MathDB
Number of ways to write number as linear combination of others

Source: IZHO 2023 P3

February 6, 2023
number theoryPolynomials

Problem Statement

Let a1,a2,⋯ ,aka_1, a_2, \cdots, a_k be natural numbers. Let S(n)S(n) be the number of solutions in nonnegative integers to a1x1+a2x2+⋯+akxk=na_1x_1 + a_2x_2 + \cdots + a_kx_k = n. Suppose S(n)≠0S(n) \neq 0 for all big enough nn. Show that for all sufficiently large nn, we have S(n+1)<2S(n)S(n+1) < 2S(n).
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