## 1310.61 – Never 100

Find a set of numbers, all less than 100, such that none of them add up to 100. For example, here is such a set: $\{93, 4, 2\}$. All these numbers are less than 100 and no combination of them sums to 100. Find the largest such set.

## Solution

There is a trick here. Make all the sums too big, i.e., bigger than 100. The answer is $\{50, 51, 52, . . . , 99\}.$ This set has 50 numbers in it. Any set with more than 50 numbers will have to include two numbers that add up to 100.