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Problems
Contests
National and Regional Contests
Czech Republic Contests
Czech and Slovak Olympiad III A
1958 Czech and Slovak Olympiad III A
1958 Czech and Slovak Olympiad III A
Part of
Czech and Slovak Olympiad III A
Subcontests
(4)
4
1
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3D loci of points
Consider positive numbers
d
,
v
d,v
d
,
v
such that
d
>
v
d>v
d
>
v
. Moreover, consider two perpendicular skew lines
p
,
q
p,q
p
,
q
of distance
v
v
v
(that is direction vectors of both lines are orthogonal and
min
X
∈
p
,
Y
∈
q
X
Y
=
v
\min_{X\in p,Y\in q}XY = v
min
X
∈
p
,
Y
∈
q
X
Y
=
v
). Finally, consider all line segments
P
Q
PQ
PQ
such that
P
∈
p
,
Q
∈
q
,
P
Q
=
d
P\in p, Q\in q, PQ=d
P
∈
p
,
Q
∈
q
,
PQ
=
d
. a) Find the locus of all points
P
P
P
. b) Find the locus of all midpoints of segments
P
Q
PQ
PQ
.
3
1
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Trigonometric inequality
Find all real
x
x
x
such that
2
+
5
2
cos
x
≤
sin
x
.
\sqrt{2+\frac{5}{2}\cos x}\leq\sin x.
2
+
2
5
cos
x
≤
sin
x
.
2
1
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Constructive geometry
Construct a triangle
A
B
C
ABC
A
BC
given the magnitude of the angle
B
C
A
BCA
BC
A
and lengths of height
h
c
h_c
h
c
and median
m
c
m_c
m
c
. Discuss conditions of solvability.
1
1
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Equation with parameter
Find all real solutions of equation
x
+
2
p
−
x
2
=
8
x + \sqrt{2p - x^2} = 8
x
+
2
p
−
x
2
=
8
with real parameter
p
p
p
.