MathDB

1

Part of 2016 Middle European Mathematical Olympiad

Problems(2)

inequality on n variables greater than -1

Source: MEMO 2016 I1

8/24/2016
Let n≥2n \ge 2 be an integer, and let x1,x2,…,xnx_1, x_2, \ldots, x_n be reals for which:
(a) xj>−1x_j > -1 for j=1,2,…,nj = 1, 2, \ldots, n and
(b) x1+x2+…+xn=n.x_1 + x_2 + \ldots + x_n = n.
Prove that ∑j=1n11+xj≥∑j=1nxj1+xj2 \sum_{j = 1}^{n} \frac{1}{1 + x_j} \ge \sum_{j = 1}^{n} \frac{x_j}{1 + x_j^2} and determine when does the equality occur.
inequalitiesinequalities proposed
cyclic system of equations in 3 variables

Source: MEMO 2016 T1

8/25/2016
Find all triples (a,b,c)(a, b, c) of real numbers such that a2+ab+c=0, a^2 + ab + c = 0, b2+bc+a=0,b^2 + bc + a = 0, c2+ca+b=0.c^2 + ca + b = 0.
system of equationsalgebraalgebra proposedquadratics