There are n sticks which have distinct integer length. Suppose that it's possible to form a non-degenerate triangle from any 3 distinct sticks among them. It's also known that there are sticks of lengths 5 and 12 among them. What is the largest possible value of n under such conditions?
By the triangle inequality, 5 must be the shortest length. In fact, the next shortest must be at least 8. Filling in gives the set {5,8,9,10,11,12} but you can't use 13. So this set has n=6.
If the second shortest length is 9, you can go up to 13, etc. you get get following maximal sets:
{5,9,10,11,12,13}
{5,10,11,12,13,14}
{5,11,12,13,14,15}
{5,12,13,14,15,16}
Basically, by setting the smallest set at 5, the other sticks must have a span of no more than 4. Having a 12 in the set limits the possible sets but not the largest n.
In general if a is the smallest stick then maximum n=a+1.
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Posted by Jer
on 2023-07-11 09:45:55 |
solution
