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Turkey NMO 2008 1st Round - P19 (Algebra)

Source:

August 25, 2012
functionquadraticsalgebrafunctional equation

Problem Statement

Let f:(0,∞)→(0,∞)f:(0,\infty) \rightarrow (0,\infty) be a function such that 10⋅x+yxy=f(x)⋅f(y)−f(xy)−90 10\cdot \frac{x+y}{xy}=f(x)\cdot f(y)-f(xy)-90 for every x,y∈(0,∞)x,y \in (0,\infty). What is f(111)f(\frac 1{11})?
<spanclass=′latex−bold′>(A)</span> 1<spanclass=′latex−bold′>(B)</span> 11<spanclass=′latex−bold′>(C)</span> 21<spanclass=′latex−bold′>(D)</span> 31<spanclass=′latex−bold′>(E)</span> There is more than one solution <span class='latex-bold'>(A)</span>\ 1 \qquad<span class='latex-bold'>(B)</span>\ 11 \qquad<span class='latex-bold'>(C)</span>\ 21 \qquad<span class='latex-bold'>(D)</span>\ 31 \qquad<span class='latex-bold'>(E)</span>\ \text{There is more than one solution}
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