MathDB

P4

Part of 2023 4th Memorial "Aleksandar Blazhevski-Cane"

Problems(2)

A familiar segment sum condition in a cyclic quadrilateral

Source: 4th Memorial Mathematical Competition "Aleksandar Blazhevski - Cane"- Junior D2 P5/ Senior D2 P4

12/21/2022
Let ABCDABCD be a cyclic quadrilateral such that AB=AD+BCAB = AD + BC and CD<ABCD < AB. The diagonals ACAC and BDBD intersect at PP, while the lines ADAD and BCBC intersect at QQ. The angle bisector of ∠APB\angle APB meets ABAB at TT. Show that the circumcenter of the triangle CTDCTD lies on the circumcircle of the triangle CQDCQD.
Proposed by Nikola Velov
geometrycyclic quadrilateralsegment sumangle bisectorCircumcentercircumcircle
Hidden elliptic curve in a junior level problem

Source: 4th Memorial Mathematical Competition &quot;Aleksandar Blazhevski - Cane&quot;- Junior D2 P4

12/21/2022
Does the equation z(y−x)(x+y)=x3z(y-x)(x+y)=x^3 have finitely many solutions in the set of positive integers?
Proposed by Nikola Velov
number theoryJuniorconstruction