MathDB

Inequality for real numbers

Source: Romanian District Olympiad 2014, Grade 7, P1

June 15, 2014

inequalitiesinequalities proposed

Problem Statement

[*]Prove that for any real numbers

a

and

b

the following inequality holds:

\left( a^{2}+1\right) \left( b^{2}+1\right) +50\geq2\left( 2a+1\right)\left( 3b+1\right)

[*]Find all positive integers

n

and

p

such that:

\left( n^{2}+1\right) \left( p^{2}+1\right) +45=2\left( 2n+1\right)\left( 3p+1\right)