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Olympic toán toàn quốc - Việt nam 2003 sưu tầm từ internet | Toán học, Olympic toán toàn quốc - Việt nam 2003 Bài từ Tủ sách Khoa học VLOS A1. Let R be the reals and f: R ’! R a function such that f( cot x ) = cos 2x + sin 2x for all 0 6. B1. Find the largest positive integer n such that the following equations have integer solutions in x, y1, y2, . , yn: (x + 1)2 + y12 = (x + 2)2 + y22 = . = (x + n)2 + yn2. B2. Define p(x) = 4x3 - 2x2 - 15x + 9, q(x) = 12x3 + 6x2 - 7x + 1. Show that each polynomial has just three distinct real roots. Let A be the largest root of p(x) and B the largest root of q(x). Show that A2 + 3 B2 = 4. B3. Let R+ be the set of positive reals and let F be the set of all functions f : R+ ’! R+ such that f(3x) e" f( f(2x) ) + x for all x. Find the largest A such that f(x) e" A x for all f in F and all x in R+.
