MathDB
Problems
Contests
National and Regional Contests
Turkey Contests
National Olympiad First Round
2012 National Olympiad First Round
30
30
Part of
2012 National Olympiad First Round
Problems
(1)
Turkish NMO First Round - 2012 Problem - 30 {Number Theory}
Source:
7/1/2012
How many integer triples
(
x
,
y
,
z
)
(x,y,z)
(
x
,
y
,
z
)
are there satisfying
x
3
+
y
3
=
x
2
y
z
+
x
y
2
z
+
2
x^3+y^3=x^2yz+xy^2z+2
x
3
+
y
3
=
x
2
yz
+
x
y
2
z
+
2
?
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
a
n
>
5
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
a
n
>
4
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
a
n
>
3
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
a
n
>
2
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
a
n
>
1
<span class='latex-bold'>(A)</span>\ 5 \qquad <span class='latex-bold'>(B)</span>\ 4 \qquad <span class='latex-bold'>(C)</span>\ 3 \qquad <span class='latex-bold'>(D)</span>\ 2 \qquad <span class='latex-bold'>(E)</span>\ 1
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
A
)
<
/
s
p
an
>
5
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
B
)
<
/
s
p
an
>
4
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
C
)
<
/
s
p
an
>
3
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
D
)
<
/
s
p
an
>
2
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
(
E
)
<
/
s
p
an
>
1