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x_1 + x^2_2+ x^3_3+...+ x^k_k= n

Source: 2022 Swedish Mathematical Competition p5

March 24, 2024
number theory

Problem Statement

Prove that for every pair of positive integers kk and nn, there exists integer x1x_1, x2x_2,......, xkx_k with 0≤xj≤2k−1⋅nk0 \le x_j \le 2^{k-1}\cdot \sqrt[k]{n} for j=1j = 1, 22, ......, kk, and such that x1+x22+x33+...+xkk=n.x_1 + x^2_2+ x^3_3+...+ x^k_k= n.
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