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National and Regional Contests
PEN Problems
PEN C Problems
PEN C Problems
Part of
PEN Problems
Subcontests
(6)
6
1
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C 6
Let
a
,
b
,
c
a, b, c
a
,
b
,
c
be integers and let
p
p
p
be an odd prime with
p
∤
a
and
p
∤
b
2
−
4
a
c
.
p \not\vert a \;\; \text{and}\;\; p \not\vert b^{2}-4ac.
p
∣
a
and
p
∣
b
2
−
4
a
c
.
Show that
∑
k
=
1
p
(
a
k
2
+
b
k
+
c
p
)
=
−
(
a
p
)
.
\sum_{k=1}^{p}\left( \frac{ak^{2}+bk+c}{p}\right) =-\left( \frac{a}{p}\right).
k
=
1
∑
p
(
p
a
k
2
+
bk
+
c
)
=
−
(
p
a
)
.
5
1
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C 5
Let
p
p
p
be an odd prime and let
Z
p
Z_{p}
Z
p
denote (the field of) integers modulo
p
p
p
. How many elements are in the set
{
x
2
:
x
∈
Z
p
}
∩
{
y
2
+
1
:
y
∈
Z
p
}
?
\{x^{2}: x \in Z_{p}\}\cap \{y^{2}+1: y \in Z_{p}\}?
{
x
2
:
x
∈
Z
p
}
∩
{
y
2
+
1
:
y
∈
Z
p
}?
4
1
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C 4
Let
M
M
M
be an integer, and let
p
p
p
be a prime with
p
>
25
p>25
p
>
25
. Show that the set
{
M
,
M
+
1
,
⋯
,
M
+
3
⌊
p
⌋
−
1
}
\{M, M+1, \cdots, M+ 3\lfloor \sqrt{p} \rfloor -1\}
{
M
,
M
+
1
,
⋯
,
M
+
3
⌊
p
⌋
−
1
}
contains a quadratic non-residue to modulus
p
p
p
.
3
1
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C 3
Let
p
p
p
be an odd prime number. Show that the smallest positive quadratic nonresidue of
p
p
p
is smaller than
p
+
1
\sqrt{p}+1
p
+
1
.
2
1
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C 2
The positive integers
a
a
a
and
b
b
b
are such that the numbers
15
a
+
16
b
15a+16b
15
a
+
16
b
and
16
a
−
15
b
16a-15b
16
a
−
15
b
are both squares of positive integers. What is the least possible value that can be taken on by the smaller of these two squares?
1
1
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C 1
Find all positive integers
n
n
n
that are quadratic residues modulo all primes greater than
n
n
n
.