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Poland - Second Round
2020 Poland - Second Round
5.
5.
Part of
2020 Poland - Second Round
Problems
(1)
Number Theory
Source: Poland - Second Round 2020 P5
2/8/2020
Let
p
>
p>
p
>
be a prime number and
S
S
S
be a set of
p
+
1
p+1
p
+
1
integers. Prove that there exist pairwise distinct numbers
a
1
,
a
2
,
.
.
.
,
a
p
−
1
∈
S
a_1,a_2,...,a_{p-1}\in S
a
1
,
a
2
,
...
,
a
p
−
1
∈
S
that
a
1
+
2
a
2
+
3
a
3
+
.
.
.
+
(
p
−
1
)
a
p
−
1
a_1+2a_2+3a_3+...+(p-1)a_{p-1}
a
1
+
2
a
2
+
3
a
3
+
...
+
(
p
−
1
)
a
p
−
1
is divisible by
p
p
p
.
number theory
prime numbers