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National and Regional Contests
Latvia Contests
Latvia BW TST
2022 Latvia Baltic Way TST
P14
P14
Part of
2022 Latvia Baltic Way TST
Problems
(1)
Picking numbers from a set to satisfy congruence
Source: Latvian TST for Baltic Way 2022 P14
11/24/2022
Let
A
A
A
be a set of
20
20
20
distinct positive integers which are all no greater than
397
397
397
. Prove that for any positive integer
n
n
n
it is possible to pick four (not necessarily distinct) elements
x
1
,
x
2
,
x
3
,
x
4
x_1, x_2, x_3, x_4
x
1
,
x
2
,
x
3
,
x
4
of
A
A
A
satisfying
x
1
≠
x
2
x_1 \neq x_2
x
1
=
x
2
and
(
x
1
−
x
2
)
n
≡
x
3
−
x
4
(
m
o
d
397
)
.
(x_1-x_2)n\equiv x_3-x_4 \pmod{397}.
(
x
1
−
x
2
)
n
≡
x
3
−
x
4
(
mod
397
)
.
number theory