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National and Regional Contests
Russia Contests
All-Russian Olympiad Regional Round
2023 All-Russian Olympiad Regional Round
11.4
11.4
Part of
2023 All-Russian Olympiad Regional Round
Problems
(1)
Process with pairs
Source: ARO Regional stage 2023 11.4
2/16/2023
We write pairs of integers on a blackboard. Initially, the pair
(
1
,
2
)
(1,2)
(
1
,
2
)
is written. On a move, if
(
a
,
b
)
(a, b)
(
a
,
b
)
is on the blackboard, we can add
(
−
a
,
−
b
)
(-a, -b)
(
−
a
,
−
b
)
or
(
−
b
,
a
+
b
)
(-b, a+b)
(
−
b
,
a
+
b
)
. In addition, if
(
a
,
b
)
(a, b)
(
a
,
b
)
and
(
c
,
d
)
(c, d)
(
c
,
d
)
are written on the blackboard, we can add
(
a
+
c
,
b
+
d
)
(a+c, b+d)
(
a
+
c
,
b
+
d
)
. Can we reach
(
2022
,
2023
)
(2022, 2023)
(
2022
,
2023
)
?
number theory
Russia