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N7
N7
Part of
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Problems
(1)
Partition into sets with equal sum of powers in F_p
Source: IMOC 2021 N7
8/11/2021
Let
p
p
p
be a given odd prime. Find the largest integer
k
′
k'
k
′
such that it is possible to partition
{
1
,
2
,
⋯
,
p
−
1
}
\{1,2,\cdots,p-1\}
{
1
,
2
,
⋯
,
p
−
1
}
into two sets
X
,
Y
X,Y
X
,
Y
such that for any
k
k
k
with
0
≤
k
≤
k
′
0 \le k \le k'
0
≤
k
≤
k
′
,
∑
a
∈
X
a
k
≡
∑
b
∈
Y
b
k
(
m
o
d
p
)
\sum_{a \in X}a^k \equiv \sum_{b \in Y}b^k \pmod p
a
∈
X
∑
a
k
≡
b
∈
Y
∑
b
k
(
mod
p
)
houkai
modular arithmetic
number theory