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4

Part of 2013 Kosovo National Mathematical Olympiad

Problems(4)

Kosovo National Mathematical Olympiad 2013 Grade 9 #4

Source:

3/4/2013
Find all value of parameter aa such that equations x2−ax+1=0x^2-ax+1=0 and x2−x+a=0x^2-x+a=0 have at least one same solution. For this value aa find same solution of this equations(real or imaginary).
parameterization
Kosovo Natinonal Mathematical 2013 Grade 10 #4

Source:

3/5/2013
Let be nn positive integer than calculate:
1⋅1!+2⋅2!+...+n⋅n!1\cdot 1!+2\cdot2!+...+n\cdot n!
induction
Kosovo National Mathematical Olympiad 2013 Grade 11 #4

Source:

3/4/2013
Let be a,b,ca,b,c three positive integer.Prove that 44 divide a2+b2+c2a^2+b^2+c^2 only and only if a,b,ca,b,c are even.
Kosovo National Mathematical Olympiad 2013 Grade 12 #4

Source:

3/4/2013
Calculate: 3535...\sqrt{3\sqrt{5\sqrt{3\sqrt{5...}}}}
algebra