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2022 CHMMC Winter (2022-23)
8
CHMMC 2022 Winter / 2022-23 Team #8
CHMMC 2022 Winter / 2022-23 Team #8
Source:
August 10, 2023
algebra
Problem Statement
Suppose
a
3
x
3
−
x
2
+
a
1
x
−
7
=
0
a_3x^3 - x^2 + a_1x - 7 = 0
a
3
x
3
−
x
2
+
a
1
x
−
7
=
0
is a cubic polynomial in x whose roots
α
,
β
,
γ
\alpha,\beta, \gamma
α
,
β
,
γ
are positive real numbers satisfying
225
α
2
α
2
+
7
=
144
β
2
β
2
+
7
=
100
γ
2
γ
2
+
7
.
\frac{225\alpha^2}{\alpha^2 +7}=\frac{144\beta^2}{\beta^2 +7}=\frac{100\gamma^2}{\gamma^2 +7}.
α
2
+
7
225
α
2
=
β
2
+
7
144
β
2
=
γ
2
+
7
100
γ
2
.
Find
a
1
a_1
a
1
.
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