MathDB

3

Part of 2010 South East Mathematical Olympiad

Problems(2)

2010 smo (3)

Source: 2010 China South East Mathematical Olympiad

7/18/2011
The incircle of triangle ABCABC touches BCBC at DD and ABAB at FF, intersects the line ADAD again at HH and the line CFCF again at KK. Prove that FD×HKFH×DK=3\frac{FD\times HK}{FH\times DK}=3
geometrygeometry unsolved
2010 SMO (7)

Source:

11/2/2010
Let nn be a positive integer. The real numbers a1,a2,⋯ ,ana_1,a_2,\cdots,a_n and r1,r2,⋯ ,rnr_1,r_2,\cdots,r_n are such that a1≤a2≤⋯≤ana_1\leq a_2\leq \cdots \leq a_n and 0≤r1≤r2≤⋯≤rn0\leq r_1\leq r_2\leq \cdots \leq r_n. Prove that ∑i=1n∑j=1naiajmin⁡(ri,rj)≥0\sum_{i=1}^n\sum_{j=1}^n a_i a_j \min (r_i,r_j)\geq 0
inequalities unsolvedinequalities