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Find number of 9-tuples

Source: CWMI 2017 Q5

August 14, 2017
algebracombinatorics

Problem Statement

Let a1,a2,⋯ ,a9a_1,a_2,\cdots ,a_9 be 99 positive integers (not necessarily distinct) satisfying: for all 1≤i<j<k≤91\le i<j<k\le 9, there exists l(1≤l≤9)l (1\le l\le 9) distinct from i,ji,j and jj such that ai+aj+ak+al=100a_i+a_j+a_k+a_l=100. Find the number of 99-tuples (a1,a2,⋯ ,a9)(a_1,a_2,\cdots ,a_9) satisfying the above conditions.
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