62nd Ukrainian National Mathematical Olympiad, Third Round, First Tour

Question

There are n 3 segments, their lengths in centimeters are distinct positive integers. It's known that it's possible to form a nondegenerate triangle from any three of these n segments. Suppose that among these segments there are segments with lengths 5 cm and 12 cm. What's the largest value n can attain?

Accepted Answer

Reorder the segments by their lengths, so that

a_{1} < a_{2} < \dots < a_{n}

. Clearly, any three segments form a triangle if and only if the sum of the lengths of the smallest two segments is larger than the length of the longest segment. So, the smallest segment except from the given two can't have a length smaller than

8

, as in that case we wouldn't be able to form a triangle from segments

5

,

a \leq 7

and

12

, as

5 + 7 \leq 12

.

So,

a_{1} = 5

and

a_{2} \geq 8

. Clearly,

a_{2} \leq 12

and the number of segments can't exceed the number of elements of the set

{5, a_{2}, a_{2} + 1, a_{2} + 2, a_{2} + 3, a_{2} + 4}

, so

n \leq 6

.

Note that the set of segments with lengths

{5, 8, 9, 10, 11, 12}

satisfies the conditions. Therefore, the answer is

n = 6

.