Difference between revisions of "2007 Cyprus MO/Lyceum/Problem 12"

Revision as of 19:45, 6 May 2007

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 $y=0$ .

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

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

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

@@ Line 15: / Line 15: @@
 <math>\displaystyle (-1)+f(-1)+(-1)=-4</math>
-<math>\displaystyle f(-1)=-2\Rightarrow\mathrm{ D}</math>
+<math>\displaystyle f(-1)=-2\Longrightarrow\mathrm{ D}</math>
 ==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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Difference between revisions of "2007 Cyprus MO/Lyceum/Problem 12"

Revision as of 19:45, 6 May 2007

Problem

Solution

See also