IMO 1972 LL USS46

Numbers 1, 2, . . . , 16 are written in a 4 imes4 square matrix so that

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IMO 1972 LL USS46

Origin: USS

Problem

Numbers 1, 2, . . . , 16 are written in a 4\times4 square matrix so that the sum of the numbers in every row, every column, and every diagonal is the same and furthermore that the numbers 1 and 16 lie in opposite corners. Prove that the sum of any two numbers symmetric with respect to the center of the square equals 17.