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National and Regional Contests
Sweden Contests
Swedish Mathematical Competition
1973 Swedish Mathematical Competition
5
5
Part of
1973 Swedish Mathematical Competition
Problems
(1)
f(x)q(x)-p(x) of form \sum\limits_{2n<k\leq 3n} (a^k + x^k) have same p(x)/q(x)
Source: 1973 Swedish Mathematical Competition p5
3/26/2021
f
(
x
)
f(x)
f
(
x
)
is a polynomial of degree
2
n
2n
2
n
. Show that all polynomials
p
(
x
)
p(x)
p
(
x
)
,
q
(
x
)
q(x)
q
(
x
)
of degree at most
n
n
n
such that
f
(
x
)
q
(
x
)
−
p
(
x
)
f(x)q(x)-p(x)
f
(
x
)
q
(
x
)
−
p
(
x
)
has the form
∑
2
n
<
k
≤
3
n
(
a
k
+
x
k
)
\sum\limits_{2n<k\leq 3n} (a^k + x^k)
2
n
<
k
≤
3
n
∑
(
a
k
+
x
k
)
have the same
p
(
x
)
/
q
(
x
)
p(x)/q(x)
p
(
x
)
/
q
(
x
)
.
algebra
polynomial