Subcontests
(16)Max and min value with constraint
Let p1,p2,...,pn be n≥2 fixed positive real numbers. Let x1,x2,...,xn be nonnegative real numbers such that x1p1+x2p2+...+xnpn=1.
Determine the (a) maximal; (b) minimal possible value of x12+x22+...+xn2. Given angles of triangle, prove third order segment equality
Let ABC be a triangle with angles ∠A=80∘,∠B=70∘,∠C=30∘. Let P be a point on the bisector of ∠BAC satisfying ∠BPC=130∘. Let PX,PY,PZ be the perpendiculars drawn from P to the sides BC,AC,AB, respectively.
Prove that the following equation with segment lengths is satisfied
AY3+BZ3+CX3=AZ3+BX3+CY3.