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Maximal value of least common multiples

Source: Bosnia and Herzegovina Team Selection Test 2005 problem 3 day 1

October 6, 2017
least common multiplemaximizationnumber theory

Problem Statement

Let nn be a positive integer such that n≥2n \geq 2. Let x1,x2,...,xnx_1, x_2,..., x_n be nn distinct positive integers and SiS_i sum of all numbers between them except xix_i for i=1,2,...,ni=1,2,...,n. Let f(x1,x2,...,xn)=GCD(x1,S1)+GCD(x2,S2)+...+GCD(xn,Sn)x1+x2+...+xn.f(x_1,x_2,...,x_n)=\frac{GCD(x_1,S_1)+GCD(x_2,S_2)+...+GCD(x_n,S_n)}{x_1+x_2+...+x_n}. Determine maximal value of f(x1,x2,...,xn)f(x_1,x_2,...,x_n), while (x1,x2,...,xn)(x_1,x_2,...,x_n) is an element of set which consists from all nn-tuples of distinct positive integers.
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