MathDB

1

Part of 2011 JBMO Shortlist

Problems(3)

circles inside a square

Source: JBMO 2011 Shortlist C1

10/14/2017
Inside of a square whose side length is 11 there are a few circles such that the sum of their circumferences is equal to 1010. Show that there exists a line that meets at least four of these circles.
JBMOcombinatorics
2011 JBMO Shortlist G1

Source: 2011 JBMO Shortlist G1

10/8/2017
Let ABCABC be an isosceles triangle with AB=ACAB=AC. On the extension of the side CA{CA} we consider the point D{D} such that AD<AC{AD<AC}. The perpendicular bisector of the segment BD{BD} meets the internal and the external bisectors of the angle ∠BAC\angle BAC at the points E{E}and Z{Z}, respectively. Prove that the points A,E,D,Z{A, E, D, Z} are concyclic.
geometryJBMO
1005^x + 2011^y = 1006^z

Source: JBMO 2011 Shortlist N1

10/14/2017
Solve in positive integers the equation 1005x+2011y=1006z1005^x + 2011^y = 1006^z.
JBMOnumber theoryalgebrainteger equationDiophantine equationmodular arithmetic