Problems – We Solve Problems
Filter Problems
Showing 1 to 20 of 23 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?

#### 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?

#### Counting in two ways , Partitions into pairs and groups bijections , Pigeonhole principle (other)

You are given 25 numbers. The sum of any 4 of these numbers is positive. Prove that the sum of all 25 numbers is also positive.

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

You are given 11 different natural numbers that are less than or equal to 20. Prove that it is always possible to choose two numbers where one is divisible by the other.

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

100 people are sitting around a round table. More than half of them are men. Prove that there are two males sitting opposite one another.

#### Division with remainder , Factoring polynomials , Partitions into pairs and groups bijections , Pigeonhole principle (other)

Prove that within a group of 52 whole numbers there will be two whose difference of squares is divisible by 100.

#### Counting in two ways , Partitions into pairs and groups bijections , Pigeonhole principle (other)

The total age of a group of 7 people is 332 years. Prove that it is possible to choose three members of this group so that the sum of their ages is no less than 142 years.

My Problem Set reset
No Problems selected