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Pvove sum less than 3Cy

Source: 2018 China Southeast MO Grade 11 P8

July 31, 2018
algebra

Problem Statement

Given a positive real C≥1C \geq 1 and a sequence a1,a2,a3,⋯a_1, a_2, a_3, \cdots satisfying for any positive integer n,n, an≥0a_n \geq 0 and for any real x≥1x \geq 1, ∣xlg⁡x−∑k=1[x][xk]ak∣≤Cx,\left|x\lg x-\sum_{k=1}^{[x]}\left[\frac{x}{k}\right]a_k \right| \leq Cx, where [x][x] is defined as the largest integer that does not exceed xx. Prove that for any real y≥1y \geq 1, ∑k=1[y]ak<3Cy.\sum_{k=1}^{[y]}a_k < 3Cy.
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