Fourty seventh Tournament of Towns, 2025 – 2026

In a class, every student either lies or tells the truth depending on the day of the week (possibly different days for different students), and no student tells the truth for two consecutive days. On April 1st, 2nd, 3rd, and 4th, every student was asked whether he will be telling the truth tomorrow. On April 1st, all students responded ''yes''; on April 2nd, half of the students responded ''yes''; on April 3rd, one third of the students. What fraction of the students told the truth on April 4th?

The path from place A to B follows first a paved road and then a forest trail. Alice and Bob started walking towards each other at the same time from places A and B respectively and met on the paved road. Next day, they also started towards each other at the same time, but now Alice started from B, and Bob — from A, and they again met on the paved road, 6 kilometers away from the place of their previous meeting. It is known that each of them walks on the paved road at constant speed, but Alice walks 1.5 times faster than Bob. It is also known that on the forest trail, each of them walks slower than on the paved road, but Alice still walks 1.5 times faster than Bob. How far their meeting point on the first day was from A?

A rectangular board, consisting of 1 × 1 squares, is covered by two layers of 1 × 2 tiles (each tile follows the grid lines of the board; each of two layers completely covers the board). Let us call a top layer tile special if its two halves are on top of two different tiles of the bottom layer, with different orientations (i.e., one horizontal and the other one vertical). Is the number of special tiles necessarily even?

Baron Munchausen has colored every positive integer number using one of three colors, red, white, or blue (all colors were used). He claims that the sum of any 99 red numbers is blue, and the sum of any 99 blue numbers is red (the same number can appear more than once in the sum). Can it be that he is telling the truth?

Does there exist an acute angle α such that some rectangle can be cut into isosceles trapezoids, each of them having α as one of its angles?

We are given a row of boxes labeled 1, 2, 3, ... (left to right). We place numbers 1, 2, ..., 2025 (in this order) in these boxes following these rules:

• Each two numbers in the same box must be relatively prime.
• Each next number is placed in the leftmost box allowed by the previous rule.

After all numbers are placed, we count how many numbers are in each box.

In a convex pentagon ABCDE, sides AE, BC, and DE are equal; also, angles A, B, C, and D are all equal. Prove that points A, B, C, D, and E lie on a circle.

30 boys and 30 girls stand in a circle. Prove that it is possible to choose 10 boys and 10 girls such that no two of the chosen people stand next to each other.

Alice and Bob are drawing on the board. Alice has drawn several circles with different centers, coloring each of the center points either red or blue. It so happened that if any two of these circles are tangent to each other, then they are externally tangent, and their centers have different colors. Is it true that Bob can always replace each circle by a different one with the same center so that the following condition holds: if two old circles were tangent to each other, then the corresponding new circles are also tangent, but now they are internally tangent?

An n × n table is filled with integers 0 to n so that in each row and in each column all entries are different. Let us call a table cell lucky if the union of its row and column contains all numbers 0 to n.

In a 12-digit number A, any two adjacent digits form a 2-digit prime number, and all eleven of these primes are distinct.

Angles B and C of a quadrilateral ABCD are both equal to 120°. There is a point on side AD such that from this point, all other sides are seen at the same angle. Find the angle between the diagonals of the quadrilateral.

Alice and Bob are playing on a 100 × 100 square table, each cell of which contains either a plus sign or a minus sign. They take turns crossing out rows and columns of the table: Alice on her turn can cross out any remaining row of the table; similarly, Bob on his turn can cross out any remaining column. They repeat that until only one cell remains. If that cell contains the plus sign, Alice wins; if the minus sign, Bob wins. Does there exist a table such that, no matter if Alice or Bob moves first, there is a winning strategy for

Prove that there exists a positive integer $N$ such that there are exactly 1000 squares strictly between N^3 and (N+1)^3.

Alice has 60 cards, numbered 1 through 60; each card contains a real number. Bob doesn't know these numbers, but can choose any 17 numbers between 1 and 60 and ask Alice for the sum of the real numbers written on these cards. Can Bob determine the sum of the real numbers on all 60 cards by asking

We are given a positive integer k. Someone places n coins of weights 1, 2, ..., n in a circle on the table; all coins look identical. It is known that the weights go in increasing order, but we do not know the starting point and whether they increase clockwise or counterclockwise. You have a balance scale that allow you to take any two coins and find out which one is heavier (you can only place one coin on each side of the scale). Baron Munchausen claims that you can make at most k such weighings and know for certain the weight of at least one coin. What is the largest $n$ for which it is true?

Alice and Bob are playing on a 100 × 100 square table, each cell of which contains either a plus sign or a minus sign. They take turns crossing out rows and columns of the table: Alice on her turn can cross out any remaining row of the table; similarly, Bob on his turn can cross out any remaining column. They repeat that until only one cell remains. If that cell contains the plus sign, Alice wins; if the minus sign, Bob wins. Does there exist a table such that, no matter if Alice or Bob moves first, there is a winning strategy for

In a convex quadrilateral, we draw a segment through the midpoint of each diagonal parallel to the other diagonal, with endpoints on the sides of the quadrilateral. Show that the endpoints of these two segments form a quadrilateral that has a pair of parallel sides.

N prisoners are held in cells placed in a circle. Tomorrow each prisoner will roll a die; after that, he would make a guess, predicting the numbers rolled by each of the 5 prisoners in cells clockwise from his own. If at least one prisoner guesses at least one of the numbers correctly, all prisoners will be freed. Today, the prisoners can talk and decide on the common strategy, but after midnight, no communication will be possible. How should they act to guarantee their release if

Prove that there exists a positive integer N such that there are exactly 1000 squares strictly between N^3 and (N+1)^3.

Each side of a regular N-gon contains a robot. Each robot is moving along the side with constant speed that may be different for different robots; upon reaching the vertex, it turns around instantly and continues moving along the same side in the opposite direction. When two robots meet at a vertex, they produce a spark. Can it happen that at each vertex, the sparks appear with the same non-zero frequency if

A snail is crawling on the plane, starting and ending at the same point. Its path consists of n straight segments, without self-intersections. It is known that the snail can only move in three directions: up, right and down-left (at 45° angle to the horizontal). Prove that n is odd. (Adjacent segments cannot have the same direction.)

We are given a positive integer k. Someone places n coins of weights 1, 2, ..., n in a circle on the table; all coins look identical. It is known that the weights go in increasing order, but we do not know the starting point and whether they increase clockwise or counterclockwise. You have a balance scale that allow you to place any number of coins on each of the two sides of the scale and determine which side is heavier. Baron Munchausen claims that you can make at most $k$ such weighings and know for certain the weight of at least one coin. What is the largest $n$ for which it is true?