Question

A triangle can be formed having side lengths 4, 5 and 8. It is impossible, however, to construct a triangle with side lengths 4, 5 and 9. Ron has 8 sticks, each having an integer length. He observes that he cannot form a triangle using any three of these sticks as side lengths. The shortest possible lengths of the longest of the 8 sticks is

(A) 20
(B) 21
(C) 22
(D) 23
(E) 24

Answer 1

The sum of two sides of a triangle will always be greater than the third side. Not equal to not less than but greater than.

So we need 8 smallest integers such that no triangles cab be formed.
The lowest integer can be 1. Then next one can also be 1. If I choose the third one as 2 you will not be able to form a triangle because length of two sides 1+1 adds to 2 and that is not possible for a triangle. So we have 1, 1, 2.
The next can be 1+2. So keep adding the last two numbers to get the next number:
1, 1, 2, 3, 5, 8, 13, 21
That is 8 sticks that you cannot form a triangle with. The longest of the 8 sticks is 21.