MathDB

Problem 2

Part of 2017 Vietnamese Southern Summer School contest

Problems(3)

Simple cyclic inequality with simple constraint

Source: Southern Summer School, gr. 10

7/9/2017
Let a,b,ca,b,c be the positive real numbers satisfying a2+b2+c2=3a^2+b^2+c^2=3. Prove that: ab(a+c)+bc(b+a)+ca(c+b)≥32.\frac{a}{b(a+c)}+\frac{b}{c(b+a)}+\frac{c}{a(c+b)}\geq \frac{3}{2}.
inequalitiescyclic inequalityFraction
Functional equations over the reals

Source: Southern Summer School, gr. 11

7/9/2017
Find all functions f:R↦Rf:\mathbb{R}\mapsto \mathbb{R} satisfy: f(x2)+f(xy)=f(x)f(y)+yf(x)+xf(x+y)f(x^2)+f(xy)=f(x)f(y)+yf(x)+xf(x+y) for all real numbers x,yx,y.
algebrafunctional equationfunction
Roots of three-degree polynomials

Source: Southern Summer School, gr. 12

7/9/2017
Let P,QP,Q be the polynomials: x3−4x2+39x−46,x3+3x2+4x−3,x^3-4x^2+39x-46, x^3+3x^2+4x-3, respectively. 1. Prove that each of P,QP, Q has an unique real root. Let them be α,β\alpha,\beta, respectively. 2. Prove that {α}>{β}2\{ \alpha\}>\{ \beta\} ^2, where {x}=x−⌊x⌋\{ x\}=x-\lfloor x\rfloor is the fractional part of xx.
algebrapolynomialrootsequation