4

Part of 2009 239 Open Mathematical Olympiad

Problems(2)

Source: 239 2009 S4

Natural numbers

a

b

are given such that the number

P = \frac{[a, b]}{a + 1} + \frac{[a, b]}{b + 1}

Is a prime. Prove that

4P + 5

is the square of a natural number.

Source: 239 2009 J4

The natural numbers

x, y > 1

, are such that

x^2 + xy -y

is the square of a natural number. Prove that

x + y + 1

is a composite number.