Here is a problem of my own invention; I hope you enjoy it:
At a festival, every guest presented a gift to every older guest. The ten youngest guests combined gave thirteen times as many gifts as they received. What was the total number of gifts presented at the festival?
Suppose there are n guests in total. The youngest guest gives a gift to everyone else, that is n-1 people, but they receive none. The second youngest gives to n-2 people, and receives only 1 gift (from the youngest guest). Continuing in this fashion, we find that the total number of gifts presented by the ten youngest guests is
(n – 1) + (n – 2) + … + (n – 10) = 10n – 55,
and the number of gifts they receive is
1 + 2 + … + 9 = 45.
We are told that the first of these totals is thirteen times the second. So
10n – 55 = 13 x 45 = 585,
and we have that n = 64. Finally, the total number of gifts presented across the entire festival is
1 + 2 + … + 63.
Here we use the fact that the sum of the first k positive integers is k(k+1)/2, to obtain a final answer of
63 x 64 / 2 = 2016.