MathDB

3

Part of 2010 China National Olympiad

Problems(2)

China Mathematics Olympiads (National Round) 2010 Problem 3

Source:

11/28/2010
Given complex numbers a,b,ca,b,c, we have that ∣az2+bz+c∣≤1|az^2 + bz +c| \leq 1 holds true for any complex number z,∣z∣≤1z, |z| \leq 1. Find the maximum value of ∣bc∣|bc|.
inequalitiestrigonometrycalculusfunctioncomplex numbersalgebra unsolvedalgebra
b_i=ka_i

Source: Chinese Mathematical Olympiad 2010 Problem 6

1/26/2010
Suppose a1,a2,a3,b1,b2,b3a_1,a_2,a_3,b_1,b_2,b_3 are distinct positive integers such that (n \plus{} 1)a_1^n \plus{} na_2^n \plus{} (n \minus{} 1)a_3^n|(n \plus{} 1)b_1^n \plus{} nb_2^n \plus{} (n \minus{} 1)b_3^n holds for all positive integers nn. Prove that there exists k∈Nk\in N such that b_i \equal{} ka_i for i \equal{} 1,2,3.
algebrapolynomialnumber theory proposednumber theory