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 The 17th United States of America Mathematics Olympiad 1988年第17届美国数学奥林匹克 The repeating decimal 0.ab ... k pq ... u = m/n, where m and n are relatively prime integers, and there is at least one decimal before the repeating part. Show that n is divisible by 2 or 5 (or both). [For example, 0.01136 = 0.01136363636 ... = 1/88 and 88 is divisible by 2.] The cubic x3 + ax2 + bx + c has real coefficients and three real roots r ≥ s ≥ t. Show that k = a2 - 3b ≥ 0 and that √k <= r - t. Let X be the set {1, 2, ... , 20} and let P be the set of all 9-element subsets of X. Show that for any map f: P → X we can find a 10-element subset Y of X, such that f(Y - {k}) ≠ k for any k in Y. ABC is a triangle with incenter I. Show that the circumcenters of IAB, IBC, ICA lie on a circle whose center is the circumcenter of ABC. Let p(x) be the polynomial (1 - x)a (1 - x2)b (1 - x3)c ... (1 - x32)k, where a, b, ..., k are integers. When expanded in powers of x, the coefficient of x1 is -2 and the coefficients of x2, x3, ... , x32 are all zero. Find k. 点击此处查看相关视频讲解 在方框内输入单词或词组