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Two numbers leaving same remainder

Source: P4 - RMO Maharashtra and Goa 2019

November 10, 2019
number theory

Problem Statement

Let a1,a2,⋯ ,a6,a7a_1,a_2,\cdots,a_6,a_7 be seven positive integers. Let SS be the set of all numbers of the form ai2+aj2a_i^2+a_j^2 where 1≤i<j≤71\leq i<j\leq 7. Prove that there exist two elements of SS which have the same remainder on dividing by 3636.
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