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Bulgaria National Olympiad
1998 Bulgaria National Olympiad
2
Prove that P(x,y)=P(y,x)
Prove that P(x,y)=P(y,x)
Source: Bulgaria 1998 (round 4)
September 24, 2013
algebra
polynomial
algebra unsolved
Problem Statement
The polynomials
P
n
(
x
,
y
)
,
n
=
1
,
2
,
.
.
.
P_n(x,y), n=1,2,...
P
n
(
x
,
y
)
,
n
=
1
,
2
,
...
are defined by
P
1
(
x
,
y
)
=
1
,
P
n
+
1
(
x
,
y
)
=
(
x
+
y
−
1
)
(
y
+
1
)
P
n
(
x
,
y
+
2
)
+
(
y
−
y
2
)
P
n
(
x
,
y
)
P_1(x,y)=1, P_{n+1}(x,y)=(x+y-1)(y+1)P_n(x,y+2)+(y-y^2)P_n(x,y)
P
1
(
x
,
y
)
=
1
,
P
n
+
1
(
x
,
y
)
=
(
x
+
y
−
1
)
(
y
+
1
)
P
n
(
x
,
y
+
2
)
+
(
y
−
y
2
)
P
n
(
x
,
y
)
Prove that
P
n
(
x
,
y
)
=
P
n
(
y
,
x
)
P_{n}(x,y)=P_{n}(y,x)
P
n
(
x
,
y
)
=
P
n
(
y
,
x
)
for all
x
,
y
∈
R
x,y \in \mathbb{R}
x
,
y
∈
R
and
n
n
n
.
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