Try this beautiful problem from the Pre-RMO, 2017 based on Sides of Quadrilateral.
What is the number of triples (a,b,c) of positive integers such that (i) a<b<c<10 and (ii) a,b,c,10 form the sides of a quadrilateral?
Largest number of triples
Quadrilateral
Distance
Answer: is 73.
PRMO, 2017, Question 20
Geometry Vol I to IV by Hall and Stevens
a+b+c>10
(a+b+c) can be
a b c
1 2 8,9
1 3 7,8,9
1 4 6 ,7,8,9
1 5 6,7,8,9
1 6 7,8,9
1 7 8,9
1 8 9
2 3 6,7,8,9
2 4 5,6,7,8,9
2 5 6,7,8,9
2 6 7,8,9
2 7 8,9
2 8 9
3 4 5,6,7,8,9
3 5 6,7,8,9
3 6 7,8,9
3 7 8,9
3 8 9
4 5 6,7,8,9
4 6 7,8,9
4 7 8,9
4 8 9
5 6 7,8,9
5 7 8,9
5 8 9
6 7 8,9
6 8 9
7 8 9
Total 73 cases.
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