MathDB

2

Part of 2013 Bosnia And Herzegovina - Regional Olympiad

Problems(4)

Regional Olympiad - FBH 2013 Grade 9 Problem 2

Source: Regional Olympiad - Federation of Bosnia and Herzegovina 2013

9/24/2018
In triangle ABCABC, ∠ACB=50∘\angle ACB=50^{\circ} and ∠CBA=70∘\angle CBA=70^{\circ}. Let DD be a foot of perpendicular from point AA to side BCBC, OO circumcenter of ABCABC and EE antipode of AA in circumcircle ABCABC. Find ∠DAE\angle DAE
geometrycircumcircleantipode
Regional Olympiad - FBH 2013 Grade 10 Problem 2

Source: Regional Olympiad - Federation of Bosnia and Herzegovina 2013

9/24/2018
In circle with radius 1010, point MM is on chord PQPQ such that PM=5PM=5 and MQ=10MQ=10. Through point MM we draw chords ABAB and CDCD, and points XX and YY are intersection points of chords ADAD and BCBC with chord PQPQ (see picture), respectively. If XM=3XM=3 find MYMY
https://services.artofproblemsolving.com/download.php?id=YXR0YWNobWVudHMvYy9kLzBiMmFmM2ViOGVmOTlmZDA5NGY2ZWY4MjM1YWI0ZDZjNjJlNzA1LnBuZw==&rn=Z2VvbWV0cmlqYS5wbmc=
geometrycirclechord
Regional Olympiad - FBH 2013 Grade 11 Problem 2

Source: Regional Olympiad - Federation of Bosnia and Herzegovina 2013

9/24/2018
Find all integers aa, bb, cc and dd such that a2+5b2−2c2−2cd−3d2=0a^2+5b^2-2c^2-2cd-3d^2=0
number theoryequation
Regional Olympiad - FBH 2013 Grade 12 Problem 2

Source: Regional Olympiad - Federation of Bosnia and Herzegovina 2013

9/24/2018
If xx and yy are real numbers, prove that 4x2+1y2+2\frac{4x^2+1}{y^2+2} is not integer
number theoryInteger