Difference between revisions of "2007 Cyprus MO/Lyceum/Problem 12"
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==Problem== | ==Problem== | ||
− | The function <math>f : \ | + | The function <math>f : \mathbb{R} \rightarrow \mathbb{R}</math> has the properties <math>f(0) = -1</math> and <math>f(xy)+f(x)+f(y)=x+y+xy+k\ \ \ \forall x,y \in \Re</math>, where <math>k \in \Re</math> is a constant. The value of <math>f(-1)</math> is |
<math> \mathrm{(A) \ } 1\qquad \mathrm{(B) \ } -1\qquad \mathrm{(C) \ } 0\qquad \mathrm{(D) \ } -2\qquad \mathrm{(E) \ } 3</math> | <math> \mathrm{(A) \ } 1\qquad \mathrm{(B) \ } -1\qquad \mathrm{(C) \ } 0\qquad \mathrm{(D) \ } -2\qquad \mathrm{(E) \ } 3</math> | ||
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First, to determine the value of <math>k</math>, let <math>x=y=0</math>. | First, to determine the value of <math>k</math>, let <math>x=y=0</math>. | ||
− | <math>f(0\cdot0)+f(0)+f(0)=0+0+0\cdot0+k</math> | + | <math>f(0\cdot0)+f(0)+f(0)=0+0+0\cdot0+k</math>, so <math>\displaystyle k = (-1)+(-1)+(-1) = - 3</math>. |
− | <math>(-1) | + | Now, to determine the value of <math>\displaystyle f(-1)</math>, let <math>x=-1</math> and <math>y=0</math>. |
− | <math> | + | <math>\displaystyle f(-1\cdot0)+f(-1)+f(0)=-1+0+0\cdot0-3</math> |
− | + | <math>\displaystyle (-1)+f(-1)+(-1)=-4</math> | |
− | <math> | + | <math>\displaystyle f(-1)=-2\Longrightarrow\mathrm{ D}</math> |
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==See also== | ==See also== | ||
− | + | {{CYMO box|year=2007|l=Lyceum|num-b=11|num-a=13}} | |
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− | + | [[Category:Introductory Algebra Problems]] |
Latest revision as of 00:19, 19 January 2024
Problem
The function has the properties
and
, where
is a constant. The value of
is
Solution
First, to determine the value of , let
.
, so
.
Now, to determine the value of , let
and
.
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 |