MathDB

1

Part of 2013 Czech-Polish-Slovak Match

Problems(2)

ABCD is cyclic with BC=CD

Source: Czech - Polish - Slovak Match 2013: P1

5/24/2014
Suppose ABCDABCD is a cyclic quadrilateral with BC=CDBC = CD. Let ω\omega be the circle with center CC tangential to the side BDBD. Let II be the centre of the incircle of triangle ABDABD. Prove that the straight line passing through II, which is parallel to ABAB, touches the circle ω\omega.
geometrygeometric transformationhomothetyLaTeXcyclic quadrilateralgeometry unsolved
integer trinomial as a perfect square

Source: Czech-Polish-Slovak Match 2013 day 2 P1

9/29/2017
Let aa and bb be integers, where bb is not a perfect square. Prove that x2+ax+bx^2 + ax + b may be the square of an integer only for finite number of integer values of xx.
(Martin Panák)
Perfect Squaretrinomialnumber theoryInteger Polynomial