MathDB

5

Part of 2016 South East Mathematical Olympiad

Problems(2)

Easy function about divisor

Source: China South East Mathematical Olympiad 2016 Grade 10 Prob. 5

7/31/2016
Let nn is positive integer, DnD_n is a set of all positive divisor of nn and f(n)=∑d∈Dn11+df(n)=\sum_{d\in D_n}{\frac{1}{1+d}} Prove that for all positive integer mm, ∑i=1mf(i)<m\sum_{i=1}^{m}{f(i)} <m
functionnumber theory
constant $\alpha$

Source: 2016 China South East Mathematical Olympiad Grade 11 Problem 5

8/6/2016
Let a constant α\alpha as 0<α<10<\alpha<1, prove that: (1)(1) There exist a constant C(α)C(\alpha) which is only depend on α\alpha such that for every x≥0x\ge 0, ln⁡(1+x)≤C(α)xα\ln(1+x)\le C(\alpha)x^\alpha. (2)(2) For every two complex numbers z1,z2z_1,z_2, ∣ln⁡∣z1z2∣∣≤C(α)(∣z1−z2z2∣α+∣z2−z1z1∣α)|\ln|\frac{z_1}{z_2}||\le C(\alpha)\left(|\frac{z_1-z_2}{z_2}|^\alpha+|\frac{z_2-z_1}{z_1}|^\alpha\right).
complex numbersinequalitiesalgebranatural logarithmChina Southeast MO