MathDB

3

Part of 2014 Bosnia And Herzegovina - Regional Olympiad

Problems(4)

Angle bissector

Source: French TST 2004 pb.2; German contest (Bundeswettbewerb) 2003, 1st round

5/25/2004
Let ABCDABCD be a parallelogram. Let MM be a point on the side ABAB and NN be a point on the side BCBC such that the segments AMAM and CNCN have equal lengths and are non-zero. The lines ANAN and CMCM meet at QQ. Prove that the line DQDQ is the bisector of the angle ∡ADC\measuredangle ADC. Alternative formulation. Let ABCDABCD be a parallelogram. Let MM and NN be points on the sides ABAB and BCBC, respectively, such that AM=CN≠0AM=CN\neq 0. The lines ANAN and CMCM intersect at a point QQ. Prove that the point QQ lies on the bisector of the angle ∡ADC\measuredangle ADC.
trigonometrygeometryangle bisectorgeometry proposed
Regional Olympiad - FBH 2014 Grade 9 Problem 3

Source: Regional Olympiad - Federation of Bosnia and Herzegovina 2014

9/24/2018
In triangle ABCABC (b≥c)(b \geq c), point EE is the midpoint of shorter arc BCBC. If DD is the point such that EDED is the diameter of circumcircle ABCABC, prove that ∠DEA=12(β−γ)\angle DEA = \frac{1}{2}(\beta-\gamma)
geometrycircumcircle
Regional Olympiad - FBH 2014 Grade 11 Problem 3

Source: Regional Olympiad - Federation of Bosnia and Herzegovina 2014

9/24/2018
Excircle of triangle ABCABC to side ABAB of triangle ABCABC touches side ABAB in point DD. Determine ratio AD:BDAD : BD if ∠CAB=2∠ADC\angle CAB = 2 \angle ADC
ratiogeometryexcircle
Regional Olympiad - FBH 2014 Grade 12 Problem 3

Source: Regional Olympiad - Federation of Bosnia and Herzegovina 2014

9/24/2018
Find all integers nn such that n4−8n+15n^4-8n+15 is product of two consecutive integers
consecutivenumber theory