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max number so that 11 divides sums of products of naturals

Source: JBMO 2016 Shortlist N2

October 14, 2017
JBMOnumber theorydivides

Problem Statement

Find the maximum number of natural numbers x1,x2,...,xmx_1,x_2, ... , x_m satisfying the conditions: a) No xi−xj,1≤i<j≤mx_i - x_j , 1 \le i < j \le m is divisible by 1111, and b) The sum x2x3...xm+x1x3...xm+⋅⋅⋅+x1x2...xm−1x_2x_3 ...x_m + x_1x_3 ... x_m + \cdot \cdot \cdot + x_1x_2... x_{m-1} is divisible by 1111.
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