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Czech-Polish-Slovak Match
2014 Czech-Polish-Slovak Match
1
1
Part of
2014 Czech-Polish-Slovak Match
Problems
(1)
trigonometric existence of triangle given equation w sides
Source: Czech-Polish-Slovak Match 2014 day 1 P1
9/29/2017
Prove that if the positive real numbers
a
,
b
,
c
a, b, c
a
,
b
,
c
satisfy the equation
a
4
+
b
4
+
c
4
+
4
a
2
b
2
c
2
=
2
(
a
2
b
2
+
a
2
c
2
+
b
2
c
2
)
,
a^4 + b^4 + c^4 + 4a^2b^2c^2 = 2 (a^2b^2 + a^2c^2 + b^2c^2),
a
4
+
b
4
+
c
4
+
4
a
2
b
2
c
2
=
2
(
a
2
b
2
+
a
2
c
2
+
b
2
c
2
)
,
then there is a triangle
A
B
C
ABC
A
BC
with internal angles
α
,
β
,
γ
\alpha, \beta, \gamma
α
,
β
,
γ
such that
sin
α
=
a
,
sin
β
=
b
,
sin
γ
=
c
.
\sin \alpha = a, \qquad \sin \beta = b, \qquad \sin \gamma= c.
sin
α
=
a
,
sin
β
=
b
,
sin
γ
=
c
.
Trigonometric Equations
polynomial equation
Triangle
geometry