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2007 Cyprus MO/Lyceum/Problem 12

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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 \displaystyle 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, so \displaystyle k = (-1)+(-1)+(-1) = - 3.

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

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

\displaystyle (-1)+f(-1)+(-1)=-4

\displaystyle f(-1)=-2\Longrightarrow\mathrm{ D}

See also

2007 Cyprus MO, Lyceum (Problems)
Preceded by
Problem 11 		Followed by
Problem 13
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30
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