MathDB
Problems
Contests
National and Regional Contests
Moldova Contests
Moldova Team Selection Test
2019 Moldova Team Selection Test
8
8
Part of
2019 Moldova Team Selection Test
Problems
(1)
very beautiful number theory
Source: Moldova TST 2019
3/10/2019
For any positive integer
k
k
k
denote by
S
(
k
)
S(k)
S
(
k
)
the number of solutions
(
x
,
y
)
∈
Z
+
×
Z
+
(x,y)\in \mathbb{Z}_+ \times \mathbb{Z}_+
(
x
,
y
)
∈
Z
+
×
Z
+
of the system
{
⌈
x
⋅
d
y
⌉
⋅
x
d
=
⌈
(
y
+
1
)
2
⌉
∣
x
−
y
∣
=
k
,
\begin{cases} \left\lceil\frac{x\cdot d}{y}\right\rceil\cdot \frac{x}{d}=\left\lceil\left(\sqrt{y}+1\right)^2\right\rceil \\ \mid x-y\mid =k , \end{cases}
{
⌈
y
x
⋅
d
⌉
⋅
d
x
=
⌈
(
y
+
1
)
2
⌉
∣
x
−
y
∣=
k
,
where
d
d
d
is the greatest common divisor of positive integers
x
x
x
and
y
.
y.
y
.
Determine
S
(
k
)
S(k)
S
(
k
)
as a function of
k
k
k
. (Here
⌈
z
⌉
\lceil z\rceil
⌈
z
⌉
denotes the smalles integer number which is bigger or equal than
z
.
z.
z
.
)
number theory