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-103946

Problems Algebra and arithmetic 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 $A B + A C + B C$ ?

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-103987

Problems Methods Pigeonhole principle Pigeonhole principle (other)

10–12 2.0

a) Prove that in any football team there are two players who were born on the same day of the week.

b) Prove that in the population of London, which is almost 9 million, there will be ten thousand people who celebrate their birthday on the same day.

Problem #PRU-104037

Problems Algebra and arithmetic Arithmetics. Mental maths Set theory and logic Joke problem

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-104046

Problems Combinatorics

10–13 2.0

A toddler has $25$ lego pieces in a box:

In how many ways are there to choose three pieces to play with?
In how many ways can he choose three pieces for the foundation, main walls and roof? Note that the order is important.

Problem #PRU-104060

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

11–13 2.0

Gabby is standing on a river bank. She has two clay jars: one – for 5 litres, and about the second Gabby remembers only that it holds either 3 or 4 litres. Help Gabby determine the capacity of the second jar. (Looking into the jar, you cannot figure out how much water is in it.)

Problem #PRU-104072

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

10–12 2.0

On Brennan’s birthday, the postman Daniel wants to find out how old Brennan is. Sarah says that Brennan is over 11 years old, and Matt claims that he is more than 10 years old. How old is Brennan, if it is known that exactly one of them was mistaken? Justify your answer.

Problem #PRU-104074

Problems Set theory and logic Set theory Inclusion-exclusion principle

11–13 2.0

In the garden of Sandra and Lewis 2006 rose bushes were growing. Lewis watered half of all the bushes, and Sandra watered half of all the bushes. At the same time, it turned out that exactly three bushes, the most beautiful, were watered by both Sandra and Lewis. How many rose bushes have not been watered?