2007 Cyprus MO/Lyceum/Problem 12

Problem

The function $f : \Re \rightarrow \Re$ has the properties $f(0) = -1$ and $f(xy)+f(x)+f(y)=x+y+xy+k \forall x,y \in \Re$ , where $k \in \Re$ is a constant. The value of $f(-1)$ is

$\mathrm{(A) \ } 1\qquad \mathrm{(B) \ } -1\qquad \mathrm{(C) \ } 0\qquad \mathrm{(D) \ } -2\qquad \mathrm{(E) \ } 3$

Solution

First, to determine the value of $k$ , let $x=y=0$ .

$f(0\cdot0)+f(0)+f(0)=0+0+0\cdot0+k$

$(-1)+(-1)+(-1)=k$

$k=-3$

Now, to determine the value of $f(-1)$ , let $x=-1$ and $y=0$ .

$f(-1\cdot0)+f(-1)+f(0)=-1+0+0\cdot0-3$

$(-1)+f(-1)+(-1)=-4$

$f(-1)=-2\Rightarrow\mathrm{ D}$

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2007 Cyprus MO/Lyceum/Problem 12

Problem

Solution

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