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National and Regional Contests
Vietnam Contests
Vietnam National Olympiad
1981 Vietnam National Olympiad
2
find all m
find all m
Source:
June 22, 2006
algebra
polynomial
calculus
integration
number theory unsolved
number theory
Problem Statement
Consider the polynomials
f
(
p
)
=
p
12
−
p
11
+
3
p
10
+
11
p
3
−
p
2
+
23
p
+
30
;
f(p) = p^{12} - p^{11} + 3p^{10} + 11p^3 - p^2 + 23p + 30;
f
(
p
)
=
p
12
−
p
11
+
3
p
10
+
11
p
3
−
p
2
+
23
p
+
30
;
g
(
p
)
=
p
3
+
2
p
+
m
.
g(p) = p^3 + 2p + m.
g
(
p
)
=
p
3
+
2
p
+
m
.
Find all integral values of
m
m
m
for which
f
f
f
is divisible by
g
g
g
.
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