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Final Mathematical Cup
2019 Final Mathematical Cup
2
2
Part of
2019 Final Mathematical Cup
Problems
(1)
a_0+a_1+a_2+...+a_n is divisible by 2 , P(m) =2018, a_i >0, m=(-1+\sqrt{17})/2
Source: 1st Final Mathematical Cup 2019 FMC , juniors p2
10/6/2020
Let
m
=
−
1
+
17
2
m=\frac{-1+\sqrt{17}}{2}
m
=
2
−
1
+
17
. Let the polynomial
P
(
x
)
=
a
n
x
n
+
a
n
−
1
x
n
−
1
+
.
.
.
+
a
1
x
+
a
0
P(x)=a_nx^n+a_{n-1}x^{n-1}+...+a_1x+a_0
P
(
x
)
=
a
n
x
n
+
a
n
−
1
x
n
−
1
+
...
+
a
1
x
+
a
0
is given, where
n
n
n
is a positive integer, the coefficients
a
0
,
a
1
,
a
2
,
.
.
.
,
a
n
a_0,a_1,a_2,...,a_n
a
0
,
a
1
,
a
2
,
...
,
a
n
are positive integers and
P
(
m
)
=
2018
P(m) =2018
P
(
m
)
=
2018
. Prove that the sum
a
0
+
a
1
+
a
2
+
.
.
.
+
a
n
a_0+a_1+a_2+...+a_n
a
0
+
a
1
+
a
2
+
...
+
a
n
is divisible by
2
2
2
.
Sum
polynomial
algebra