Problem:
Define an ordered quadruple of integers (a,b,c,d) to be interesting if 1≤ a<b<c<d≤10 and a+d>b+c. How many interesting ordered quadruples are there?
Solution:
For a given value of a,1≤a≤6, an ordered quadruple is interesting if and only if it has the form (a,a+i,a+j,a+k) with 0<i<j,i+j<k, and 4≤k≤9. The number of ordered triples (i,j,k) meeting these conditions for k=4,5,6,7,8, and 9 is 1,2,4,6,9, and 12, respectively, and the number of possible values of a is 10−k. Thus the number of interesting ordered quadruples is 6⋅1+5⋅2+4⋅4+3⋅6+2⋅9+1⋅12=80​.
The problems on this page are the property of the MAA's American Mathematics Competitions