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12
Math Prize 2016 Problem 12
Math Prize 2016 Problem 12
Source:
September 12, 2016
Math Prize for Girls
polynomial
Problem Statement
Let
b
1
b_1
b
1
,
b
2
b_2
b
2
,
b
3
b_3
b
3
,
c
1
c_1
c
1
,
c
2
c_2
c
2
, and
c
3
c_3
c
3
be real numbers such that for every real number
x
x
x
, we have
x
6
−
x
5
+
x
4
−
x
3
+
x
2
−
x
+
1
=
(
x
2
+
b
1
x
+
c
1
)
(
x
2
+
b
2
x
+
c
2
)
(
x
2
+
b
3
x
+
c
3
)
.
x^6 - x^5 + x^4 - x^3 + x^2 - x + 1 = (x^2 + b_1 x + c_1)(x^2 + b_2 x + c_2)(x^2 + b_3 x + c_3).
x
6
−
x
5
+
x
4
−
x
3
+
x
2
−
x
+
1
=
(
x
2
+
b
1
x
+
c
1
)
(
x
2
+
b
2
x
+
c
2
)
(
x
2
+
b
3
x
+
c
3
)
.
Compute
b
1
c
1
+
b
2
c
2
+
b
3
c
3
b_1 c_1 + b_2 c_2 + b_3 c_3
b
1
c
1
+
b
2
c
2
+
b
3
c
3
.
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