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 The 15th United States of America Mathematics Olympiad 1986年第15届美国数学奥林匹克 Do there exist 14 consecutive positive integers each divisible by a prime less than 13 ? What about 21 consecutive positive integers each divisible by a prime less than 17 ? Five professors attended a lecture. Each fell asleep just twice. For each pair there was a moment when both were asleep. Show that there was a moment when three of them were asleep. What is the smallest n > 1 for which the average of the first n (non-zero) squares is a square ? A T-square allows you to construct a straight line through two points and a line perpendicular to a given line through a given point. Circles C and C' intersect at X and Y. XY is a diameter of C. P is a point on C' inside C. Using only a T-square, find points Q,R on C such that QR is perpendicular to XY and PQ is perpendicular to PR. A partition of n is an increasing sequence of integers with sum n. For example, the partitions of 5 are: 1, 1, 1, 1, 1; 1, 1, 1, 2; 1, 1, 3; 1, 4; 5; 1, 2, 2; and 2, 3. If p is a partition, f(p) = the number of 1s in p, and g(p) = the number of distinct integers in the partition. Show that ∑ f(p) = ∑ g(p), where the sum is taken over all partitions of n. 点击此处查看相关视频讲解 在方框内输入单词或词组
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