Problems

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 $a\ne b(\mathrm{mod}101)$. 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 $1+x+x^2+x^3=2^y$.

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.