MathDB

2

Part of 2016 South East Mathematical Olympiad

Problems(2)

Hard geometry

Source: China South East Mathematical Olympiad 2016 Grade 10 Prob.2

7/30/2016
Suppose PABPAB and PCDPCD are two secants of circle OO. Lines AD∩BC=QAD \cap BC=Q. Point TT lie on segment BQBQ and point KK is intersection of segment PTPT with circle OO, S=QK∩PAS=QK\cap PA Given that ST∥PQST \parallel PQ, prove that B,S,K,TB,S,K,T lie on a circle.
geometry
China South East Mathematical Olympiad 2016 Grade11 Q2

Source: China Nanchang

7/30/2016
Let nn be positive integer,x1,x2,⋯ ,xnx_1,x_2,\cdots,x_n be positive real numbers such that x1x2⋯xn=1x_1x_2\cdots x_n=1 . Prove that∑i=1nxix12+x22+⋯xi2≥n+12n\sum\limits_{i = 1}^{n}x_i\sqrt{x^2_1+x^2_2+\cdots x^2_i}\ge\frac{n+1}{2}\sqrt{n}
inequalities