IMO 1969 LL NET48

Let x1, x2, x3, x4, and x5 be positive integers satisfying

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IMO 1969 LL NET48

Origin: NET

Problem

Let x1, x2, x3, x4, and x5 be positive integers satisfying x1 +x2 +x3 +x4 +x5 = 1000, x1 −x2 +x3 −x4 +x5 > 0, x1 +x2 −x3 +x4 −x5 > 0, −x1 +x2 +x3 −x4 +x5 > 0, x1 −x2 +x3 +x4 −x5 > 0, −x1 +x2 −x3 +x4 +x5 > 0. (a) Find the maximum of (x1 + x3)x2+x4. (b) In how many different ways can we choose x1, . . . , x5 to obtain the desired maximum?