A student did not notice the multiplication sign between two three-digit numbers and wrote one six-digit number, which turned out to be exactly seven times their product. Determine these numbers.
The student did not notice the multiplication sign between two seven-digit numbers and wrote one fourteen-digit number, which turned out to be three times bigger than their product. Determine these numbers.
Solve the equation \((x + 1)^3 = x^3\).
Solve the equation \(3x + 5y = 7\) in integers. Make sure that you’ve found all integer solutions.
Determine all the integer solutions for the equation \(21x + 48y = 6\).
Solve the equations \(x^2 = 14 + y^2\) in integers.
Determine all solutions of the equation \((n + 2)! - (n + 1)! - n! = n^2 + n^4\) in natural numbers.
Find all functions \(f (x)\) defined for all real values of \(x\) and satisfying the equation \(2f (x) + f (1 - x) = x^2\).
In a group of friends, each two people have exactly five common acquaintances. Prove that the number of pairs of friends is divisible by 3.
How many rational terms are contained in the expansion of
a) \((\sqrt 2 + \sqrt[4]{3})^{100}\);
b) \((\sqrt 2 + \sqrt[3]{3})^{300}\)?