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National and Regional Contests
Czech Republic Contests
Czech and Slovak Olympiad III A
1959 Czech and Slovak Olympiad III A
1959 Czech and Slovak Olympiad III A
Part of
Czech and Slovak Olympiad III A
Subcontests
(4)
3
1
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Coaxial right circular conical frustums
Consider a piece of material in the shape of a right circular conical frustum with radii
R
,
r
,
R
>
r
R,r,R>r
R
,
r
,
R
>
r
. A cavity in the shape of another coaxial right circular conical frustum was drilled into the material (see the picture). That way only half of the original volume of material remained. Compute radii
R
′
,
r
′
R',r'
R
′
,
r
′
of the cavity. Decide for which ratio
R
/
r
R/r
R
/
r
the problem has a solution. https://cdn.artofproblemsolving.com/attachments/b/f/12f579458b7cf0fc31849b319e6f58e50b0363.png
4
1
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System of trigonometric equations
Find all pair
(
x
,
y
)
(x, y)
(
x
,
y
)
in degrees such that \begin{align*} &\sin (x + 150^\circ) = \cos (y - 75^\circ), \\ &\cos x + \sin (y - 225^\circ) + \frac{\sqrt3}{2} = 0. \end{align*}
2
1
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Positive symmetric expressions
Let
a
,
b
,
c
a, b, c
a
,
b
,
c
be real numbers such that
a
+
b
+
c
>
0
a+b+c > 0
a
+
b
+
c
>
0
,
a
b
+
b
c
+
c
a
>
0
ab+bc+ca > 0
ab
+
b
c
+
c
a
>
0
,
a
b
c
>
0
abc > 0
ab
c
>
0
. Show that
a
,
b
,
c
a, b, c
a
,
b
,
c
are all positive.
1
1
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Construction of triangle
Construct a triangle
A
B
C
ABC
A
BC
with the right angle at vertex
C
C
C
given lengths of its medians
m
a
m_a
m
a
,
m
b
m_b
m
b
. Discuss conditions of solvability.