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Contests
National and Regional Contests
Czech Republic Contests
Czech and Slovak Olympiad III A
2018 Czech and Slovak Olympiad III A
2018 Czech and Slovak Olympiad III A
Part of
Czech and Slovak Olympiad III A
Subcontests
(6)
1
1
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$k$-great groups of people
In a group of people, there are some mutually friendly pairs. For positive integer
k
≥
3
k\ge3
k
≥
3
we say the group is
k
k
k
-great, if every (unordered)
k
k
k
-tuple of people from the group can be seated around a round table it the way that all pairs of neighbors are mutually friendly. (Since this was the 67th year of CZE/SVK MO,) show that if the group is 6-great, then it is 7-great as well. Bonus (not included in the competition): Determine all positive integers
k
≥
3
k\ge3
k
≥
3
for which, if the group is
k
k
k
-great, then it is
(
k
+
1
)
(k+1)
(
k
+
1
)
-great as well.
3
1
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Circumcenters
In triangle
A
B
C
ABC
A
BC
let be
D
D
D
an intersection of
B
C
BC
BC
and the
A
A
A
-angle bisector. Denote
E
,
F
E,F
E
,
F
the circumcenters of
A
B
D
ABD
A
B
D
and
A
C
D
ACD
A
C
D
respectively. Assuming that the circumcenter of
A
E
F
AEF
A
EF
lies on the line
B
C
BC
BC
what is the possible size of the angle
B
A
C
BAC
B
A
C
?
5
1
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Isosceles trapezoid
Let
A
B
C
D
ABCD
A
BC
D
an isosceles trapezoid with the longer base
A
B
AB
A
B
. Denote
I
I
I
the incenter of
Δ
A
B
C
\Delta ABC
Δ
A
BC
and
J
J
J
the excenter relative to the vertex
C
C
C
of
Δ
A
C
D
\Delta ACD
Δ
A
C
D
. Show that the lines
I
J
IJ
I
J
and
A
B
AB
A
B
are parallel.
6
1
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3-coloring of numbers 1, 2, ..., n
Determine the least positive integer
n
n
n
with the following property – for every 3-coloring of numbers
1
,
2
,
…
,
n
1,2,\ldots,n
1
,
2
,
…
,
n
there are two (different) numbers
a
,
b
a,b
a
,
b
of the same color such that
∣
a
−
b
∣
|a-b|
∣
a
−
b
∣
is a perfect square.
4
1
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Perfect square under odd conditions
Let
a
,
b
,
c
a,b,c
a
,
b
,
c
be integers which are lengths of sides of a triangle,
gcd
(
a
,
b
,
c
)
=
1
\gcd(a,b,c)=1
g
cd
(
a
,
b
,
c
)
=
1
and all the values \frac{a^2+b^2-c^2}{a+b-c}, \frac{b^2+c^2-a^2}{b+c-a}, \frac{c^2+a^2-b^2}{c+a-b} are integers as well. Show that
(
a
+
b
−
c
)
(
b
+
c
−
a
)
(
c
+
a
−
b
)
(a+b-c)(b+c-a)(c+a-b)
(
a
+
b
−
c
)
(
b
+
c
−
a
)
(
c
+
a
−
b
)
or
2
(
a
+
b
−
c
)
(
b
+
c
−
a
)
(
c
+
a
−
b
)
2(a+b-c)(b+c-a)(c+a-b)
2
(
a
+
b
−
c
)
(
b
+
c
−
a
)
(
c
+
a
−
b
)
is a perfect square.
2
1
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All possible values
Let
x
,
y
,
z
x,y,z
x
,
y
,
z
be real numbers such that the numbers \frac{1}{|x^2+2yz|}, \frac{1}{|y^2+2zx|}, \frac{1}{|z^2+2xy|} are lengths of sides of a (non-degenerate) triangle. Determine all possible values of
x
y
+
y
z
+
z
x
xy+yz+zx
x
y
+
yz
+
z
x
.