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Problems
Contests
National and Regional Contests
Czech Republic Contests
Czech and Slovak Olympiad III A
1956 Czech and Slovak Olympiad III A
1956 Czech and Slovak Olympiad III A
Part of
Czech and Slovak Olympiad III A
Subcontests
(4)
4
1
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Locus given by moving on semicircle
Let a semicircle
A
B
AB
A
B
be given and let
X
X
X
be an inner point of the arc. Consider a point
Y
Y
Y
on ray
X
A
XA
X
A
such that
X
Y
=
X
B
XY=XB
X
Y
=
XB
. Find the locus of all points
Y
Y
Y
when
X
X
X
moves on the arc
A
B
AB
A
B
(excluding the endpoints).
2
1
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Construction of parallelogram in space
In a given plane
ϱ
\varrho
ϱ
consider a convex quadrilateral
A
B
C
D
ABCD
A
BC
D
and denote
E
=
A
C
∩
B
D
.
E=AC\cap BD.
E
=
A
C
∩
B
D
.
Moreover, consider a point
V
∉
ϱ
V\notin\varrho
V
∈
/
ϱ
. On rays
V
A
,
V
B
,
V
C
,
V
D
VA,VB,VC,VD
V
A
,
V
B
,
V
C
,
V
D
find points
A
′
,
B
′
,
C
′
,
D
′
A',B',C',D'
A
′
,
B
′
,
C
′
,
D
′
respectively such that
E
,
A
′
,
B
′
,
C
′
,
D
′
E,A',B',C',D'
E
,
A
′
,
B
′
,
C
′
,
D
′
are coplanar and
A
′
B
′
C
′
D
′
A'B'C'D'
A
′
B
′
C
′
D
′
is a parallelogram. Discuss conditions of solvability.
3
1
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Non-linear system
Find all real pairs
x
,
y
x,y
x
,
y
such that \begin{align*} x-|y+1|&=1, \\ x^2+y&=10. \end{align*}
1
1
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Trigonometric system
Find all
x
,
y
∈
(
0
,
π
2
)
x,y\in\left(0,\frac{\pi}{2}\right)
x
,
y
∈
(
0
,
2
π
)
such that \begin{align*} \frac{\cos x}{\cos y}&=2\cos^2 y, \\ \frac{\sin x}{\sin y}&=2\sin^2 y. \end{align*}