Find the last digit of the number \(1 \times 2 + 2 \times 3 + \dots + 999 \times 1000\).
Is the number 12345678926 square?
Prove there are no integer solutions for the equation \(4^k - 4^l = 10^n\).
There are 100 notes of two types: \(a\) and \(b\) pounds, and \(a \neq b \pmod {101}\). Prove that you can select several bills so that the amount received (in pounds) is divisible by 101.
If a salary is first increased by 20%, and then reduced by 20%, will the salary paid increase or decrease as a result?
In a room there are some chairs with 4 legs and some stools with 3 legs. When each chair and stool has one person sitting on it, then in the room there are a total of 39 legs. How many chairs and stools are there in the room?
Arrange in a row the numbers from 1 to 100 so that any two neighbouring ones differ by at least 50.
An \(8 \times 8\) square is painted in two colours. You can repaint any \(1 \times 3\) rectangle in its predominant colour. Prove that such operations can make the whole square monochrome.
12 teams played a volleyball tournament in one round. Two teams scored exactly 7 wins.
Prove that there are teams \(A\), \(B\), \(C\) where \(A\) won against \(B\), \(B\) won against \(C\), and \(C\) won against \(A\).
The numbers from 1 to 9999 are written out in a row. How can I remove 100 digits from this row so that the remaining number is a) maximal b) minimal?