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Problems
Contests
National and Regional Contests
Bulgaria Contests
Bulgaria National Olympiad
1998 Bulgaria National Olympiad
2
2
Part of
1998 Bulgaria National Olympiad
Problems
(2)
Prove that P(x,y)=P(y,x)
Source: Bulgaria 1998 (round 4)
9/24/2013
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
.
algebra
polynomial
algebra unsolved
A is odd
Source: bulgaria 1998
8/12/2014
let m and n be natural numbers such that:
3
m
∣
(
m
+
3
)
n
+
1
3m|(m+3)^n+1
3
m
∣
(
m
+
3
)
n
+
1
Prove that
(
m
+
3
)
n
+
1
3
m
\frac{(m+3)^n+1}{3m}
3
m
(
m
+
3
)
n
+
1
is odd
modular arithmetic
number theory