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Part of 2011 Kosovo National Mathematical Olympiad

Problems(4)

Prove that the left number is even. [KMO #3, Grade 9]

Source:

A little boy wrote the numbers

1,2,\cdots,2011

on a blackboard. He picks any two numbers

x,y

, erases them with a sponge and writes the number

|x-y|

. This process continues until only one number is left. Prove that the number left is even.

invariantcombinatorics proposedcombinatorics

KMO #3 (Grade 10) [inequality with logarithms]

Source:

Prove that the following inequality holds:

\left( \log_{24}48 \right)^2+ \left( \log_{12}54 \right)^2>4

inequalitieslogarithmsinequalities proposed

KMO #3 (Grade 11) [Maximal value]

Source:

Find maximal value of the function

f(x)=8-3\sin^2 (3x)+6 \sin (6x)

functiontrigonometryinequalitiescalculusalgebrasystem of equationsinequalities proposed

Kosovo Mathematical Olympiad, #3. (Grade 12)

Source:

a,b,c

are real positive numbers prove that the inequality holds:

\frac{\sqrt{a^3+b^3}}{a^2+b^2}+\frac{\sqrt{b^3+c^3}}{b^2+c^2}+\frac{\sqrt{c^3+a^3}}{c^2+a^2} \ge \frac{6(ab+bc+ac)}{(a+b+c)\sqrt{(a+b)(b+c)(c+a)}}

inequalitiesinequalities proposed