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Problems
Contests
National and Regional Contests
Czech Republic Contests
Czech and Slovak Olympiad III A
2014 Czech and Slovak Olympiad III A
2014 Czech and Slovak Olympiad III A
Part of
Czech and Slovak Olympiad III A
Subcontests
(6)
5
1
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3 circles given, collinearity wanted
Given is the acute triangle
A
B
C
ABC
A
BC
. Let us denote
k
k
k
a circle with diameter
A
B
AB
A
B
. Another circle, tangent to
A
B
AB
A
B
at point
A
A
A
and passing through point
C
C
C
intersects the circle
k
k
k
at point
P
,
P
≠
A
P, P \ne A
P
,
P
=
A
. Another circle which touches AB at point
B
B
B
and passes point
C
C
C
, intersects the circle
k
k
k
at point
Q
,
Q
≠
B
Q, Q \ne B
Q
,
Q
=
B
. Prove that the intersection of the line
A
Q
AQ
A
Q
and
B
P
BP
BP
lies on one of the sides of angle
A
C
B
ACB
A
CB
.(Peter Novotný)
4
1
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every viewer in i-th row gets to know just j viewers in j-th row in cinema
234
234
234
viewers came to the cinema. Determine for which
n
≥
4
n \ge 4
n
≥
4
the viewers could be can be arranged in
n
n
n
rows so that every viewer in
i
i
i
-th row gets to know just
j
j
j
viewers in
j
j
j
-th row for any
i
,
j
∈
{
1
,
2
,
.
.
.
,
n
}
,
i
≠
j
i, j \in \{1, 2,... , n\}, i\ne j
i
,
j
∈
{
1
,
2
,
...
,
n
}
,
i
=
j
. (The relationship of acquaintance is mutual.)(Tomáš Jurík)
3
1
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Cutting the chessboard
Suppose we have a
8
×
8
8\times8
8
×
8
chessboard. Each edge have a number, corresponding to number of possibilities of dividing this chessboard into
1
×
2
1\times2
1
×
2
domino pieces, such that this edge is part of this division. Find out the last digit of the sum of all these numbers.(Day 1, 3rd problem author: Michal Rolínek)
2
1
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Set of points
A segment
A
B
AB
A
B
is given in (Euclidean) plane. Consider all triangles
X
Y
Z
XYZ
X
Y
Z
such, that
X
X
X
is an inner point of
A
B
AB
A
B
, triangles
X
B
Y
XBY
XB
Y
and
X
Z
A
XZA
XZ
A
are similar (in this order of vertices), and points
A
,
B
,
Y
,
Z
A, B, Y, Z
A
,
B
,
Y
,
Z
lie on a circle in this order. Find the locus of midpoints of all such segments
Y
Z
YZ
Y
Z
.(Day 1, 2nd problem authors: Michal Rolínek, Jaroslav Švrček)
1
1
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Divisors
Let be
n
n
n
a positive integer. Denote all its (positive) divisors as
1
=
d
1
<
d
2
<
⋯
<
d
k
−
1
<
d
k
=
n
1=d_1<d_2<\cdots<d_{k-1}<d_k=n
1
=
d
1
<
d
2
<
⋯
<
d
k
−
1
<
d
k
=
n
. Find all values of
n
n
n
satisfying
d
5
−
d
3
=
50
d_5-d_3=50
d
5
−
d
3
=
50
and
11
d
5
+
8
d
7
=
3
n
11d_5+8d_7=3n
11
d
5
+
8
d
7
=
3
n
.(Day 1, 1st problem author: Matúš Harminc)
6
1
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Inequality
For arbitrary non-negative numbers
a
a
a
and
b
b
b
prove inequality
a
b
2
+
1
+
b
a
2
+
1
≥
a
+
b
a
b
+
1
\frac{a}{\sqrt{b^2+1}}+\frac{b}{\sqrt{a^2+1}}\ge\frac{a+b}{\sqrt{ab+1}}
b
2
+
1
a
+
a
2
+
1
b
≥
ab
+
1
a
+
b
, and find, where equality occurs.(Day 2, 6th problem authors: Tomáš Jurík, Jaromír Šimša)