Subcontests
(4)2 equal circles, 1 symmetric point, 1 altitude, perpendicularity wanted
Let ABC be an acute triangle with AB<AC<BC, inscribed in circle c(O,R) (with center O and radius R). Let O1 be the symmetric point of O wrt AC. Circle c1(O1,R) intersects BC at Z. If the extension of the altitude AD intersects the cicrumscribed circle c(O,R) at point E, prove that EC is perpendicular on AZ. integer solutions of x+y+z=2013 so that xyz becomes max
Numbers x,y,z are positive integers and satisfy the equation x+y+z=2013. (E)
a) Find the number of the triplets (x,y,z) that are solutions of the equation (E).
b) Find the number of the solutions of the equation (E) for which x=y.
c) Find the solution (x,y,z) of the equation (E) for which the product xyz becomes maximum. p^2+2q^2+334=[p^2,q^2] , where p,q coprime
Find all pairs of coprime positive integers (p,q) such that p2+2q2+334=[p2,q2] where [p2,q2] is the leact common multiple of p2,q2 .