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Serbia National Math Olympiad
2018 Serbia National Math Olympiad
4
Two variable polynomial divisibility
Two variable polynomial divisibility
Source: Serbia MO 2018 P4
April 2, 2018
algebra
polynomial
Olympiad
Problem Statement
Prove that there exists a uniqe
P
(
x
)
P(x)
P
(
x
)
polynomial with real coefficients such that\\
x
y
−
x
−
y
∣
(
x
+
y
)
1000
−
P
(
x
)
−
P
(
y
)
xy-x-y|(x+y)^{1000}-P(x)-P(y)
x
y
−
x
−
y
∣
(
x
+
y
)
1000
−
P
(
x
)
−
P
(
y
)
for all real
x
,
y
x,y
x
,
y
.
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