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National and Regional Contests
Ukraine Contests
Official Ukraine Selection Cycle
Ukraine Team Selection Test
2009 Ukraine Team Selection Test
6
equal sets modulo p
equal sets modulo p
Source: Ukraine TST 2009 p6
May 3, 2020
modulo
number theory
Sets
remainder
Problem Statement
Find all odd prime numbers
p
p
p
for which there exists a natural number
g
g
g
for which the sets
A
=
{
(
k
2
+
1
)
m
o
d
p
∣
k
=
1
,
2
,
…
,
p
−
1
2
}
A=\left\{ \left( {{k}^{2}}+1 \right)\,\bmod p|\,k=1,2,\ldots ,\frac{p-1}{2} \right\}
A
=
{
(
k
2
+
1
)
mod
p
∣
k
=
1
,
2
,
…
,
2
p
−
1
}
and
B
=
{
g
k
m
o
d
p
∣
k
=
1
,
2
,
.
.
.
,
p
−
1
2
}
B=\left\{ {{g}^{k}}\bmod \,p|\,k=1,2,...,\frac{p-1}{2} \right\}
B
=
{
g
k
mod
p
∣
k
=
1
,
2
,
...
,
2
p
−
1
}
are equal.
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