We solve problems

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

Find the smallest four-digit number \(CEEM\) for which there exists a solution to the rebus \(MN + PORG = CEEM\). (The same letters correspond to the same numbers, different – different.)

Problem #PRU-103883

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

11–12 2.0

A space traveller decided to visit the planet and upon arrival he met three inhabitants. The traveller asked each of the inhabitants: “How many truth tellers are there among your companions?”. The first one answered: “None.” The second one said: “One”. What did the third alien say?

Problem #PRU-103884

Problems Combinatorics Dissections, partitions, covers and tilings Dissections with certain properties Methods Examples and counterexamples. Constructive proofs Algebra Fractions Simple fractions

11–12 2.0

A rectangle is cut into several smaller rectangles, the perimeter of each of which is an integer number of meters. Is it true that the perimeter of the original rectangle is also an integer number of meters?

Problem #PRU-103885

Problems Combinatorics Dissections, partitions, covers and tilings Tilings with ordinary and domino tiles

11–12 2.0

At the disposal of a tile layer there are 10 identical tiles, each of which consists of 4 squares and has the shape of the letter L (all tiles are oriented the same way). Can he make a rectangle with a size of \(5 \times 8\)? (The tiles can be rotated, but you cannot turn them over). For example, the figure shows the wrong solution: the shaded tile is incorrectly oriented.

Problem #PRU-103888

Problems Methods Algebraic methods Proof by exhaustion Geometry Solid geometry Visual geometry in space

11–12 2.0

A square napkin was folded in half, the resulting rectangle was then folded in half again (see the figure). The resulting square was then cut with scissors (in a straight line). Could the napkin have been broken up a) into 2 parts? b) into 3 parts? c) into 4 parts? d) into 5 parts? If yes – illustrate such a cut, if not – write the word “no”.

Problem #PRU-103946

Problems Algebra Arithmetic operations. Number identities

11–12 2.0

The number \(A\) is positive, \(B\) is negative, and \(C\) is zero. What is the sign of the number \(AB + AC + BC\)?

Problem #PRU-103947

Problems Geometry Plane geometry Area

10–12 2.0

Find the area of the figures shown below.

Problem #PRU-103960

Problems Methods Pigeonhole principle Pigeonhole principle (other)

11–13 2.0

Prove that if 21 people collected 200 nuts between them, there are two people in the group who collected the same number of nuts.

Problem #PRU-103967

Problems Set theory and logic Algorithm Theory Algorithm Theory

10–12 2.0

There is a rectangular table. Two players start in turn to place on it one pound coin each, so that these coins do not overlap one another. The player who cannot make a move loses. Who will win with the correct strategy?

Problem #PRU-103969

Problems Set theory and logic Algorithm Theory Algorithm Theory

10–12 2.0

a) Two players play in the following game: on the table there are 7 two pound coins and 7 one pound coins. In a turn it is allowed to take coins worth no more than three pounds. The one who takes the last coin wins. Who will win with the correct strategy?

b) The same question, if there are 12 one pound and 12 two pound coins.