MathDB

3

Part of 2010 Irish Math Olympiad

Problems(2)

Inequality

Source: 2010 IrMO Paper 1 Problem 3

1/30/2018
Suppose x,y,zx,y,z are positive numbers such that x+y+z=1x+y+z=1. Prove that (a) xy+yz+xz≥9xyzxy+yz+xz\ge 9xyz; (b) xy+yz+xz<14+3xyzxy+yz+xz<\frac{1}{4}+3xyz;
inequalities
Obtuse Triangle with Altitude

Source: 2010 IrMO Paper 2 Problem 3

1/30/2018
In triangle ABCABC we have ∣AB∣=1|AB|=1 and ∠ABC=120∘.\angle ABC=120^\circ. The perpendicular line to ABAB at BB meets ACAC at DD such that ∣DC∣=1|DC|=1. Find the length of ADAD.
geometry