2005 AMC 10B Problem 25

Question

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?

50

51

62

65

68

Which integers can pair with another to sum to

125?

Separate the integers with no possible partner from the disjoint pairs whose members sum to

125

The pairs summing to

125

are

(25, 100), (26, 99), \ldots, (62, 63),

which is

62 - 25 + 1 = 38

pairs. From each pair,

B

may contain at most one element.

The numbers

1

through

24

cannot pair with anything in range to sum to

125,

so all

24

of them may be included.

Thus

B

has at most

38 + 24 = 62

elements, and the set

\{1, 2, \ldots, 62\}

achieves this.

Thus, C is the correct answer.

Problem 25 in other years