MathDB

4

Part of 2017 South East Mathematical Olympiad

Problems(2)

CSMO 2017 Grade 10 Problem 4

Source: CSMO 2017 Grade 10 Problem 4

8/6/2017
Let a1,a2,…,a2017a_1,a_2,\dots,a_{2017} be reals satisfied a1=a2017a_1=a_{2017}, ∣ai+ai+2−2ai+1∣≤1|a_i+a_{i+2}-2a_{i+1}|\le 1 for all i=1,2,…,2015i=1,2,\dots,2015. Find the maximum value of max⁡1≤i<j≤2017∣ai−aj∣\max_{1\le i<j\le 2017}|a_i-a_j|.
algebramaximum value
CSMO Grade 11 Problem 4

Source: China 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=0,1,2,3i = 0, 1, 2, 3. Determine the smallest positive integer mm such that f0(m)+f1(m)−f2(m)−f3(m)=2017f_0(m) + f_1(m) - f_2(m) - f_3(m) = 2017.
number theoryDivisorsCalculate