MathDB

2

Part of 2008 International Zhautykov Olympiad

Problems(2)

Good polynomials

Source: Problem 2 from ZIMO 2008

1/21/2008
A polynomial P(x) P(x) with integer coefficients is called good,if it can be represented as a sum of cubes of several polynomials (in variable x x) with integer coefficients.For example,the polynomials x^3 \minus{} 1 and 9x^3 \minus{} 3x^2 \plus{} 3x \plus{} 7 \equal{} (x \minus{} 1)^3 \plus{} (2x)^3 \plus{} 2^3 are good. a)Is the polynomial P(x) \equal{} 3x \plus{} 3x^7 good? b)Is the polynomial P(x) \equal{} 3x \plus{} 3x^7 \plus{} 3x^{2008} good? Justify your answers.
algebrapolynomialfactorizationsum of cubesalgebra proposed
Nice,but easy

Source: Problem 5 from ZIMO 2008

1/21/2008
Let A1A2 A_1A_2 be the external tangent line to the nonintersecting cirlces ω1(O1) \omega_1(O_1) and ω2(O2) \omega_2(O_2),A1∈ω1 A_1\in\omega_1,A2∈ω2 A_2\in\omega_2.Points K K is the midpoint of A1A2 A_1A_2.And KB1 KB_1 and KB2 KB_2 are tangent lines to ω1 \omega_1 and ω2 \omega_2,respectvely(B1≠A1 B_1\neq A_1,B2≠A2 B_2\neq A_2).Lines A1B1 A_1B_1 and A2B2 A_2B_2 meet in point L L,and lines KL KL and O1O2 O_1O_2 meet in point P P. Prove that points B1,B2,P B_1,B_2,P and L L are concyclic.
symmetrygeometrycircumcircletrigonometrycyclic quadrilateralgeometry proposed