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Contests
National and Regional Contests
Czech Republic Contests
Czech and Slovak Olympiad III A
2019 Czech and Slovak Olympiad III A
2019 Czech and Slovak Olympiad III A
Part of
Czech and Slovak Olympiad III A
Subcontests
(6)
6
1
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Integral arithmetic means
Assume we can fill a table
n
×
n
n\times n
n
×
n
with all numbers
1
,
2
,
…
,
n
2
−
1
,
n
2
1,2,\ldots,n^2-1,n^2
1
,
2
,
…
,
n
2
−
1
,
n
2
in such way that arithmetic means of numbers in every row and every column is an integer. Determine all such positive integers
n
n
n
.
5
1
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Numbers of a given form
Prove that there are infinitely many integers which cannot be expressed as
2
a
+
3
b
−
5
c
2^a+3^b-5^c
2
a
+
3
b
−
5
c
for non-negative integers
a
,
b
,
c
a,b,c
a
,
b
,
c
.
4
1
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Perpendicular lines
Let be
A
B
C
ABC
A
BC
an acute-angled triangle. Consider point
P
P
P
lying on the opposite ray to the ray
B
C
BC
BC
such that
∣
A
B
∣
=
∣
B
P
∣
|AB|=|BP|
∣
A
B
∣
=
∣
BP
∣
. Similarly, consider point
Q
Q
Q
on the opposite ray to the ray
C
B
CB
CB
such that
∣
A
C
∣
=
∣
C
Q
∣
|AC|=|CQ|
∣
A
C
∣
=
∣
CQ
∣
. Denote
J
J
J
the excenter of
A
B
C
ABC
A
BC
with respect to
A
A
A
and
D
,
E
D,E
D
,
E
tangent points of this excircle with the lines
A
B
AB
A
B
and
A
C
AC
A
C
, respectively. Suppose that the opposite rays to
D
P
DP
D
P
and
E
Q
EQ
EQ
intersect in
F
≠
J
F\neq J
F
=
J
. Prove that
A
F
⊥
F
J
AF\perp FJ
A
F
⊥
F
J
.
3
1
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Perfect n-th powers
Let
a
,
b
,
c
,
n
a,b,c,n
a
,
b
,
c
,
n
be positive integers such that the following conditions hold (i) numbers
a
,
b
,
c
,
a
+
b
+
c
a,b,c,a+b+c
a
,
b
,
c
,
a
+
b
+
c
are pairwise coprime, (ii) number
(
a
+
b
)
(
b
+
c
)
(
c
+
a
)
(
a
+
b
+
c
)
(
a
b
+
b
c
+
c
a
)
(a+b)(b+c)(c+a)(a+b+c)(ab+bc+ca)
(
a
+
b
)
(
b
+
c
)
(
c
+
a
)
(
a
+
b
+
c
)
(
ab
+
b
c
+
c
a
)
is a perfect
n
n
n
-th power. Prove, that the product
a
b
c
abc
ab
c
can be expressed as a difference of two perfect
n
n
n
-th powers.
2
1
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Construction of points with given property
Let be
A
B
C
D
ABCD
A
BC
D
a rectangle with
∣
A
B
∣
=
a
≥
b
=
∣
B
C
∣
|AB|=a\ge b=|BC|
∣
A
B
∣
=
a
≥
b
=
∣
BC
∣
. Find points
P
,
Q
P,Q
P
,
Q
on the line
B
D
BD
B
D
such that
∣
A
P
∣
=
∣
P
Q
∣
=
∣
Q
C
∣
|AP|=|PQ|=|QC|
∣
A
P
∣
=
∣
PQ
∣
=
∣
QC
∣
. Discuss the solvability with respect to the lengths
a
,
b
a,b
a
,
b
.
1
1
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System of equations
Find all triplets
(
x
,
y
,
z
)
∈
R
3
(x,y,z)\in\mathbb{R}^3
(
x
,
y
,
z
)
∈
R
3
such that \begin{align*} x^2-yz &= |y-z|+1, \\ y^2-zx &= |z-x|+1, \\ z^2-xy &= |x-y|+1. \end{align*}