MathDB

3

Part of 2011 Bosnia And Herzegovina - Regional Olympiad

Problems(4)

Regional Olympiad - FBH 2011 Grade 9 Problem 3

Source: Regional Olympiad - Federation of Bosnia and Herzegovina 2011

9/26/2018
Triangle AOBAOB is rotated in plane around point OO for 90∘90^{\circ} and it maps in triangle A1OB1A_1OB_1 (AA maps to A1A_1, BB maps to B1B_1). Prove that median of triangle OAB1OAB_1 of side AB1AB_1 is orthogonal to A1BA_1B
rotationgeometrymedian
Regional Olympiad - FBH 2011 Grade 10 Problem 3

Source: Regional Olympiad - Federation of Bosnia and Herzegovina 2011

9/26/2018
Let II be the incircle and OO a circumcenter of triangle ABCABC such that ∠ACB=30∘\angle ACB=30^{\circ}. On sides ACAC and BCBC there are points EE and DD, respectively, such that EA=AB=BDEA=AB=BD. Prove that DE=IODE=IO and DE⊥IODE \perp IO
geometrycircumcircle
Regional Olympiad - FBH 2011 Grade 11 Problem 3

Source: Regional Olympiad - Federation of Bosnia and Herzegovina 2011

9/27/2018
Let ADAD and BEBE be angle bisectors in triangle ABCABC. Let xx, yy and zz be distances from point MM, which lies on segment DEDE, from sides BCBC, CACA and ABAB, respectively. Prove that z=x+yz=x+y
geometryangle bisector
Regional Olympiad - FBH 2011 Grade 12 Problem 3

Source: Regional Olympiad - Federation of Bosnia and Herzegovina 2011

9/27/2018
If nn is a positive integer and n+1n+1 is divisible with 2424, prove that sum of all positive divisors of nn is divisible with 2424
Divisorsnumber theory