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Contests
National and Regional Contests
Czech Republic Contests
Czech and Slovak Olympiad III A
1962 Czech and Slovak Olympiad III A
1962 Czech and Slovak Olympiad III A
Part of
Czech and Slovak Olympiad III A
Subcontests
(4)
4
1
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Light beam reflections in a circle
Consider a circle
k
k
k
with center
S
S
S
and radius
r
r
r
. Let a point
A
≠
S
A\neq S
A
=
S
be given with
S
A
=
d
<
r
SA=d<r
S
A
=
d
<
r
. Consider a light ray emitted at point
A
A
A
, reflected at point
B
∈
k
B\in k
B
∈
k
, further reflected in point
C
∈
k
C\in k
C
∈
k
, which then passes through the original point
A
A
A
. Compute the sinus of convex angle
S
A
B
SAB
S
A
B
in terms of
d
,
r
d,r
d
,
r
and discuss conditions of solvability.
3
1
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Constant length of segments in space
Let skew lines
P
M
,
Q
N
PM, QN
PM
,
QN
be given such that
P
M
⊥
P
Q
⊥
Q
N
PM\perp PQ\perp QN
PM
⊥
PQ
⊥
QN
. Let a plane
σ
⊥
P
Q
\sigma\perp PQ
σ
⊥
PQ
containing the midpoint
O
O
O
of segment
P
Q
PQ
PQ
be given and in it a circle
k
k
k
with center
O
O
O
and given radius
r
r
r
. Consider all segments
X
Y
XY
X
Y
with endpoint
X
,
Y
X, Y
X
,
Y
on lines
P
M
,
Q
N
PM, QN
PM
,
QN
, respectively, which contain a point of
k
k
k
. Show that segments
X
Y
XY
X
Y
have the same length. Find the locus of all such points
X
X
X
.
2
1
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Set given by trigonometric inequality
Determine the set of all points
(
x
,
y
)
(x,y)
(
x
,
y
)
in two-dimensional cartesian coordinate system such that \begin{align*}0\le &\,x\le\frac{\pi}{2}, \\ \sqrt{1-\sin 2x}-\sqrt{1+\sin 2x}\le &\,y\le\sqrt{1-\cos2x}-\sqrt{1+\cos2x}.\end{align*} Draw a picture of the set.
1
1
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Squared prime
Determine all integers
x
x
x
such that
2
x
2
−
x
−
36
2x^2-x-36
2
x
2
−
x
−
36
is a perfect square of a prime.