We solve problems

Problems Methods Pigeonhole principle Pigeonhole principle (finite number of poits, lines etc.)

10 children, including Billy, attended Billy’s birthday party. It turns out that any two children picked from those at the party share a grandfather. Prove that 7 of the children share a grandfather.

Problem #PRU-115470

Problems Methods Pigeonhole principle Pigeonhole principle (other)

11–13 2.0

A class has 25 pupils. It is known that for any two girls in the class, the number of male friends they have in the class is different. What is the maximum number of girls that it is possible for there to be in the class?

Problem #PRU-115477

Problems Set theory and logic Mathematical logic Mathematical logic (other) Theory of algotithms Theory of algorithms (other)

11–13 2.0

In the king’s prison, there are five cells numbered from 1 to 5. In each cell, there is one prisoner. Kristen persuaded the king to conduct an experiment: on the wall of each cell she writes at one point a number and at midnight, each prisoner will go to the cell with the indicated number (if the number on the wall coincides with the cell number, the prisoner does not go anywhere). On the following night at midnight, the prisoners again must move from their cell to another cell according to the instructions on the wall, and they do this for five nights. If the location of prisoners in the cells for all six days (including the first) is never repeated, then Kristen will be given the title of Wisdom, and the prisoners will be released. Help Kristen write numbers in the cells.

Problem #PRU-115489

Problems Set theory and logic Mathematical logic Mathematical logic (other) Methods Algebraic methods Proof by exhaustion

10–12 2.0

One day, Claudia, Sofia and Freia noticed that they brought the same toy cars to kindergarten. Claudia has a car with a trailer, a small car and a green car without a trailer. Sofia has a car without a trailer and a small green one with a trailer, and Freia has a big car and a small blue car with a trailer. What kind of car (in terms of colour, size and availability of a trailer) did all of the girls bring to the kindergarten? Explain the answer.

Problem #PRU-115492

Problems Methods Algebraic methods Proof by exhaustion Set theory and logic Mathematical logic Puzzles

11–13 2.0

The $K U B$ is a cube. Prove that the ball, $C I R$ , is not a cube. ( $K U B$ and $C I R$ are three-digit numbers, where different letters denote different numbers).

Problem #PRU-115707

Problems Set theory and logic Mathematical logic Puzzles

11–13 2.0

Can I replace the letters with numbers in the puzzle $R E \times C T S + 1 = C E \times M S$ so that the correct equality is obtained (different letters need to be replaced by different numbers, and the same letters must correspond to the same digits)?

Problem #PRU-116087

Problems Geometry Plane geometry Triangles Similar triangles Circles Tangent lines and tangent circles Tangent lines to circles

15–16 4.0

Three circles are constructed on a triangle, with the medians of the triangle forming the diameters of the circles. It is known that each pair of circles intersects. Let $C_{1}$ be the point of intersection, further from the vertex $C$ , of the circles constructed from the medians $A M_{1}$ and $B M_{2}$ . Points $A_{1}$ and $B_{1}$ are defined similarly. Prove that the lines $A A_{1}$ , $B B_{1}$ and $C C_{1}$ intersect at the same point.

Problem #PRU-116145

Problems Algebra and arithmetic Algebraic inequalities and systems of inequalities Linear inequalites and systems of inequalities Set theory and logic Logic and set theory

12–14 2.0

Of the four inequalities $2 x > 70$ , $x < 100$ , $4 x > 25$ and $x > 5$ , two are true and two are false. Find the value of $x$ if it is known that it is an integer.

Problem #PRU-116208

Problems Set theory and logic Theory of algotithms Balance puzzles

11–13 2.0

At the vertices of the hexagon $A B C D E F$ (see Fig.) There were 6 identical balls: at $A$ – one with mass 1 g, at $B$ – 2 g, ..., at $F$ – 6 g. Callum changed the places of two balls in opposite vertices. A set of weighing scales with 2 plates is available, which let you know which plate contains the balls of greater mass. How, in one weighing, can it be determined which balls were rearranged?

Problem #PRU-116278

Problems Methods Examples and counterexamples. Constructive proofs Pigeonhole principle Pigeonhole principle (other)

13–14 4.0

Two ants crawled along their own closed route on a $7 \times 7$ board. Each ant crawled only on the sides of the cells of the board and visited each of the 64 vertices of the cells exactly once. What is the smallest possible number of cell edges, along which both the first and second ants crawled?