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

m (moved 2010 UNC Math Contest II Problems/Problem 10 to 2010 UNCO Math Contest II Problems/Problem 10: disambiguation of University of Northern Colorado with University of North Carolina)
(Solution)
 
(One intermediate revision by the same user not shown)
Line 10: Line 10:
  
 
== Solution ==
 
== Solution ==
 +
<math>\frac{n+2}{2}</math> if <math>n</math> is even;  <math>\frac{n+1}{2}</math> if <math>n</math> is odd.
  
 
== See also ==
 
== See also ==
{{UNC Math Contest box|year=2010|n=II|num-b=9|num-a=11}}
+
{{UNCO Math Contest box|year=2010|n=II|num-b=9|num-a=11}}
  
 
[[Category:Intermediate Algebra Problems]]
 
[[Category:Intermediate Algebra Problems]]

Latest revision as of 02:57, 13 January 2019

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.

See also

2010 UNCO Math Contest II (ProblemsAnswer KeyResources)
Preceded by
Problem 9
Followed by
Problem 11
1 2 3 4 5 6 7 8 9 10
All UNCO Math Contest Problems and Solutions
Invalid username
Login to AoPS