MathDB
Problems
Contests
National and Regional Contests
Turkey Contests
National Olympiad First Round
2014 National Olympiad First Round
2
P02 [Number Theory] - Turkish NMO 1st Round - 2014
P02 [Number Theory] - Turkish NMO 1st Round - 2014
Source:
May 21, 2014
Problem Statement
How many pairs of integers
(
m
,
n
)
(m,n)
(
m
,
n
)
are there such that
m
n
+
n
+
14
=
(
m
−
1
)
2
mn+n+14=\left (m-1 \right)^2
mn
+
n
+
14
=
(
m
−
1
)
2
?
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
a
)
<
/
s
p
a
n
>
16
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
b
)
<
/
s
p
a
n
>
12
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
c
)
<
/
s
p
a
n
>
8
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
d
)
<
/
s
p
a
n
>
6
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
b
o
l
d
′
>
e
)
<
/
s
p
a
n
>
2
<span class='latex-bold'>a)</span>\ 16 \qquad<span class='latex-bold'>b)</span>\ 12 \qquad<span class='latex-bold'>c)</span>\ 8 \qquad<span class='latex-bold'>d)</span>\ 6 \qquad<span class='latex-bold'>e)</span>\ 2
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
a
)
<
/
s
p
an
>
16
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
b
)
<
/
s
p
an
>
12
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
c
)
<
/
s
p
an
>
8
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
d
)
<
/
s
p
an
>
6
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
b
o
l
d
′
>
e
)
<
/
s
p
an
>
2
Back to Problems
View on AoPS