Subcontests
(4)intersection of two lines is the circumcenter of a triangle
Let ABC be an acute triangle with AB<AC<BC,c it's circumscribed circle and D,E be the midpoints of AB,AC respectively. With diameters the sides AB,AC, we draw semicircles, outer of the triangle, which are intersected by line D at points M and N respectively. Lines MB and NC intersect the circumscribed circle at points T,S respectively. Lines MB and NC intersect at point H. Prove that:
a) point H lies on the circumcircle of triangle AMN
b) lines AH and TS are perpedicular and their intersection, let it be Z, is the circimcenter of triangle AMN largest possible value of sum of third powers in a tennis tournament
12 friends play a tennis tournament, where each plays only one game with any of the other eleven. Winner gets one points. Loser getos zero points, and there is no draw. Final points of the participants are B1,B2,...,B12. Find the largest possible value of the sum Σ3=B13+B23+...+B123 . if a^2+b^2+c^2+d^2=4 and a,b,c,d > 0 prove 2 of a,b,c,d have sum <=2
Let a,b,c,d be positive real numbers such that a2+b2+c2+d2=4.
Prove that exist two of a,b,c,d with sum less or equal to 2.