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Contests
National and Regional Contests
Czech Republic Contests
Czech and Slovak Olympiad III A
1954 Czech and Slovak Olympiad III A
1954 Czech and Slovak Olympiad III A
Part of
Czech and Slovak Olympiad III A
Subcontests
(4)
4
1
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Quadrilateral in 3D space
Consider a cube
A
B
C
D
A
′
B
′
C
′
D
ABCDA'B'C'D
A
BC
D
A
′
B
′
C
′
D
(with
A
B
⊥
A
A
′
∥
B
B
′
∥
C
C
′
∥
D
D
AB\perp AA'\parallel BB'\parallel CC'\parallel DD
A
B
⊥
A
A
′
∥
B
B
′
∥
C
C
′
∥
DD
). Let
X
X
X
be an inner point of the segment
A
B
AB
A
B
and denote
Y
Y
Y
the intersection of the edge
A
D
AD
A
D
and the plane
B
′
D
′
X
B'D'X
B
′
D
′
X
. (a) Let
M
=
B
′
Y
∩
D
′
X
M=B'Y\cap D'X
M
=
B
′
Y
∩
D
′
X
. Find the locus of all
M
M
M
s. (b) Determine whether there is a quadrilateral
B
′
D
′
Y
X
B'D'YX
B
′
D
′
Y
X
such that its diagonals divide each other in the ratio 1:2.
3
1
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Logarithm of $pi$
Show that
log
2
π
+
log
4
π
<
5
2
.
\log_2\pi+\log_4\pi<\frac52.
lo
g
2
π
+
lo
g
4
π
<
2
5
.
2
1
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Right-angle triange in complex plane
Let
a
,
b
a,b
a
,
b
complex numbers. Show that if the roots of the equation
z
2
+
a
z
+
b
=
0
z^2+az+b=0
z
2
+
a
z
+
b
=
0
and 0 form a triangle with the right angle at the origin, then
a
2
=
2
b
≠
0.
a^2=2b\neq0.
a
2
=
2
b
=
0.
Also determine whether the opposite implication holds.
1
1
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Equation with parameter
Solve the equation
a
x
2
+
2
(
a
−
1
)
x
+
a
−
5
=
0
ax^2+2(a-1)x+a-5=0
a
x
2
+
2
(
a
−
1
)
x
+
a
−
5
=
0
in real numbers with respect to (real) parametr
a
a
a
.