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Minimize a function with absolute values

Source: 2022 China TST, Test 2, P4

March 29, 2022
algebrainequalitiesfunctionabsolute value

Problem Statement

Given a positive integer nn, find all nn-tuples of real number (x1,x2,…,xn)(x_1,x_2,\ldots,x_n) such that f(x1,x2,⋯ ,xn)=∑k1=02∑k2=02⋯∑kn=02∣k1x1+k2x2+⋯+knxn−1∣ f(x_1,x_2,\cdots,x_n)=\sum_{k_1=0}^{2} \sum_{k_2=0}^{2} \cdots \sum_{k_n=0}^{2} \big| k_1x_1+k_2x_2+\cdots+k_nx_n-1 \big| attains its minimum.
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