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Contests
National and Regional Contests
Czech Republic Contests
Czech and Slovak Olympiad III A
1953 Czech and Slovak Olympiad III A
1953 Czech and Slovak Olympiad III A
Part of
Czech and Slovak Olympiad III A
Subcontests
(4)
1
1
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Locus in complex plane
Find the locus of all numbers
z
∈
C
z\in\mathbb C
z
∈
C
in complex plane satisfying
z
+
z
ˉ
=
a
⋅
∣
z
∣
,
z+\bar z=a\cdot|z|,
z
+
z
ˉ
=
a
⋅
∣
z
∣
,
where
a
∈
R
a\in\mathbb R
a
∈
R
is given.
4
1
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Locus of midpoints in space
Consider skew lines
a
,
b
a,b
a
,
b
and a plane
ρ
\rho
ρ
that intersect both of the lines (but does not contain any of them). Choose such points
X
∈
a
,
Y
∈
b
X\in a,Y\in b
X
∈
a
,
Y
∈
b
that
X
Y
∥
ρ
.
XY\parallel\rho.
X
Y
∥
ρ
.
Find the locus of midpoints
M
M
M
of all segments
X
Y
,
XY,
X
Y
,
when
X
X
X
moves along line
a
a
a
.
3
1
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Well-known inequality
Prove that the inequality
(
a
1
+
⋯
+
a
n
)
(
1
a
1
+
⋯
+
1
a
n
)
≥
n
2
\left(a_1+\cdots+a_n\right)\left(\frac{1}{a_1}+\cdots+\frac{1}{a_n}\right)\ge n^2
(
a
1
+
⋯
+
a
n
)
(
a
1
1
+
⋯
+
a
n
1
)
≥
n
2
holds for any positive numbers
a
1
,
…
,
a
n
a_1,\ldots,a_n
a
1
,
…
,
a
n
and determine when equality occurs.
2
1
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Angles of a triangle
Let
α
,
β
,
γ
\alpha,\beta,\gamma
α
,
β
,
γ
be angles of a triangle. Two of them can be expressed using an auxiliary angle
φ
\varphi
φ
in a way that \alpha=\varphi+\frac\pi4, \beta=\pi-3\varphi. Show that
α
>
γ
.
\alpha>\gamma.
α
>
γ
.