MathDB

1

Part of 2006 Junior Balkan Team Selection Tests - Romania

Problems(5)

< BOD = 60^o iff k =\sqrt3 where BD /AC = AE/CD = k, right triangle

Source: 2006 Romania JBMO TST1 p1

5/16/2020
Let ABCABC be a triangle right in CC and the points D,ED, E on the sides BCBC and CACA respectively, such that BDAC=AECD=k\frac{BD}{AC} =\frac{AE}{CD} = k. Lines BEBE and ADAD intersect at OO. Show that the angle ∠BOD=60o\angle BOD = 60^o if and only if k=3k =\sqrt3.
equal ratiogeometryangleratioright triangle
a^3/bc + b^3/ca + c^3 /ba >= a + b + c for a,b,c>0

Source: 2006 Romania JBMO TST 2.1

6/2/2020
Prove that a3bc+b3ca+c3ba≥a+b+c\frac{a^3}{bc} + \frac{b^3}{ca} + \frac{c^3}{ba} \ge a + b + c, for all positive real numbers a,ba, b, and cc.
inequalitiesalgebra
ABCD is cyclic and has area equal to 8

Source: Romanian JBMO TST 2006, Day 3, Problem 1

5/16/2006
Let ABCDABCD be a cyclic quadrilateral of area 8. If there exists a point OO in the plane of the quadrilateral such that OA+OB+OC+OD=8OA+OB+OC+OD = 8, prove that ABCDABCD is an isosceles trapezoid.
geometrycyclic quadrilateralgeometry unsolved
A set with 2006 elements and intersection of subsets

Source: Romanian JBMO TST 2006, Day 4, Problem 1

5/19/2006
Let A={1,2,…,2006}A=\{1,2,\ldots, 2006\}. Find the maximal number of subsets of AA that can be chosen such that the intersection of any 2 such distinct subsets has 2004 elements.
AB x CD = AC x BD, if D on A-median and <BDC = 180^o - <BAC

Source: 2006 Romania JBMO TST5 p1

5/16/2020
Let ABCABC be a triangle and DD a point inside the triangle, located on the median of AA. Prove that if ∠BDC=180o−∠BAC\angle BDC = 180^o - \angle BAC, then AB⋅CD=AC⋅BDAB \cdot CD = AC \cdot BD.
anglesgeometrymedianratio