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N-tuples irrationality

Source: 12th Dürer Math Competition ,First Round, Category E+, P4

August 18, 2020
irrationaln-tuplescombinatorics

Problem Statement

An nn-tuple (x1,x2,…,xn)(x_1, x_2,\dots, x_n) is called unearthly if q1x1+q2x2+⋯+qnxnq_1x_1 +q_2x_2 +\dots+q_nx_n is irrational for any non-negative rational coefficients q1,q2,…,qnq_1, q_2, \dots, q_n where qiq_i’s are not all zero. Prove that it is possible to select an unearthly nn-tuple from any 2n−12n-1 distinct irrational numbers.
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