MathDB
Problems
Contests
National and Regional Contests
Poland Contests
Polish MO Finals
1985 Polish MO Finals
5
5
Part of
1985 Polish MO Finals
Problems
(1)
p(cos t, sin t) = 0, p(x,y) = (x^2 + y^2 - 1) q(x,y), polynomials
Source: 1985 Polish MO Finals p5
1/21/2020
p
(
x
,
y
)
p(x,y)
p
(
x
,
y
)
is a polynomial such that
p
(
c
o
s
t
,
s
i
n
t
)
=
0
p(cos t, sin t) = 0
p
(
cos
t
,
s
in
t
)
=
0
for all real
t
t
t
. Show that there is a polynomial
q
(
x
,
y
)
q(x,y)
q
(
x
,
y
)
such that
p
(
x
,
y
)
=
(
x
2
+
y
2
ā
1
)
q
(
x
,
y
)
p(x,y) = (x^2 + y^2 - 1) q(x,y)
p
(
x
,
y
)
=
(
x
2
+
y
2
ā
1
)
q
(
x
,
y
)
.
polynomial
algebra