Find the last digit of the number \(1 \times 2 + 2 \times 3 + \dots + 999 \times 1000\).
Is the number 12345678926 square?
Solve the equation in integers \(2x + 5y = xy - 1\).
Prove there are no integer solutions for the equation \(x^2 + 1990 = y^2\).
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.
Recall that a natural number \(x\) is called prime if \(x\) has no divisors except \(1\) and itself. Solve the equation with prime numbers \(pqr = 7(p + q + r)\).
Solve the equation with natural numbers \(1 + x + x^2 + x^3 = 2y\).
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.