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Common Points are Above Curve

Source: 1995 IrMO Paper 2 Problem 3

December 27, 2017
algebra

Problem Statement

Let SS be the square consisting of all pints (x,y)(x,y) in the plane with 0≤x,y≤10\le x,y\le 1. For each real number tt with 0<t<10<t<1, let CtC_t denote the set of all points (x,y)∈S(x,y)\in S such that (x,y)(x,y) is on or above the line joining (t,0)(t,0) to (0,1−t)(0,1-t). Prove that the points common to all CtC_t are those points in SS that are on or above the curve x+y=1\sqrt{x}+\sqrt{y}=1.
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