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Toán học, Olympic toán toàn quốc - Việt nam 2000.. Bài từ Tủ sách Khoa học VLOS.[r]
(1)Toán học, Olympic toán toàn quốc - Việt nam 2000
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A1 Define a sequence of positive reals x0, x1, x2, by x0 = b, xn+1 = �"(c - �"(c + xn)) Find all values of c such that for all b in the interval (0, c), such a sequence exists and converges to a finite limit as n tends to infinity
A2 C and C' are circles centers O and O' respectively X and X' are points on C and C' respectively such that the lines OX and O'X' intersect M and M' are variable points on C and C' respectively, such that "XOM = "X'O'M' (both measured clockwise) Find the locus of the midpoint of MM' Let OM and O'M' meet at Q Show that the circumcircle of QMM' passes through a fixed point
A3 Let p(x) = x3 + 153x2 - 111x + 38 Show that p(n) is divisible by 32000 for at least nine positive integers n less than 32000 For how many such n is it divisible?
B1 Given an angle ±� such that 0 < ±� < À�, show that there is a unique real monic quadratic x2 + ax + b which is a factor of pn(x) = sin ±� xn - sin(n±�) x + sin(n±� -±�) for all n > 2 Show that there is no linear polynomial x + c which divides pn(x) for all n > 2
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