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National and Regional Contests
Ireland Contests
Ireland National Math Olympiad
2014 Irish Math Olympiad
9
9
Part of
2014 Irish Math Olympiad
Problems
(1)
f(x) = (x - a_0)g(x)= x^{n+1} + b_1x^n + b_2x^{n-1}+...+ b_nx + b_{n+1}
Source: Irmo 2014 p2 q9
9/15/2018
Let
n
n
n
be a positive integer and
a
1
,
.
.
.
,
a
n
a_1,...,a_n
a
1
,
...
,
a
n
be positive real numbers. Let
g
(
x
)
g(x)
g
(
x
)
denote the product
(
x
+
a
1
)
⋅
.
.
.
⋅
(
x
+
a
n
)
(x + a_1)\cdot ... \cdot (x + a_n)
(
x
+
a
1
)
⋅
...
⋅
(
x
+
a
n
)
. Let
a
0
a_0
a
0
be a real number and let
f
(
x
)
=
(
x
−
a
0
)
g
(
x
)
=
x
n
+
1
+
b
1
x
n
+
b
2
x
n
−
1
+
.
.
.
+
b
n
x
+
b
n
+
1
f(x) = (x - a_0)g(x)= x^{n+1} + b_1x^n + b_2x^{n-1}+...+ b_nx + b_{n+1}
f
(
x
)
=
(
x
−
a
0
)
g
(
x
)
=
x
n
+
1
+
b
1
x
n
+
b
2
x
n
−
1
+
...
+
b
n
x
+
b
n
+
1
. Prove that all the coeffcients
b
1
,
b
2
,
.
.
.
,
b
n
+
1
b_1,b_2,..., b_{n+1}
b
1
,
b
2
,
...
,
b
n
+
1
of the polynomial
f
(
x
)
f(x)
f
(
x
)
are negative if and only if
a
0
>
a
1
+
a
2
+
.
.
.
+
a
n
a_0 > a_1 + a_2 +...+ a_n
a
0
>
a
1
+
a
2
+
...
+
a
n
.
polynomial
algebra