MathDB
Problems
Contests
National and Regional Contests
Turkey Contests
National Olympiad First Round
2000 National Olympiad First Round
28
28
Part of
2000 National Olympiad First Round
Problems
(1)
Turkey NMO 2000 1st Round - P28 (Algebra)
Source:
7/25/2012
f
1
(
x
)
=
x
2
+
x
f
2
(
x
)
=
2
x
2
−
x
f
3
(
x
)
=
x
2
+
x
g
1
(
x
)
=
x
−
2
g
2
(
x
)
=
2
x
g
3
(
x
)
=
x
+
2
\begin{array}{ rlrlrl} f_1(x)=&x^2+x & f_2(x)=&2x^2-x & f_3(x)=&x^2 +x \\ g_1(x)=&x-2 & g_2(x)=&2x \ \ & g_3(x)=&x+2 \\ \end{array}
f
1
(
x
)
=
g
1
(
x
)
=
x
2
+
x
x
−
2
f
2
(
x
)
=
g
2
(
x
)
=
2
x
2
−
x
2
x
f
3
(
x
)
=
g
3
(
x
)
=
x
2
+
x
x
+
2
If
h
(
x
)
=
x
h(x)=x
h
(
x
)
=
x
can be get from
f
i
f_i
f
i
and
g
i
g_i
g
i
by using only addition, substraction, multiplication defined on those functions where
i
∈
{
1
,
2
,
3
}
i\in\{1,2,3\}
i
∈
{
1
,
2
,
3
}
, then
F
i
=
1
F_i=1
F
i
=
1
. Otherwise,
F
i
=
0
F_i=0
F
i
=
0
. What is
(
F
1
,
F
2
,
F
3
)
(F_1,F_2,F_3)
(
F
1
,
F
2
,
F
3
)
?
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
a
n
>
(
0
,
0
,
0
)
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
a
n
>
(
0
,
0
,
1
)
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
a
n
>
(
0
,
1
,
0
)
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
a
n
>
(
0
,
1
,
1
)
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
a
n
>
None
<span class='latex-bold'>(A)</span>\ (0,0,0) \qquad<span class='latex-bold'>(B)</span>\ (0,0,1) \qquad<span class='latex-bold'>(C)</span>\ (0,1,0) \qquad<span class='latex-bold'>(D)</span>\ (0,1,1) \qquad<span class='latex-bold'>(E)</span>\ \text{None}
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
an
>
(
0
,
0
,
0
)
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
an
>
(
0
,
0
,
1
)
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
an
>
(
0
,
1
,
0
)
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
(
0
,
1
,
1
)
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
an
>
None
function