MathDB

1

Part of 2013 Kosovo National Mathematical Olympiad

Problems(4)

Kosovo National Mathematical Olympiad 2013 Grade 9 #1

Source:

3/4/2013
Prove that:
10+24+40+60=2+3+5\sqrt{10+\sqrt{24}+\sqrt{40}+\sqrt{60}}=\sqrt{2}+\sqrt3+\sqrt5
Kosovo Natinonal Mathematical 2013 Grade 10 #1

Source:

3/5/2013
Let be a,ba,b real numbers such that ∣a∣≠∣b∣|a|\neq |b| and a+ba−b+a−ba+b=6\frac{a+b}{a-b}+\frac{a-b}{a+b}=6 .
Calculate:
a3+b3a3−b3+a3−b3a3+b3\frac{a^3+b^3}{a^3-b^3}+\frac{a^3-b^3}{a^3+b^3}
Kosovo National Mathematical Olympiad 2013 Grade 12 #1

Source:

3/4/2013
Which number is bigger 2012!2012\sqrt[2012]{2012!} or 2013!2013\sqrt[2013]{2013!}.
inequalities
Kosovo National Mathematical Olympiad 2013 Grade 11 #1

Source:

3/4/2013
Let be z1z_1 and z2z_2 two complex numbers such that ∣z1+2z2∣=∣2z1+z2∣|z_1+2z_2|=|2z_1+z_2|.Prove that for all real numbers aa is true ∣z1+az2∣=∣az1+z2∣|z_1+az_2|=|az_1+z_2|
complex numbers