MathDB

3

Part of 2017 South East Mathematical Olympiad

Problems(2)

CSMO 2017 Grade 10 Problem 3

Source: Chinese Southeast Mathematical Olympiad

8/1/2017
For any positive integer nn, let DnD_n denote the set of all positive divisors of nn, and let fi(n)f_i(n) denote the size of the set Fi(n)={a∈Dn∣a≡i(mod4)}F_i(n) = \{a \in D_n | a \equiv i \pmod{4} \} where i=1,2i = 1, 2. Determine the smallest positive integer mm such that 2f1(m)−f2(m)=20172f_1(m) - f_2(m) = 2017.
modular arithmeticnumber theory
China South East Mathematical Olympiad 2017 Grade11 Q3

Source: China Jiangxi , Jul 30, 2017

7/30/2017
Let a1,a2,⋯ ,an+1>0a_1,a_2,\cdots,a_{n+1}>0. Prove that∑i−1nai∑i=1nai+1≥∑i=1naiai+1ai+ai+1⋅∑i=1n(ai+ai+1)\sum_{i-1}^{n}a_i\sum_{i=1}^{n}a_{i+1}\geq \sum_{i=1}^{n}\frac{a_i a_{i+1}}{a_i+a_{i+1}}\cdot \sum_{i=1}^{n}(a_i+a_{i+1})
inequalitiesalgebraChinaBPSQ