MathDB

3

Part of 2011 Vietnam National Olympiad

Problems(2)

Concurrency problem and calculation - [VMO 2011]

Source:

1/11/2011
Let ABAB be a diameter of a circle (O)(O) and let PP be any point on the tangent drawn at BB to (O).(O). Define AP∩(O)=C≠A,AP\cap (O)=C\neq A, and let DD be the point diametrically opposite to C.C. If DPDP meets (O)(O) second time in E,E, then,
(i) Prove that AE,BC,POAE, BC, PO concur at M.M.
(ii) If RR is the radius of (O),(O), find PP such that the area of △AMB\triangle AMB is maximum, and calculate the area in terms of R.R.
geometrytrigonometryprojective geometrygeometry proposed
Prove that P(x,y) cannot be factorised -[VMO 2011]

Source:

1/12/2011
Let n∈Nn\in\mathbb N and define P(x,y)=xn+xy+yn.P(x,y)=x^n+xy+y^n. Show that we cannot obtain two non-constant polynomials G(x,y)G(x,y) and H(x,y)H(x,y) with real coefficients such that P(x,y)=G(x,y)⋅H(x,y).P(x,y)=G(x,y)\cdot H(x,y).
algebrapolynomialalgebra proposed