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Simon Marais Mathematical Competition
2024 Simon Marais Mathematical Competition
B4
B4
Part of
2024 Simon Marais Mathematical Competition
Problems
(1)
Polynomial congruence modulo 2
Source: SMMC 2024 B4
10/12/2024
The following problem is open in the sense that the answer to part (b) is not currently known.Let
n
n
n
be an odd positive integer and let
f
n
(
x
,
y
,
z
)
=
x
n
+
y
n
+
z
n
+
(
x
+
y
+
z
)
n
.
f_n(x,y,z) = x^n + y^n + z^n + (x+y+z)^n.
f
n
(
x
,
y
,
z
)
=
x
n
+
y
n
+
z
n
+
(
x
+
y
+
z
)
n
.
(
a
)
(a)
(
a
)
Prove that there exist infinitely many values of
n
n
n
such that
f
n
(
x
,
y
,
z
)
≡
(
x
+
y
)
(
y
+
z
)
(
z
+
x
)
g
n
(
x
,
y
,
z
)
h
n
(
x
,
y
,
z
)
(
m
o
d
2
)
,
f_n(x,y,z) \equiv (x+y)(y+z)(z+x) g_n(x,y,z) h_n(x,y,z) \pmod{2},
f
n
(
x
,
y
,
z
)
≡
(
x
+
y
)
(
y
+
z
)
(
z
+
x
)
g
n
(
x
,
y
,
z
)
h
n
(
x
,
y
,
z
)
(
mod
2
)
,
for some integer polynomials
g
n
(
x
,
y
,
z
)
g_n(x,y,z)
g
n
(
x
,
y
,
z
)
and
h
n
(
x
,
y
,
z
)
h_n(x,y,z)
h
n
(
x
,
y
,
z
)
, neither of which is constant modulo 2.
(
b
)
(b)
(
b
)
Determine all values of
n
n
n
such that
f
n
(
x
,
y
,
z
)
≡
(
x
+
y
)
(
y
+
z
)
(
z
+
x
)
g
n
(
x
,
y
,
z
)
h
n
(
x
,
y
,
z
)
(
m
o
d
2
)
,
f_n(x,y,z) \equiv (x+y)(y+z)(z+x) g_n(x,y,z) h_n(x,y,z) \pmod{2},
f
n
(
x
,
y
,
z
)
≡
(
x
+
y
)
(
y
+
z
)
(
z
+
x
)
g
n
(
x
,
y
,
z
)
h
n
(
x
,
y
,
z
)
(
mod
2
)
,
for some integer polynomials
g
n
(
x
,
y
,
z
)
g_n(x,y,z)
g
n
(
x
,
y
,
z
)
and
h
n
(
x
,
y
,
z
)
h_n(x,y,z)
h
n
(
x
,
y
,
z
)
, neither of which is constant modulo 2.(Two integer polynomials are \emph{congruent modulo 2} if every coefficient of their difference is even. A polynomial is \emph{constant modulo 2} if it is congruent to a constant polynomial modulo 2.)
algebra