2006 USAMO Problems/Problem 2
Problem
For a given positive integer find, in terms of
, the minimum value of
for which there is a set of
distinct positive integers that has sum greater than
but every subset of size
has sum at most
.
Solution
Let one optimal set of integers be with
.
The two conditions can now be rewritten as and
.
Subtracting, we get that
, and hence
.
In words, the sum of the
smallest numbers must exceed the sum of the
largest ones.
Let . As all the numbers are distinct integers, we must have
, and also
.
Thus we get that , and
.
As we want the second sum to be larger, clearly we must have .
This simplifies to
.
Hence we get that:
On the other hand, for the set the sum of the largest
elements is exactly
, and the sum of the entire set is
, which is more than twice the sum of the largest set.
Hence the smallest possible is
.