MathDB

2

Part of 2004 Vietnam Team Selection Test

Problems(2)

function from VietNam

Source: VietNam TST 2004, problem 2

5/9/2004
Find all real values of α\alpha, for which there exists one and only one function f:R↦Rf: \mathbb{R} \mapsto \mathbb{R} and satisfying the equation f(x2+y+f(y))=(f(x))2+α⋅y f(x^2 + y + f(y)) = (f(x))^2 + \alpha \cdot y for all x,y∈Rx, y \in \mathbb{R}.
functioncalculusintegrationsearchalgebra unsolvedalgebra
convex hexagon

Source: VietNam TST 2004, problem 5

5/9/2004
Let us consider a convex hexagon ABCDEF. Let A1,B1,C1,D1,E1,F1A_1, B_1,C_1, D_1, E_1, F_1 be midpoints of the sides AB,BC,CD,DE,EF,FAAB, BC, CD, DE, EF,FA respectively. Denote by pp and p1p_1, respectively, the perimeter of the hexagon ABCDEF A B C D E F and hexagon A1B1C1D1E1F1 A_1B_1C_1D_1E_1F_1 . Suppose that all inner angles of hexagon A1B1C1D1E1F1 A_1B_1C_1D_1E_1F_1 are equal. Prove that p≥2⋅33⋅p1. p \geq \frac{2 \cdot \sqrt{3}}{3} \cdot p_1 . When does equality hold ?
geometryperimetertrigonometrygeometry unsolved