We solve problems

Problem #PRU-103964

Problems Number Theory Divisibility Division with remainders. Arithmetic of remainders Arithmetic of remainders Methods Pigeonhole principle Pigeonhole principle (other)

12–14 3.0

There are \(n\) integers. Prove that among them either there are several numbers whose sum is divisible by \(n\) or there is one number divisible by \(n\) itself.

Problem #PRU-103966

Problems Algebra Algebraic equations and systems of equations Higher order equations. Palindromic polynomial equations Equations of higher order (other) Calculus Real numbers Integer and fractional parts. Archimedean property

12–14 2.0

In Mongolia there are in circulation coins of 3 and 5 tugriks. An entrance ticket to the central park costs 4 tugriks. One day before the opening of the park, a line of 200 visitors queued up in front of the ticket booth. Each of them, as well as the cashier, has exactly 22 tugriks. Prove that all of the visitors will be able to buy a ticket in the order of the queue.

Problem #PRU-104001

Problems Discrete Mathematics Combinatorics Dissections, partitions, covers and tilings Various dissection problems

12–13 2.0

Kai has a piece of ice in the shape of a “corner” (see the figure). The Snow Queen demanded that Kai cut it into four equal parts. How can he do this?

Problem #PRU-104009

Problems Discrete Mathematics Set theory and logic Mathematical logic Mathematical logic (other)

12–13 2.0

The king made a test for the future groom of his daughter. He put the princess in one of three rooms, a tiger in the other, and left the last room empty. It is known that the sign on the door where the princess is sitting is true, where the tiger is – it is false, and nothing is known about the sign on the third room. The tablets are as follows:

1 – room 3 is empty

2 – the tiger is in room 1

3 – this room is empty

Can the prince correctly guess the room with the princess?

Problem #PRU-104021

Problems Geometry Solid geometry Parallelepipeds Special cases of parallelepipeds Cube Number Theory Divisibility Odd and even numbers Methods Pigeonhole principle Pigeonhole principle (other)

12–13 2.0

a) A 1 or a 0 is placed on each vertex of a cube. The sum of the 4 adjacent vertices is written on each face of the cube. Is it possible for each of the numbers written on the faces to be different?

b) The same question, but if 1 and \(-1\) are used instead.

Problem #PRU-104029

Problems Discrete Mathematics Combinatorics Geometry on grid paper

12–13 2.0

a) In the construction in the figure, move two matches so that there are five identical squares created. b) From the new figure, remove 3 matches so that only 3 squares remain.

Problem #PRU-104030

Problems Discrete Mathematics Combinatorics Geometry on grid paper

12–13 2.0

How many squares are shown in the picture?

Problem #PRU-104037

Problems Algebra and arithmetic Arithmetics. Mental maths Fun Problems Joke Problems

11–13 2.0

Gerard says: the day before yesterday I was 10 years old, and next year I will turn 13. Can this be?

Problem #PRU-104039

Problems Fun Problems Joke Problems

12–14 2.0

We call a natural number “amazing” if it has the form \(a^b + b^a\) (where \(a\) and \(b\) are natural numbers). For example, the number 57 is amazing, since \(57 = 2^5 + 5^2\). Is the number 2006 amazing?

Problem #PRU-104041

Problems Fun Problems Joke Problems

12–13 2.0

Homework. Cut a hole in an exercise book of a size so that you yourself can climb through it.