MathDB

2

Part of 2016 Azerbaijan BMO TST

Problems(3)

Balkan TSTp3.2

Source: Azerbaijan Balkan TST 2016 no 3

10/20/2016
İn triangle ABCABC the bisector of ∠BAC\angle BAC intersects the side BCBC at the point DD.The circle ω\omega passes through AA and tangent to the side BCBC at DD.ACAC and ω\omega intersects at MM second time , BMBM and ω\omega intersects at PP second time. Prove that point PP lies on median of triangle ABDABD.
geometry
No perfect square

Source:

2/20/2016
Set AA consists of natural numbers such that these numbers can be expressed as 2x2+3y2,2x^2+3y^2, where xx and yy are integers. (x2+y2≠0)(x^2+y^2\not=0) a)a) Prove that there is no perfect square in the set A.A. b)b) Prove that multiple of odd number of elements of the set AA cannot be a perfect square.
number theory
Balkan TSTp4.2

Source: Azerbaijan Balkan TST 2016 no 4

10/20/2016
There are 100100 students who praticipate at exam.Also there are 2525 members of jury.Each student is checked by one jury.Known that every student likes 1010 jury a)a) Prove that we can select 77 jury such that any student likes at least one jury. b)b) Prove that we can make this every student will be checked by the jury that he likes and every jury will check at most 1010 students.
combinatoricsProbabilistic MethodHall s marriage theorem