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Problems
Contests
National and Regional Contests
Vietnam Contests
Vietnam National Olympiad
2013 Vietnam National Olympiad
1
system of equations
system of equations
Source: Problem 1 VMO 2013 day 1
January 11, 2013
linear algebra
matrix
trigonometry
algebra proposed
algebra
Problem Statement
Solve with full solution:
{
(
sin
x
)
2
+
1
(
sin
x
)
2
+
(
cos
y
)
2
+
1
(
cos
y
)
2
=
20
y
x
+
y
(
sin
y
)
2
+
1
(
sin
y
)
2
+
(
cos
x
)
2
+
1
(
cos
x
)
2
=
20
x
x
+
y
\left\{\begin{matrix}\sqrt{(\sin x)^2+\frac{1}{(\sin x)^2}}+\sqrt{(\cos y)^2+\frac{1}{(\cos y)^2}}=\sqrt\frac{20y}{x+y} \\\sqrt{(\sin y)^2+\frac{1}{(\sin y)^2}}+\sqrt{(\cos x)^2+\frac{1}{(\cos x)^2}}=\sqrt\frac{20x}{x+y}\end{matrix}\right.
⎩
⎨
⎧
(
sin
x
)
2
+
(
s
i
n
x
)
2
1
+
(
cos
y
)
2
+
(
c
o
s
y
)
2
1
=
x
+
y
20
y
(
sin
y
)
2
+
(
s
i
n
y
)
2
1
+
(
cos
x
)
2
+
(
c
o
s
x
)
2
1
=
x
+
y
20
x
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