MathDB

2

Part of 2013 Kosovo National Mathematical Olympiad

Problems(4)

Kosovo National Mathematical Olympiad 2013 Grade 9 #2

Source:

3/4/2013
Find all integer nn such that n−5n-5 divide n2+n−27n^2+n-27.
algebrapolynomialcalculusintegration
Kosovo Natinonal Mathematical 2013 Grade 10 #2

Source:

3/5/2013
Three numbers have sum kk (where k∈Rk\in \mathbb{R}) such that the numbers are arethmetic progression.If First of two numbers remain the same and to the third number we add k6\frac{k}{6} than we have geometry progression. Find those numbers?
geometry
Kosovo National Mathematical Olympiad 2013 Grade 11 #2

Source:

3/4/2013
Solve equation 27⋅33sin⁡x=9cos⁡2x27\cdot3^{3\sin x}=9^{\cos^2x} where x∈[0,2π)x\in [0,2\pi )
trigonometry
Kosovo National Mathematical Olympiad 2013 Grade 12 #2

Source:

3/4/2013
Math teacher wrote in a table a polynomial P(x)P(x) with integer coefficients and he said: "Today my daughter have a birthday.If in polynomial P(x)P(x) we have x=ax=a where aa is the age of my daughter we have P(a)=aP(a)=a and P(0)=pP(0)=p where pp is a prime number such that p>ap>a." How old is the daughter of math teacher?
algebrapolynomial