AMC10 2005 B

AMC10 2005 B · Q25

AMC10 2005 B · Q25. It mainly tests Basic counting (rules of product/sum), Casework.

A subset B of the set of integers from 1 to 100, inclusive, has the property that no two elements of B sum to 125. What is the maximum possible number of elements in B?

{1,2,…,100}的一个子集B，具有B中无两个元素和为125的性质。B的最大可能元素个数是多少？

(A) 50 50

(B) 51 51

(D) 65 65

(E) 68 68

Answer

Correct choice: (C)

正确答案：（C）

Solution

(C) Several pairs of numbers from 1 to 100 sum to 125. These pairs are (25, 100), (26, 99), . . . , (62, 63). Set $B$ can have at most one number from each of these $62-25+1=38$ pairs. In addition, $B$ can contain all of the numbers $1, 2, . . . , 24$ since these cannot be paired with any of the available numbers to sum to 125. So $B$ has at most $38+24=62$ numbers. The set containing the first 62 positive integers, for example, is one of these maximum sets.

（C）从 1 到 100 的数中，有若干对数的和为 125。这些数对是 (25, 100)、(26, 99)、……、(62, 63)。集合 $B$ 在这些 $62-25+1=38$ 对数中，每一对最多只能取一个数。此外，$B$ 还可以包含所有的 $1, 2, \ldots, 24$，因为它们无法与任何可用的数配对使和为 125。因此，$B$ 最多有 $38+24=62$ 个数。例如，包含前 62 个正整数的集合就是这样的一个最大集合。

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