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compare \sqrt{3}+\sqrt{10+2\sqrt{3}} with \sqrt{5+\sqrt{22}}+\sqrt{8-\sqrt{22}}}

Source: Spanish Mathematical Olympiad 1990 P1

August 2, 2018
algebraradical

Problem Statement

Prove that x+y+xy\sqrt{x}+\sqrt{y}+\sqrt{xy} is equal tox+y+xy+2yx \sqrt{x}+\sqrt{y+xy+2y\sqrt{x}}
and compare the numbers 3+10+23\sqrt{3}+\sqrt{10+2\sqrt{3}} and 5+22+8−22+215−322\sqrt{5+\sqrt{22}}+\sqrt{8- \sqrt{22}+2\sqrt{15-3\sqrt{22}}}.
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