MathDB

2

Part of 2017 Bosnia And Herzegovina - Regional Olympiad

Problems(4)

Regional Olympiad - FBH 2017 Grade 9 Problem 2

Source: Regional Olympiad - Federation of Bosnia and Herzegovina 2017

9/19/2018
Prove that numbers 1,2,...,161,2,...,16 can be divided in sequence such that sum of any two neighboring numbers is perfect square
Perfect Squarearrangingcombinatoricsnumber theory
Regional Olympiad - FBH 2017 Grade 10 Problem 2

Source: Regional Olympiad - Federation of Bosnia and Herzegovina 2017

9/19/2018
It is given triangle ABCABC. Let internal and external angle bisector of angle ∠BAC\angle BAC intersect line BCBC in points DD and EE, respectively, and circumcircle of triangle ADEADE intersects line ACAC in point FF. Prove that FDFD is angle bisector of ∠BFC\angle BFC
geometryangle bisectorcircumcircle
Regional Olympiad - FBH 2017 Grade 12 Problem 2

Source: Regional Olympiad - Federation of Bosnia and Herzegovina 2017

9/19/2018
In triangle ABCABC on side ACAC are points KK, LL and MM such that BKBK is an angle bisector of ∠ABL\angle ABL, BLBL is an angle bisector of ∠KBM\angle KBM and BMBM is an angle bisector of ∠LBC\angle LBC, respectively. Prove that 4⋅LM<AC4 \cdot LM <AC and 3⋅∠BAC−∠ACB<180∘3\cdot \angle BAC - \angle ACB < 180^{\circ}
geometryangle bisector
Regional Olympiad - FBH 2017 Grade 11 Problem 2

Source: Regional Olympiad - Federation of Bosnia and Herzegovina 2017

9/19/2018
Let ABCABC be an isosceles triangle such that AB=ACAB=AC. Find angles of triangle ABCABC if ABBC=1+2cos⁡2π7\frac{AB}{BC}=1+2\cos{\frac{2\pi}{7}}
geometryisoscelesanglescosine