MathDB

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Part of 2011 Kosovo National Mathematical Olympiad

Problems(4)

Compare x^y and y^x, [KMO #1. (Grade 9)]

Source:

3/13/2011
Let x=(1+1n)nx = \left( 1 + \frac{1}{n}\right)^n and y=(1+1n)n+1y = \left( 1 + \frac{1}{n}\right)^{n+1} where n∈Nn \in \mathbb{N}. Which one of the numbers xyx^y, yxy^x is bigger ?
logarithms
KMO #1 (Grade 10) [p,q rational numbers]

Source:

3/13/2011
Suppose that the roots p,qp,q of the equation x2−x+c=0x^2-x+c=0 where c∈Rc \in \mathbb{R}, are rational numbers. Prove that the roots of the equation x2+px−q=0x^2+px-q=0 are also rational numbers.
algebra proposedalgebra
KMO #1 (Grade 11) [ f(cosx)=?]

Source:

3/13/2011
It is given the function f:R→Rf:\mathbb{R} \to \mathbb{R} such that it holds f(sin⁡x)=sin⁡(2011x)f(\sin x)=\sin (2011x). Find the value of f(cos⁡x)f(\cos x).
functiontrigonometryalgebra proposedalgebra
Kosovo Mathematical Olympiad, #1 (Grade 12) [Complex Number]

Source:

3/13/2011
The complex numbers z1z_1 and z2z_2 are given such that z1=−1+iz_1=-1+i and z2=2+4iz_2=2+4i. Find the complex number z3z_3 such that z1,z2,z3z_1,z_2,z_3 are the points of an equilateral triangle. How many solutions do we have ?
complex numbersalgebra proposedalgebra