MathDB

3

Part of 2021 Brazil National Olympiad

Problems(2)

Floor of powers of an irrational equals square minus constant

Source: Brazilian Mathematical Olympiad 2021, Level 3, Problem 3

2/7/2022
Find all positive integers kk for which there is an irrational α>1\alpha>1 and a positive integer NN such that ⌊αn⌋\left\lfloor\alpha^{n}\right\rfloor is a perfect square minus kk for every integer nn with n>Nn>N.
number theoryfloor functionpowersirrational numberPerfect SquaresBrazilian Math OlympiadBrazil
Geometry is not real

Source: Brazil National Olympiad Junior 2021 #3

2/7/2022
Let ABCABC be a scalene triangle and ω\omega is your incircle. The sides BC,CABC,CA and ABAB are tangents to ω\omega in X,Y,ZX,Y,Z respectively. Let MM be the midpoint of BCBC and DD is the intersection point of BCBC with the angle bisector of ∠BAC\angle BAC. Prove that ∠BAX=∠MAC\angle BAX=\angle MAC if and only if YZYZ passes by the midpoint of ADAD.
geometrycircumcircleangle bisectorprojective geometry