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max {a_1, a_2,. . . , a_{12} } - min {a_1, a_2,. . . , a_{12}\} = 20

Source: 2004 Moldova JBMO TST p4

February 20, 2021
number theoryconsecutive

Problem Statement

Different non-zero natural numbers a1,a2,...,a12_1, a_2,. . . , a_{12} satisfy the condition: all positive differences other than two numbers aia_i and aja_j form many 2020 consecutive natural numbers. a) Show that max⁡{a1,a2,...,a12}−min⁡{a1,a2,...,a12}=20\max \{a_1, a_2,. . . , a_{12}\} - \min \{a_1, a_2,. . . , a_{12}\} = 20. b)Determine 1212 natural numbers with the property from the statement.
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