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National and Regional Contests
Serbia Contests
Serbia Team Selection Test
2017 Serbia Team Selection Test
2
2
Part of
2017 Serbia Team Selection Test
Problems
(1)
Interesting process
Source: Serbia TST 2017 #2
5/21/2017
Initally a pair
(
x
,
y
)
(x, y)
(
x
,
y
)
is written on the board, such that exactly one of it's coordinates is odd. On such a pair we perform an operation to get pair
(
x
2
,
y
+
x
2
)
(\frac x 2, y+\frac x 2)
(
2
x
,
y
+
2
x
)
if
2
∣
x
2|x
2∣
x
and
(
x
+
y
2
,
y
2
)
(x+\frac y 2, \frac y 2)
(
x
+
2
y
,
2
y
)
if
2
∣
y
2|y
2∣
y
. Prove that for every odd
n
>
1
n>1
n
>
1
there is a even positive integer
b
<
n
b<n
b
<
n
such that starting from the pair
(
n
,
b
)
(n, b)
(
n
,
b
)
we will get the pair
(
b
,
n
)
(b, n)
(
b
,
n
)
after finitely many operations.
combinatorics