Prove that some integer is in at least 3 rows and at least 3 columns.. Find all possible values for angle B.[r]
(1)Tốn học, Đề thi tốn vơ địch giới, 2001
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A1 ABC is acute-angled O is its circumcenter X is the foot of the perpendicular from A to BC Angle C e" angle B + 30o Prove that angle A + angle COX < 90o
A2 a, b, c are positive reals Let a' = �"(a2 + 8bc), b' = �"(b2 + 8ca), c' = �"(c2 + 8ab) Prove that a/a' + b/b' + c/c' e"
A3 Integers are placed in each of the 441 cells of a 21 x 21 array Each row and each column has at most different integers in it Prove that some integer is in at least rows and at least columns
B1 Let n1, n2, , nm be integers where m is odd Let x = (x1, , xm) denote a permutation of the integers 1, 2, , m Let f(x) = x1n1 + x2n2 + + xmnm Show that for some distinct permutations a, b the difference f(a) - f(b) is a multiple of m!
B2 ABC is a triangle X lies on BC and AX bisects angle A Y lies on CA and BY bisects angle B Angle A is 60o AB + BX = AY + YB Find all possible values for angle B
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