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# of Triples Satisfying Condition

Source: 2013 AIME I Problem 15

March 15, 2013
modular arithmeticAMCAIMEAIME Iarithmetic sequence

Problem Statement

Let NN be the number of ordered triples (A,B,C)(A,B,C) of integers satisfying the conditions
(a) 0≤A<B<C≤990\leq A<B<C\leq99, (b) there exist integers aa, bb, and cc, and prime pp where 0≤b<a<c<p0\leq b < a < c < p, (c) pp divides A−aA-a, B−bB-b, and C−cC-c, and (d) each ordered triple (A,B,C)(A,B,C) and each ordered triple (b,a,c)(b,a,c) form arithmetic sequences.
Find NN.
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