MathDB

1

Part of 2023 Germany Team Selection Test

Problems(5)

Polynomial values of positive squares

Source: German TST 2023 AIMO 2, Problem 1

11/2/2023
Let PP be a polynomial with integer coefficients. Assume that there exists a positive integer nn with P(n2)=2022P(n^2)=2022. Prove that there cannot be a positive rational number rr with P(r2)=2024P(r^2)=2024.
algebrapolynomial
Tangents, a perpendicular bisector and a parallel line

Source: German TSTST (VAIMO) 2022 P4

7/15/2023
Let ABCABC be an acute triangle and let ω\omega be its circumcircle. Let the tangents to ω\omega through B,CB,C meet each other at point PP. Prove that the perpendicular bisector of ABAB and the parallel to ABAB through PP meet at line ACAC.
geometrycircumcircleperpendicular bisector
Primes with difference 2 dividing 2^n-1

Source: German TST 2023 AIMO 3, Problem 1

11/2/2023
Does there exist a positive odd integer nn so that there are primes p1p_1, p2p_2 dividing 2n−12^n-1 with p1−p2=2p_1-p_2=2?
power of 2number theoryprimes
Geometry contest 2014 Problem 5

Source:

12/17/2014
In a triangle △ABC\triangle ABC with orthocenter HH, let BHBH and CHCH intersect ACAC and ABAB at EE and FF, respectively. If the tangent line to the circumcircle of △ABC\triangle ABC passing through AA intersects BCBC at PP, MM is the midpoint of AHAH, and EFEF intersects BCBC at GG, then prove that PMPM is parallel to GHGH.
Proposed by Sreejato Bhattacharya
geometrycircumcircletrigonometrypower of a pointradical axisexterior angle
Orthocenter moving on line

Source: German TST 2023 AIMO 7, Problem 1

11/2/2023
Let ABCABC be a acute angled triangle and let AD,BE,CFAD,BE,CF be its altitudes. X≠A,BX \not=A,B and Y≠A,CY \not=A,C lie on sides ABAB and ACAC, respectively, so that ADXYADXY is a cyclic quadrilateral. Let HH be the orthocenter of triangle AXYAXY.
Prove that HH lies on line EFEF.
geometrycyclic quadrilateral