Triangle Theory
Let be a set of 5 distinct integers {a, b, c, d, e} such that any subset of size 3 derived from this set can be possible sides for a scalene oblique triangle, and that for any 3 elements chosen, the GCD is 1.
Let be the set of the areas of the triangles whose sides correspond to the elements of the subsets derived from set .
Determine the minimum value of the sum of the square of the elements of .
The answer is 3766.5.
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1 solution
By virtue of the triangle inequality, we have to make sure that the elements of the set are made small as possible. Since it is stated that they are integers, then the numbers are probably consecutive.
1 cannot be the smallest number in set S because it will not conform with the inequality. The same goes for 2 and 3 (i.e, {2, 3, 4, 5, 6} and {3, 4, 5, 6, 7}). So, the minimum possible values for S will be {4,5,6,7,8}. However, it is also stated that any three elements derived from this set must have a GCD of 1, which is clearly not the case when we select {4,6,8}. Thus, we reject this set and select the next one, {5,6,7,8,9}.
Now, we know that there are 5 C 3 = 1 0 possible subsets which have 3 elements. So, we know that set Q has 10 elements. By heron's formula, given by
A = s ( s − a ) ( s − b ) ( s − c )
where s represents half the triangle's perimeter.
we will be able to get the sum of squares of the respective areas of these triangles. We do this by squaring the formula above and add up the 10 values together.
In that case, the answer found is 3 7 6 6 . 5