MathDB

3

Part of 1995 Irish Math Olympiad

Problems(2)

inequality

Source: Ireland 1995

7/1/2009
Points A,X,D A,X,D lie on a line in this order, point B B is on the plane such that ∠ABX>120∘ \angle ABX>120^{\circ}, and point C C is on the segment BX BX. Prove the inequality: 2AD \ge \sqrt{3} (AB\plus{}BC\plus{}CD).
inequalitiesgeometry proposedgeometry
Common Points are Above Curve

Source: 1995 IrMO Paper 2 Problem 3

12/27/2017
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.
algebra