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Korea National Olympiad
1997 Korea National Olympiad
6
Polynomial Problem
Polynomial Problem
Source: 1997 Korea National Olympiad #6
March 18, 2018
algebra
polynomial
Problem Statement
Find all polynomial
P
(
x
,
y
)
P(x,y)
P
(
x
,
y
)
for any reals
x
,
y
x,y
x
,
y
such that (i)
x
100
+
y
100
≤
P
(
x
,
y
)
≤
101
(
x
100
+
y
100
)
x^{100}+y^{100}\le P(x,y)\le 101(x^{100}+y^{100})
x
100
+
y
100
≤
P
(
x
,
y
)
≤
101
(
x
100
+
y
100
)
(ii)
(
x
−
y
)
P
(
x
,
y
)
=
(
x
−
1
)
P
(
x
,
1
)
+
(
1
−
y
)
P
(
1
,
y
)
.
(x-y)P(x,y)=(x-1)P(x,1)+(1-y)P(1,y).
(
x
−
y
)
P
(
x
,
y
)
=
(
x
−
1
)
P
(
x
,
1
)
+
(
1
−
y
)
P
(
1
,
y
)
.
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