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 $B?$

$50$

$51$

$62$

$65$

$68$

Answer: C

Concepts:subsets pairing and grouping extremal argument

Difficulty rating: 1720

Solution:

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.

2005 AMC 10B Problem 25

Answer: C

Difficulty rating: 1720

Solution:

Problem 25 in Other Years