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a = ( p +\sqrt{p^2 + q} )^2 is irrational

Source: - 2001 Cuba MO 2.5

September 15, 2024
algebrairrational

Problem Statement

Let pp and qq be two positive integers such that 1≤q≤p1 \le q \le p. Also let a=(p+p2+q)2a = \left( p +\sqrt{p^2 + q} \right)^2. a) Prove that the number aa is irrational. b) Show that {a}>0.75\{a\} > 0.75.
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