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Contests
National and Regional Contests
Czech Republic Contests
Czech and Slovak Olympiad III A
1965 Czech and Slovak Olympiad III A
1965 Czech and Slovak Olympiad III A
Part of
Czech and Slovak Olympiad III A
Subcontests
(4)
4
1
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Water in a hollow cube
Consider a container of a hollow cube
A
B
G
C
D
E
P
F
ABGCDEPF
A
BGC
D
EPF
(where
A
B
G
C
ABGC
A
BGC
,
D
E
P
F
DEPF
D
EPF
are squares and
A
D
∥
B
E
∥
G
P
∥
C
F
AD\parallel BE\parallel GP\parallel CF
A
D
∥
BE
∥
GP
∥
CF
). The cube is placed on a table in a way that the space diagonal
A
P
=
1
AP=1
A
P
=
1
is perpendicular to the table. Then, water is poured into the cube. Denote
x
x
x
the length of part of
A
P
AP
A
P
submerged in water. Determine the volume of water
y
y
y
in terms of
x
x
x
when a)
0
<
x
≤
1
3
0 < x \leq\frac13
0
<
x
≤
3
1
, b)
1
3
<
x
≤
1
2
\frac13 < x \leq\frac12
3
1
<
x
≤
2
1
.
3
1
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Parametrized radicals
Find all real roots
x
x
x
of the equation
x
2
−
2
x
−
1
+
x
2
+
2
x
−
1
=
p
,
\sqrt{x^2-2x-1}+\sqrt{x^2+2x-1}=p,
x
2
−
2
x
−
1
+
x
2
+
2
x
−
1
=
p
,
where
p
p
p
is a real parameter.
2
1
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Construction of a right triangle
Line segment
A
M
=
d
>
0
AM=d>0
A
M
=
d
>
0
is given in the plane. Furthermore, a positive number
v
v
v
is given. Construct a right triangle
A
B
C
ABC
A
BC
with hypotenuse
A
B
AB
A
B
, altitude to the hypotenuse of the length
v
v
v
and the leg
B
C
BC
BC
being divided by
M
M
M
in ration
M
B
/
M
C
=
2
/
3
MB/MC=2/3
MB
/
MC
=
2/3
. Discuss conditions of solvability in terms of
d
,
v
d, v
d
,
v
.
1
1
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Divisibility by 19
Show that the number
5
2
n
+
1
2
n
+
2
+
3
n
+
2
2
2
n
+
1
5^{2n+1}2^{n+2}+3^{n+2}2^{2n+1}
5
2
n
+
1
2
n
+
2
+
3
n
+
2
2
2
n
+
1
is divisible by
19
19
19
for every non-negative integer
n
n
n
.