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Contests
National and Regional Contests
Czech Republic Contests
Czech and Slovak Olympiad III A
1966 Czech and Slovak Olympiad III A
1966 Czech and Slovak Olympiad III A
Part of
Czech and Slovak Olympiad III A
Subcontests
(4)
4
1
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Triangles in space
Two triangles
A
B
C
,
A
B
D
ABC,ABD
A
BC
,
A
B
D
(with the common side
c
=
A
B
c=AB
c
=
A
B
) are given in space. Triangle
A
B
C
ABC
A
BC
is right with hypotenuse
A
B
AB
A
B
,
A
B
D
ABD
A
B
D
is equilateral. Denote
φ
\varphi
φ
the dihedral angle between planes
A
B
C
,
A
B
D
ABC,ABD
A
BC
,
A
B
D
. 1) Determine the length of
C
D
CD
C
D
in terms of
a
=
B
C
,
b
=
C
A
,
c
a=BC,b=CA,c
a
=
BC
,
b
=
C
A
,
c
and
φ
\varphi
φ
. 2) Determine all values of
φ
\varphi
φ
such that the tetrahedron
A
B
C
D
ABCD
A
BC
D
has four sides of the same length.
3
1
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Triangles with vertices on sides of square
A square
A
B
C
D
,
A
B
=
s
=
1
ABCD,AB=s=1
A
BC
D
,
A
B
=
s
=
1
is given in the plane with its center
S
S
S
. Furthermore, points
E
,
F
E,F
E
,
F
are given on the rays opposite to
C
B
,
D
A
CB,DA
CB
,
D
A
, respectively,
C
E
=
a
,
D
F
=
b
CE=a,DF=b
CE
=
a
,
D
F
=
b
. Determine all triangles
X
Y
Z
XYZ
X
Y
Z
such that
X
,
Y
,
Z
X,Y,Z
X
,
Y
,
Z
lie in this order on segments
C
D
,
A
D
,
B
C
CD,AD,BC
C
D
,
A
D
,
BC
and
E
,
S
,
F
E,S,F
E
,
S
,
F
lie on lines
X
Y
,
Y
Z
,
Z
X
XY,YZ,ZX
X
Y
,
Y
Z
,
ZX
respectively. Discuss conditions of solvability in terms of
a
,
b
,
s
a,b,s
a
,
b
,
s
and unknown
x
=
C
X
x=CX
x
=
CX
.
1
1
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Inequalities with absolute value
Consider a system of inequalities \begin{align*}y-x&\ge|x+1|-|x-1|, \\ |y&-x|-y+x\ge2.\end{align*} Draw solutions of each inequality in the plane separately and highlight solution of the system.
2
1
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n circles divide the plane, intersect in pairs, no point lies on 3
Into how many regions do
n
n
n
circles divide the plane, if each pair of circles intersects in two points and no point lies on three circles?