Problems – We Solve Problems
Filter Problems
Showing 1 to 20 of 29 entries

#### Extremal principle (other) , Partitions into pairs and groups bijections , Pigeonhole principle (other)

Ben noticed that all 25 of his classmates have a different number of friends in this class. How many friends does Ben have?

#### Partitions into pairs and groups bijections , Pigeonhole principle (other) , The fundamental theorm of arithmetic. Prime factorisation.

The product of a group of 48 natural numbers has exactly 10 prime factors. Prove that the product of some four of the numbers in the group will always give a square number.

#### Partitions into pairs and groups bijections , Pigeonhole principle (other) , The fundamental theorm of arithmetic. Prime factorisation.

The product of 1986 natural numbers has exactly 1985 different prime factors. Prove that either one of these natural numbers, or the product of several of them, is the square of a natural number.

#### Arithmetic of remainders , Partitions into pairs and groups bijections , Pigeonhole principle (other)

The sum of 100 natural numbers, each of which is no greater than 100, is equal to 200. Prove that it is possible to pick some of these numbers so that their sum is equal to 100.

#### Decimal number system , Partitions into pairs and groups bijections , Pigeonhole principle (other)

Prove that amongst any 11 different decimal fractions of infinite length, there will be two whose digits in the same column – 10ths, 100s, 1000s, etc – coincide $($are the same$)$ an infinite number of times.

#### Decimal number system , Partitions into pairs and groups bijections , Pigeonhole principle (other)

We are given 51 two-digit numbers – we will count one-digit numbers as two-digit numbers with a leading 0. Prove that it is possible to choose 6 of these so that no two of them have the same digit in the same column.

#### Divisibility of a number. General properties , Partitions into pairs and groups bijections , Pigeonhole principle (other)

From the set of numbers 1 to 2n, n + 1 numbers are chosen. Prove that among the chosen numbers there are two, one of which is divisible by another.

#### Partitions into pairs and groups bijections , Pigeonhole principle (other)

26 numbers are chosen from the numbers $1, 2, 3,…, 49, 50$. Will there always be two numbers chosen whose difference is 1?

#### Partitions into pairs and groups bijections , Pigeonhole principle (finite number of poits, lines etc.) , Regular polygons

2001 vertices of a regular 5000-gon are painted. Prove that there are three coloured vertices lying on the vertices of an isosceles triangle.

#### Area inequalities , Partitions into pairs and groups bijections , Pigeonhole principle (area and volume)

In a square which has sides of length 1 there are 100 figures, the total area of which sums to more than 99. Prove that in the square there is a point which belongs to all of these figures.

#### Discrete distribution , Partitions into pairs and groups bijections

Two people toss a coin: one tosses it 10 times, the other – 11 times.
What is the probability that the second person’s coin showed heads more times than the first?

#### Decimal number system , Partitions into pairs and groups bijections , Pigeonhole principle (other)

Is it possible to find 57 different two digit numbers, such that no sum of any two of them was equal to 100?

My Problem Set reset
No Problems selected