MathDB

2

Part of 2011 Kosovo National Mathematical Olympiad

Problems(4)

Is f injective ? [KMO #2, (Grade 9)]

Source:

3/13/2011
It is given the function f:(R−{0})→Rf: \left(\mathbb{R} - \{0\}\right) \to \mathbb{R} such that f(x)=x+1xf(x)=x+\frac{1}{x}. Is this function injective ? Justify your answer.
function
KMO #2 (Grade 10) [floor function equation]

Source:

3/13/2011
Find all solutions to the equation:
(⌊x+73⌋)2−⌊x−94⌋=16 \left(\left\lfloor x+\frac{7}{3} \right\rfloor \right)^2-\left\lfloor x-\frac{9}{4} \right\rfloor = 16
functionfloor functionalgebra proposedalgebra
KMO #2 (Grade 11) [Transformation Problem]

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3/13/2011
Is it possible that by using the following transformations: f(x)↦x2⋅f(1x+1)   or   f(x)↦(x−1)2⋅f(1x−1) f(x) \mapsto x^2 \cdot f \left(\frac{1}{x}+1 \right) \ \ \ \text{or} \ \ \ f(x) \mapsto (x-1)^2 \cdot f\left(\frac{1}{x-1} \right) the function f(x)=x2+5x+4f(x)=x^2+5x+4 is sent to the function g(x)=x2+10x+8g(x)=x^2+10x+8 ?
functionquadraticsalgebra proposedalgebra
Kosovo Mathematical Olympiad, #2. (Grade 12)

Source:

3/13/2011
It is given the function f:(R−{0})×(R−{0})→Rf:\left( \mathbb{R} - \{0\} \right) \times \left( \mathbb{R}-\{0\} \right) \to \mathbb{R} such that f(a,b)=∣∣b−a∣∣ab∣+b+aab−1∣+∣b−a∣∣ab∣+b+aab+1f(a,b)= \left| \frac{|b-a|}{|ab|}+\frac{b+a}{ab}-1 \right|+ \frac{|b-a|}{|ab|}+ \frac{b+a}{ab}+1 where a,b≠0a,b \not=0. Prove that: f(a,b)=4⋅max{1a,1b,12} f(a,b)=4 \cdot \text{max} \left\{\frac{1}{a},\frac{1}{b},\frac{1}{2} \right\}
functionalgebra proposedalgebra