MathDB
Problems
Contests
National and Regional Contests
China Contests
(China) National High School Mathematics League
2006 China Second Round Olympiad
3
3
Part of
2006 China Second Round Olympiad
Problems
(2)
solve the system of equations
Source: China second round 2006 p3
3/10/2012
Solve the system of equations in real numbers:
{
x
−
y
+
z
−
w
=
2
x
2
−
y
2
+
z
2
−
w
2
=
6
x
3
−
y
3
+
z
3
−
w
3
=
20
x
4
−
y
4
+
z
4
−
w
4
=
66
\begin{cases} x-y+z-w=2 \\ x^2-y^2+z^2-w^2=6 \\ x^3-y^3+z^3-w^3=20 \\ x^4-y^4+z^4-w^4=66 \end{cases}
⎩
⎨
⎧
x
−
y
+
z
−
w
=
2
x
2
−
y
2
+
z
2
−
w
2
=
6
x
3
−
y
3
+
z
3
−
w
3
=
20
x
4
−
y
4
+
z
4
−
w
4
=
66
algebra
system of equations
algebra unsolved
2006 China Second Round Olympiad Test 1 #3
Source:
9/28/2014
Suppose
A
=
x
∣
5
x
−
a
≤
0
A = {x|5x-a \le 0}
A
=
x
∣5
x
−
a
≤
0
,
B
=
x
∣
6
x
−
b
>
0
B = {x|6x-b > 0}
B
=
x
∣6
x
−
b
>
0
,
a
,
b
∈
N
a,b \in \mathbb{N}
a
,
b
∈
N
, and
A
∩
B
∩
N
=
2
,
3
,
4
A \cap B \cap \mathbb{N} = {2,3,4}
A
∩
B
∩
N
=
2
,
3
,
4
. The number of such pairs
(
a
,
b
)
(a,b)
(
a
,
b
)
is
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
a
n
>
20
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
a
n
>
25
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
a
n
>
30
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
a
n
>
42
{ <span class='latex-bold'>(A)</span>\ 20\qquad<span class='latex-bold'>(B)</span>\ 25\qquad<span class='latex-bold'>(C)</span>\ 30\qquad<span class='latex-bold'>(D)</span>} 42\qquad
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
an
>
20
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
an
>
25
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
an
>
30
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
42
inequalities