We solve problems

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

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-103899

Problems Combinatorics Discrete geometry

12–12 2.0

A rabbit, preparing for the arrival of guests, hung lightbulbs in three corners of his polygonal hole. Winnie the Pooh and Piglet came and noticed that lights did not illuminate all the pots of honey which were in the rabbit hole. When they reached for some honey, two of the lightbulbs broke. The rabbit moved the remaining light bulb into some corner so that the whole hole was lit up. Is this possible? (If yes, draw an example, if not, justify the answer.)

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-103955

Problems Set theory and logic Algorithm Theory Game Theory Winning and loosing positions

12–13 2.0

Ben and Joe play chess. In addition to a chessboard, they have one rook, which they put in the lower right corner, and they move it in turns. It can only be moved upwards or to the left (for any number of cells). The player who can not make a move, loses. Joe goes first. Who will win with the correct method?

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-103964

Problems Algebra 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-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.