MathDB

4

Part of 2018 Bosnia And Herzegovina - Regional Olympiad

Problems(4)

Regional Olympiad - FBH 2018 Grade 10 Problem 4

Source: Regional Olympiad - Federation of Bosnia and Herzegovina 2018

9/18/2018
Let PP be a point on circumcircle of triangle ABCABC on arc BC⌢\stackrel{\frown}{BC} which does not contain point AA. Let lines ABAB and CPCP intersect at point EE, and lines ACAC and BPBP intersect at FF. If perpendicular bisector of side ABAB intersects ACAC in point KK, and perpendicular bisector of side ACAC intersects side ABAB in point JJ, prove that: (CEBF)2=AJ⋅JEAK⋅KF{\left(\frac{CE}{BF}\right)}^2=\frac{AJ\cdot JE}{AK \cdot KF}
geometrycircumcircleperpendicular bisector
Regional Olympiad - FBH 2018 Grade 9 Problem 4

Source: Regional Olympiad - Federation of Bosnia and Herzegovina 2018

9/18/2018
Prove that among arbitrary 1313 points in plane with coordinates as integers, such that no three are collinear, we can pick three points as vertices of triangle such that its centroid has coordinates as integers.
analytic geometrygeometrycombinatorics
Regional Olympiad - FBH 2018 Grade 11 Problem 4

Source: Regional Olympiad - Federation of Bosnia and Herzegovina 2018

9/18/2018
We observe that number 10001=73⋅13710001=73\cdot137 is not prime. Show that every member of infinite sequence 10001,100010001,1000100010001,...10001, 100010001, 1000100010001,... is not prime
SequenceprimeCompositeinfinitely many solutionsnumber theory
Regional Olympiad - FBH 2018 Grade 12 Problem 4

Source: Regional Olympiad - Federation of Bosnia and Herzegovina 2018

9/18/2018
Let ABCDABCD be a cyclic quadrilateral and let k1k_1 and k2k_2 be circles inscribed in triangles ABCABC and ABDABD. Prove that external common tangent of those circles (different from ABAB) is parallel with CDCD
geometrycyclic quadrilateralincircletangentparallel