Difference between revisions of "2010 UNCO Math Contest II Problems/Problem 10"

Problem

Let $S=\left\{1,2,3,\cdots ,n\right\}$ where $n \ge 4$. What is the maximum number of elements in a subset $A$ of $S$, which has at least three elements, such that $a+b>c$ for all $a, b, c$ in $A$? As an example, the subset $A=\left \{2,3,4\right \}$ of $S=\left\{1,2,3,4,5 \right\}$ has the property that the sum of any two elements is strictly bigger than the third element, but the subset $\left \{2,3,4,5 \right \}$ does not since $2+3$ is $\underline{not}$ greater than $5$. Since there is no subset of size $4$ satisfying these conditions, the answer for $n=5$ is $3$.

Solution

$\frac{n+2}{2}$ if $n$ is even;  $\frac{n+1}{2}$ if $n$ is odd.