MathDB

1

Part of 2013 International Zhautykov Olympiad

Problems(2)

Prove that the line O1O2 passes through the point N

Source: International Zhautykov Olympiad 2013 - D1 - P1

1/17/2013
Given a trapezoid ABCDABCD (AD∥BCAD \parallel BC) with ∠ABC>90∘\angle ABC > 90^\circ . Point MM is chosen on the lateral side ABAB. Let O1O_1 and O2O_2 be the circumcenters of the triangles MADMAD and MBCMBC, respectively. The circumcircles of the triangles MO1DMO_1D and MO2CMO_2C meet again at the point NN. Prove that the line O1O2O_1O_2 passes through the point NN.
geometrytrapezoidcircumcirclegeometric transformationgeometry unsolved
The equation p(x) = 1/n has no rational roots

Source: International Zhautykov Olympiad 2013 - D2 - P1

1/17/2013
A quadratic trinomial p(x)p(x) with real coefficients is given. Prove that there is a positive integer nn such that the equation p(x)=1np(x) = \frac{1}{n} has no rational roots.
quadraticsnumber theoryrelatively primealgebra unsolvedalgebra