MathDB

2

Part of 2012 Kosovo National Mathematical Olympiad

Problems(4)

Kosovo MO 2012 Grade 9, Problem 2

Source: Kosovo MO 2012 Grade 9, Problem 2

6/7/2021
If a>1,b>1a>1,b>1 are the legths of the catheti of an right triangle and cc the length of its hypotenuse, prove that a+b≤c2a+b\leq c\sqrt 2
geometryInequality
Kosovo MO 2012 Grade 10, Problem 2

Source: Kosovo MO 2012 Grade 10, Problem 2

6/7/2021
Let a,b,ca,b,c be the lengths of the sides of a triangle. Prove that, ∣ab+bc+ca−ba−cb−ac∣<1\left|\frac {a}{b}+\frac {b}{c}+\frac {c}{a}-\frac {b}{a}-\frac {c}{b}-\frac {a}{c}\right|<1
geometryInequality
Kosovo MO 2012 Grade 11, Problem 2

Source: Kosovo MO 2012 Grade 11, Problem 2

6/7/2021
The bisector of acute angle α\alpha of the right triangle ABCABC splits the side aa in two segments with lengths mm and nn. Find the acute angle β\beta and the lenths of the other two sides by knowing mm and nn.
geometry
Kosovo MO 2012 Grade 12, Problem 2

Source: Kosovo MO 2012 Grade 12, Problem 2

6/7/2021
In a sphere S0S_0 we radius rr a cube K0K_0 has been inscribed. Then in the cube K0K_0 another sphere S1S_1 has been inscribed and so on to infinity. Calculate the volume of all spheres created in this way.
geometry