Subcontests
(2)Easy for a #6 -- Prove AP and AQ are isogonal
Let P be a point inside triangle ABC, and suppose lines AP, BP, CP meet the circumcircle again at T, S, R (here T=A, S=B, R=C). Let U be any point in the interior of PT. A line through U parallel to AB meets CR at W, and the line through U parallel to AC meets BS again at V. Finally, the line through B parallel to CP and the line through C parallel to BP intersect at point Q. Given that RS and VW are parallel, prove that ∠CAP=∠BAQ. n-variable product of kth powers [Taiwan 2014 Quizzes]
Let ai>0 for i=1,2,…,n and suppose a1+a2+⋯+an=1. Prove that for any positive integer k,
(a1k+a1k1)(a2k+a2k1)…(ank+ank1)≥(nk+nk1)n.