We solve problems

Problems Set theory and logic Theory of algotithms Balance puzzles

At the vertices of the hexagon \(ABCDEF\) (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 \(7\times7\) 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?

Problem #PRU-116301

Problems Geometry Plane geometry Geometrical inequalities Area inequalities Methods Pigeonhole principle Pigeonhole principle (finite number of poits, lines etc.) Area

13–14 4.0

101 random points are chosen inside a unit square, including on the edges of the square, so that no three points lie on the same straight line. Prove that there exist some triangles with vertices on these points, whose area does not exceed 0.01.

Problem #PRU-116539

Problems Methods Algebraic methods Partitions into pairs and groups; bijections Pigeonhole principle Pigeonhole principle (other)

13–15 3.0

There are a thousand tickets with numbers 000, 001, ..., 999 and a hundred boxes with the numbers 00, 01, ..., 99. A ticket is allowed to be dropped into a box if the number of the box can be obtained from the ticket number by erasing one of the digits. Is it possible to arrange all of the tickets into 50 boxes?

Problem #PRU-116589

Problems Algebra and arithmetic Number theory. Divisibility Divisibility of a number. General properties Sequences Recurrent relations Linear recurrent relations Calculus Number sequences Number sequences (other)

13–15 3.0

The sequence of numbers \(a_1, a_2, \dots\) is given by the conditions \(a_1 = 1\), \(a_2 = 143\) and

for all \(n \geq 2\).

Prove that all members of the sequence are integers.

Problem #PRU-116661

Problems Algebra and arithmetic Word problems Solving problems by inequalities. Going through the cases. Sequences Progressions Arithmetic progression Methods Pigeonhole principle Pigeonhole principle (other) Proof by contradiction

10–12 3.0

Can 100 weights of masses 1, 2, 3, ..., 99, 100 be arranged into 10 piles of different masses so that the following condition is fulfilled: the heavier the pile, the fewer weights in it?

Problem #PRU-116842

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

11–12 2.0

Going to school, Michael found everything he needed under the pillow, under the sofa, on the table or under the table. The items he needed to find were a notebook, a cheat sheet, an mp3 player and sneakers. Under the table, he did not find a notebook or an mp3 player. His cheat sheet never lies on the floor. The mp3 player was neither on the table nor under the sofa. What was lying where, if there was only one object in each of the places?

Problem #PRU-116978

Problems Methods Pigeonhole principle Pigeonhole principle (other)

10–12 3.0

When cleaning her children’s room, a mother found \(9\) socks. In a group of any \(4\) of the socks at least two belonged to the same child. In a group of any \(5\) of the socks no more than \(3\) had the same owner. How many children are there in the room and how many socks belong to each child?

Problem #PRU-21970

Problems Methods Pigeonhole principle Pigeonhole principle (other)

10–12 2.0

A bag contains balls of two different colours – black and white. What is the minimum number of balls you need to remove, without looking, to guarantee that within the removed balls at least two are the same colour.

Problem #PRU-21971

Problems Methods Pigeonhole principle Pigeonhole principle (other)

11–12 2.0

A forest contains a million fir trees. It is known that any given tree has at most 600,000 needles. Prove that there will be two trees with the same number of needles.