MathDB

4

Part of 2011 Irish Math Olympiad

Problems(2)

Four Points Form a Rectangle

Source: 2011 IrMO Paper 1 Problem 4

2/8/2018
The incircle C1\mathcal{C}_1 of triangle ABCABC touches the sides ABAB and ACAC at the points DD and EE, respectively. The incircle C2\mathcal{C}_2 of the triangle ADEADE touches the sides ABAB and ACAC at the points PP and QQ, and intersects the circle C1\mathcal{C}_1 at the points MM and nn. Prove that
(a) the center of the circle C2\mathcal{C}_2 lies on the circle C1\mathcal{C}_1.
(b) the four points M,N,P,QM,N,P,Q in appropriate order form a rectangle if and only if twice the radius of C1\mathcal{C}_1 is three times the radius of C2\mathcal{C}_2.
geometryrectangle
Inequalities

Source: 2011 IrMO Paper 2 Problem 4

2/8/2018
Suppose that x,yx,y and zz are positive numbers such that 1=2xyz+xy+yz+zx1=2xyz+xy+yz+zx Prove that (i) 34≤xy+yz+zx<1\frac{3}{4}\le xy+yz+zx<1 (ii) xyz≤18xyz\le \frac{1}{8} Using (i) or otherwise, deduce that x+y+z≥32x+y+z\ge \frac{3}{2} and derive the case of equality.
inequalities