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Contests
National and Regional Contests
Czech Republic Contests
Czech and Slovak Olympiad III A
1961 Czech and Slovak Olympiad III A
1961 Czech and Slovak Olympiad III A
Part of
Czech and Slovak Olympiad III A
Subcontests
(4)
4
1
Hide problems
Extrema of area of triangle
Consider a unit square
A
B
C
D
ABCD
A
BC
D
and a (variable) equilateral triangle
X
Y
Z
XYZ
X
Y
Z
such that
X
,
Z
X, Z
X
,
Z
lie on rays
A
B
,
D
C
,
AB, DC,
A
B
,
D
C
,
respectively, and
Y
Y
Y
lies on segment
A
D
AD
A
D
. Compute the area of triangle
X
Y
Z
XYZ
X
Y
Z
in terms of
x
=
A
X
x=AX
x
=
A
X
and determine its maximum and minimum.
3
1
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Cyclists on circular track
Two cyclists start moving simultaneously in opposite directions on a circular circuit. The first cyclist maintains a constant speed
c
1
c_1
c
1
meters per second, the second maintains
c
2
c_2
c
2
meters per second. How many times did they meet when the first cyclist completed
n
n
n
laps? Compute for
c
1
=
10
,
c
2
=
7
,
n
=
11
c_1=10,c_2=7,n=11
c
1
=
10
,
c
2
=
7
,
n
=
11
.
2
1
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Square construction
Let a right isosceles triangle
A
P
Q
APQ
A
PQ
with the hypotenuse
A
P
AP
A
P
be given in plane. Construct such a square
A
B
C
D
ABCD
A
BC
D
that the lines
B
C
,
C
D
BC, CD
BC
,
C
D
contain points
P
,
Q
,
P, Q,
P
,
Q
,
respectively. Compute the length of side
A
B
=
b
AB = b
A
B
=
b
in terms of
A
Q
=
a
AQ=a
A
Q
=
a
.
1
1
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Non-decresing sequence of consecutive positive integers
Consider an infinite sequence
1
,
2
,
2
,
3
,
3
,
3
,
4
,
4
,
4
,
4
,
5
,
…
,
n
,
…
,
n
⏟
n
times
,
…
.
1, 2, 2, 3, 3, 3, 4, 4, 4, 4, 5, \ldots, \underbrace{n,\ldots,n}_{n\text{ times}},\ldots.
1
,
2
,
2
,
3
,
3
,
3
,
4
,
4
,
4
,
4
,
5
,
…
,
n
times
n
,
…
,
n
,
…
.
Find the 1000th term of the sequence.