MathDB

3

Part of 2013 Kosovo National Mathematical Olympiad

Problems(4)

Kosovo National Mathematical Olympiad 2013 Grade 9 #3

Source:

3/4/2013
For all real numbers aa prove that 3(a4+a2+1)≥(a2+a+1)23(a^4+a^2+1)\geq (a^2+a+1)^2
Kosovo Natinonal Mathematical 2013 Grade 10 #3

Source:

3/5/2013
How many positive integers which are less or equal with 20132013 such that 33 or 55 divide the number.
Kosovo National Mathematical Olympiad 2013 Grade 11 #3

Source:

3/4/2013
Prove that solution of equation y=x2+ax+by=x^2+ax+b and x=y2+cy+dx=y^2+cy+d it belong a circle.
system of equationsanalytic geometry
Kosovo National Mathematical Olympiad 2013 Grade 12 #3

Source:

3/4/2013
Find all numbers xx such that:
1+2⋅2x+3⋅3x<6x1+2\cdot2^x+3\cdot3^x<6^x
inequalities