don’t know the language. You know that ”!” and ”?” stand for addition and subtraction, but you don’t know which is which. Each of these two symbols can be written between two arguments, but for subtraction you don’t know if the left argument is subtracted from the right or vice versa. So, for instance, a?b could mean any of a − b, b − a, and a + b. You don’t know how to write any numbers, but variables and parenthesis work as usual. Given two arguments a and b, how can you write an expression that equals 20a − 18b? (12 points) Nikolay Belukhov

Kvant 2019, M2558
Tags: Line , game , game strategy , combinatorics , Tournament of Towns , ToT , Kvant
$2019$ point grasshoppers sit on a line. At each move one of the grasshoppers jumps over another one and lands at the point the same distance away from it. Jumping only to the right, the grasshoppers are able to position themselves so that some two of them are exactly $1$ mm apart. Prove that the grasshoppers can achieve the same, jumping only to the left and starting from the initial position. (Sergey Dorichenko)

2020 Tournament Of Towns, 2
Tags: ToT
$ What~ is~ the~ maximum~ number~ of~ distinct~ integers~ in~ a~ row~ such~ that~ the~sum~ of~ any~ 11~ consequent~ integers~ is~ either~ 100~ or~ 101~?$ I'm posting this problem for people to discuss

2019 Tournament Of Towns, 1
Tags: game , game strategy , combinatorics , Tournament of Towns , ToT
The King gives the following task to his two wizards. The First Wizard should choose $7$ distinct positive integers with total sum $100$ and secretly submit them to the King. To the Second Wizard he should tell only the fourth largest number. The Second Wizard must figure out all the chosen numbers. Can the wizards succeed for sure? The wizards cannot discuss their strategy beforehand. (Mikhail Evdokimov)

2019 Tournament Of Towns, 3
Tags: number theory , Tournament of Towns , ToT , Divisibility
The product of two positive integers $m$ and $n$ is divisible by their sum. Prove that $m + n \le n^2$. (Boris Frenkin)

2024/2025 TOURNAMENT OF TOWNS, P3
Tags: geometry , rectangle , ToT
A point $K$ is chosen on the side $CD$ of a rectangle $ABCD$. From the vertex $B$, the perpendicular $BH$ is dropped to the segment $AK$. The segments $AK$ and $BH$ divide the rectangle into three parts such that each of them has the inscribed circle (see figure). Prove that if the circles tangent to $CD$ are equal then the third circle is also equal to them.