MathDB

4

Part of 2011 Kosovo National Mathematical Olympiad

Problems(3)

Find the length of AB [KMO #4, Grade 9]

Source:

3/13/2011
In triangle ABCABC medians of triangle BEBE and ADAD are perpendicular to each other. Find the length of AB‾\overline{AB}, if BC‾=6\overline{BC}=6 and AC‾=8\overline{AC}=8
Pythagorean Theoremgeometrygeometry proposed
KMO #4 (Grade 11) [Area of the square]

Source:

3/13/2011
A point PP is given in the square ABCDABCD such that PA‾=3\overline{PA}=3, PB‾=7\overline{PB}=7 and PD‾=5\overline{PD}=5. Find the area of the square.
geometryrotationtrigonometrygeometry proposed
Kosovo Mathematical Olympiad, #4. (Grade 12)

Source:

3/13/2011
It is given a convex hexagon A1A2⋯A6A_1A_2 \cdots A_6 such that all its interior angles are same valued (congruent). Denote by a1=A1A2‾,  a2=A2A3‾, ⋯ ,a6=A6A1‾.a_1= \overline{A_1A_2},\ \ a_2=\overline{A_2A_3},\ \cdots , a_6=\overline{A_6A_1}.
a)a) Prove that holds: a1−a4=a2−a5=a3−a6 a_1-a_4=a_2-a_5=a_3-a_6 b)b) Prove that if a1,a2,a3,...,a6a_1,a_2,a_3,...,a_6 satisfy the above equation, we can construct a convex hexagon with its same-valued (congruent) interior angles.
geometry proposedgeometry