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National and Regional Contests
Turkey Contests
Turkey Team Selection Test
2020 Turkey Team Selection Test
9
9
Part of
2020 Turkey Team Selection Test
Problems
(1)
Number of equivalent's solutions
Source: 2020 Turkey TST P9
3/14/2020
For
a
,
n
a,n
a
,
n
positive integers we show number of different integer 10-tuples
(
x
1
,
x
2
,
.
.
.
,
x
10
)
(x_1,x_2,...,x_{10})
(
x
1
,
x
2
,
...
,
x
10
)
on
(
m
o
d
n
)
(mod n)
(
m
o
d
n
)
satistfying
x
1
x
2
.
.
.
x
10
=
a
(
m
o
d
n
)
x_1x_2...x_{10}=a (mod n)
x
1
x
2
...
x
10
=
a
(
m
o
d
n
)
with
f
(
a
,
n
)
f(a,n)
f
(
a
,
n
)
. Let
a
,
b
a,b
a
,
b
given positive integers , a) Prove that there exist a positive integer
c
c
c
such that for all
n
∈
Z
+
n\in \mathbb{Z^+}
n
∈
Z
+
f
(
a
,
c
n
)
f
(
b
,
c
n
)
\frac {f(a,cn)}{f(b,cn)}
f
(
b
,
c
n
)
f
(
a
,
c
n
)
is constant b) Find all
(
a
,
b
)
(a,b)
(
a
,
b
)
pairs such that minumum possible value of
c
c
c
is 27 where
c
c
c
satisfying condition in
(
a
)
(a)
(
a
)
number theory