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Swedish Mathematical Competition
2020 Swedish Mathematical Competition
5
5
Part of
2020 Swedish Mathematical Competition
Problems
(1)
p divides sum of combinations (p-1,k)a^k
Source: 2020 Swedish Mathematical Competition p5
5/1/2021
Find all integers
a
a
a
such that there is a prime number of
p
≥
5
p\ge 5
p
≥
5
that divides
(
p
−
1
2
)
{p-1 \choose 2}
(
2
p
−
1
)
+
(
p
−
1
3
)
a
+ {p-1 \choose 3} a
+
(
3
p
−
1
)
a
+
(
p
−
1
4
)
a
2
+{p-1 \choose 4} a^2
+
(
4
p
−
1
)
a
2
+ ...+
(
p
−
1
p
−
3
)
a
p
−
5
.
{p-1 \choose p-3} a^{p-5} .
(
p
−
3
p
−
1
)
a
p
−
5
.
number theory
combinatorics
divides
divisible