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Korea Junior Mathematics Olympiad
2010 Korea Junior Math Olympiad
5
sin^2 x + sin^2 y + sin^2 z < 1 if tan x + tan y + tan z = 2
sin^2 x + sin^2 y + sin^2 z < 1 if tan x + tan y + tan z = 2
Source: KJMO 2010 p5
May 2, 2019
trigonometry
Trigonometric inequality
algebra
inequalities
Problem Statement
If reals
x
,
y
,
z
x, y, z
x
,
y
,
z
satises
t
a
n
x
+
t
a
n
y
+
t
a
n
z
=
2
tan x + tan y + tan z = 2
t
an
x
+
t
an
y
+
t
an
z
=
2
and
0
<
x
,
y
,
z
<
π
2
.
0 < x, y,z < \frac{\pi}{2}.
0
<
x
,
y
,
z
<
2
π
.
Prove that
s
i
n
2
x
+
s
i
n
2
y
+
s
i
n
2
z
<
1.
sin^2 x + sin^2 y + sin^2 z < 1.
s
i
n
2
x
+
s
i
n
2
y
+
s
i
n
2
z
<
1.
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