Combinatorics AIME Difficult
2013

Problem - 194

Let $N$ be the number of ordered triples $(A,B,C)$ of integers satisfying the conditions:

• $0\le A < B < C \le 99$,
• there exist integers $a$, $b$, and $c$, and prime $p$ where $0\le b < a < c < p$,
• $p$ divides $(A-a)$, $(B-b)$, and $(C-c)$, and
• each ordered triple $(A,B,C)$ and each ordered triple $(b,a,c)$ form arithmetic sequences.

Find $N$.