MathDB

1

Part of 2009 International Zhautykov Olympiad

Problems(2)

Pretty simple diophantine equation, ZIMO-09

Source: International Zhautykov Olympiad 2009, day 1, problem 1

1/17/2009
Find all pairs of integers (x,y) (x,y), such that x^2 \minus{} 2009y \plus{} 2y^2 \equal{} 0
quadraticsalgebranumber theory proposednumber theory
Four concyclic points on each of two parabola's, ZIMO - 09

Source: International Zhautykov Olympiad 2009, day 2, problem 4.

1/17/2009
On the plane, a Cartesian coordinate system is chosen. Given points A1,A2,A3,A4 A_1,A_2,A_3,A_4 on the parabola y \equal{} x^2, and points B1,B2,B3,B4 B_1,B_2,B_3,B_4 on the parabola y \equal{} 2009x^2. Points A1,A2,A3,A4 A_1,A_2,A_3,A_4 are concyclic, and points Ai A_i and Bi B_i have equal abscissas for each i \equal{} 1,2,3,4. Prove that points B1,B2,B3,B4 B_1,B_2,B_3,B_4 are also concyclic.
analytic geometryconicsparabolaalgebrapolynomialVietafunction