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2010 UNCO Math Contest II Problems/Problem 10

Revision as of 15:20, 19 October 2014 by Timneh (talk | contribs) (Created page with "== Problem == Let <math>S=\left\{1,2,3,\cdots ,n\right\}</math> where <math>n \ge 4</math>. What is the maximum number of elements in a subset <math>A</math> of <math>S</math>,...")
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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

See also

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