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International Mathematical Olympic Revenge
2018 International Olympic Revenge
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1
Part of
2018 International Olympic Revenge
Problems
(1)
Problem 1 - IMOR 2018
Source: 2nd IMOR - 2018
7/13/2018
Let
p
p
p
be a prime number, and
X
X
X
be the set of cubes modulo
p
p
p
, including
0
0
0
. Denote by
C
2
(
k
)
C_2(k)
C
2
(
k
)
the number of ordered pairs
(
x
,
y
)
∈
X
×
X
(x, y) \in X \times X
(
x
,
y
)
∈
X
×
X
such that
x
+
y
≡
k
(
m
o
d
p
)
x + y \equiv k \pmod p
x
+
y
≡
k
(
mod
p
)
. Likewise, denote by
C
3
(
k
)
C_3(k)
C
3
(
k
)
the number of ordered pairs
(
x
,
y
,
z
)
∈
X
×
X
×
X
(x, y, z) \in X \times X \times X
(
x
,
y
,
z
)
∈
X
×
X
×
X
such that
x
+
y
+
z
≡
k
(
m
o
d
p
)
x + y + z \equiv k \pmod p
x
+
y
+
z
≡
k
(
mod
p
)
. Prove that there are integers
a
,
b
a, b
a
,
b
such that for all
k
k
k
not in
X
X
X
, we have
C
3
(
k
)
=
a
⋅
C
2
(
k
)
+
b
.
C_3(k) = a\cdot C_2(k) + b.
C
3
(
k
)
=
a
⋅
C
2
(
k
)
+
b
.
Proposed by Murilo Corato, Brazil.
IMOR
number theory