MathDB
Problems
Contests
International Contests
Balkan MO Shortlist
2014 Balkan MO Shortlist
N2
BMO 2014 SL N2
BMO 2014 SL N2
Source: Balkan MO 2014 Shortlist
October 10, 2016
number theory
prime numbers
Problem Statement
N
2
\boxed{N2}
N
2
Let
p
p
p
be a prime numbers and
x
1
,
x
2
,
.
.
.
,
x
n
x_1,x_2,...,x_n
x
1
,
x
2
,
...
,
x
n
be integers.Show that if
x
1
n
+
x
2
n
+
.
.
.
+
x
p
n
≡
0
(
m
o
d
p
)
x_1^n+x_2^n+...+x_p^n\equiv 0 \pmod{p}
x
1
n
+
x
2
n
+
...
+
x
p
n
≡
0
(
mod
p
)
for all positive integers n then
x
1
≡
x
2
≡
.
.
.
≡
x
p
(
m
o
d
p
)
.
x_1\equiv x_2 \equiv...\equiv x_p \pmod{p}.
x
1
≡
x
2
≡
...
≡
x
p
(
mod
p
)
.
Back to Problems
View on AoPS