MathDB
Problems
Contests
International Contests
Czech-Polish-Slovak Match
2009 Czech-Polish-Slovak Match
2009 Czech-Polish-Slovak Match
Part of
Czech-Polish-Slovak Match
Subcontests
(6)
5
1
Hide problems
n-tuple (a_k) to represent any (b_k) as m∙b_k = c_k^a_k
The
n
n
n
-tuple
(
a
1
,
a
2
,
…
,
a
n
)
(a_1,a_2,\ldots,a_n)
(
a
1
,
a
2
,
…
,
a
n
)
of integers satisfies the following: (i)
1
≤
a
1
<
a
2
<
⋯
<
a
n
≤
50
1\le a_1<a_2<\cdots < a_n\le 50
1
≤
a
1
<
a
2
<
⋯
<
a
n
≤
50
(ii) for each
n
n
n
-tuple
(
b
1
,
b
2
,
…
,
b
n
)
(b_1,b_2,\ldots,b_n)
(
b
1
,
b
2
,
…
,
b
n
)
of positive integers, there exist a positive integer
m
m
m
and an
n
n
n
-tuple
(
c
1
,
c
2
,
…
,
c
n
)
(c_1,c_2,\ldots,c_n)
(
c
1
,
c
2
,
…
,
c
n
)
of positive integers such that
m
b
i
=
c
i
a
i
for
i
=
1
,
2
,
…
,
n
.
mb_i=c_i^{a_i}\qquad\text{for } i=1,2,\ldots,n.
m
b
i
=
c
i
a
i
for
i
=
1
,
2
,
…
,
n
.
Prove that
n
≤
16
n\le 16
n
≤
16
and determine the number of
n
n
n
-tuples
(
a
1
,
a
2
,
…
,
a
n
(a_1,a_2,\ldots,a_n
(
a
1
,
a
2
,
…
,
a
n
) satisfying these conditions for
n
=
16
n=16
n
=
16
.
2
1
Hide problems
Infinite sequence with exactly 2009 different numbers
For positive integers
a
a
a
and
k
k
k
, define the sequence
a
1
,
a
2
,
…
a_1,a_2,\ldots
a
1
,
a
2
,
…
by
a
1
=
a
,
and
a
n
+
1
=
a
n
+
k
⋅
ϱ
(
a
n
)
for
n
=
1
,
2
,
…
a_1=a,\qquad\text{and}\qquad a_{n+1}=a_n+k\cdot\varrho(a_n)\qquad\text{for } n=1,2,\ldots
a
1
=
a
,
and
a
n
+
1
=
a
n
+
k
⋅
ϱ
(
a
n
)
for
n
=
1
,
2
,
…
where
ϱ
(
m
)
\varrho(m)
ϱ
(
m
)
denotes the product of the decimal digits of
m
m
m
(for example,
ϱ
(
413
)
=
12
\varrho(413)=12
ϱ
(
413
)
=
12
and
ϱ
(
308
)
=
0
\varrho(308)=0
ϱ
(
308
)
=
0
). Prove that there are positive integers
a
a
a
and
k
k
k
for which the sequence
a
1
,
a
2
,
…
a_1,a_2,\ldots
a
1
,
a
2
,
…
contains exactly
2009
2009
2009
different numbers.
1
1
Hide problems
Functional equation: ( 1 + y f(x) )( 1 - y f(x+y) ) = 1
Let
R
+
\mathbb{R}^+
R
+
denote the set of positive real numbers. Find all functions
f
:
R
+
→
R
+
f : \mathbb{R}^+\to\mathbb{R}^+
f
:
R
+
→
R
+
that satisfy
(
1
+
y
f
(
x
)
)
(
1
−
y
f
(
x
+
y
)
)
=
1
\Big(1+yf(x)\Big)\Big(1-yf(x+y)\Big)=1
(
1
+
y
f
(
x
)
)
(
1
−
y
f
(
x
+
y
)
)
=
1
for all
x
,
y
∈
R
+
x,y\in\mathbb{R}^+
x
,
y
∈
R
+
.
6
1
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At least 4n√n quadrilaterals with concurrent diagonals
Let
n
≥
16
n\ge 16
n
≥
16
be an integer, and consider the set of
n
2
n^2
n
2
points in the plane:
G
=
{
(
x
,
y
)
∣
x
,
y
∈
{
1
,
2
,
…
,
n
}
}
.
G=\big\{(x,y)\mid x,y\in\{1,2,\ldots,n\}\big\}.
G
=
{
(
x
,
y
)
∣
x
,
y
∈
{
1
,
2
,
…
,
n
}
}
.
Let
A
A
A
be a subset of
G
G
G
with at least
4
n
n
4n\sqrt{n}
4
n
n
elements. Prove that there are at least
n
2
n^2
n
2
convex quadrilaterals whose vertices are in
A
A
A
and all of whose diagonals pass through a fixed point.
3
1
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Point of concurrency independent of incircle
Let
ω
\omega
ω
denote the excircle tangent to side
B
C
BC
BC
of triangle
A
B
C
ABC
A
BC
. A line
ℓ
\ell
ℓ
parallel to
B
C
BC
BC
meets sides
A
B
AB
A
B
and
A
C
AC
A
C
at points
D
D
D
and
E
E
E
, respectively. Let
ω
′
\omega'
ω
′
denote the incircle of triangle
A
D
E
ADE
A
D
E
. The tangent from
D
D
D
to
ω
\omega
ω
(different from line
A
B
AB
A
B
) and the tangent from
E
E
E
to
ω
\omega
ω
(different from line
A
C
AC
A
C
) meet at point
P
P
P
. The tangent from
B
B
B
to
ω
′
\omega'
ω
′
(different from line
A
B
AB
A
B
) and the tangent from
C
C
C
to
ω
′
\omega'
ω
′
(different from line
A
C
AC
A
C
) meet at point
Q
Q
Q
. Prove that, independent of the choice of
ℓ
\ell
ℓ
, there is a fixed point that line
P
Q
PQ
PQ
always passes through.
4
1
Hide problems
Distance between midpoints independent of reflection lines
Given a circle, let
A
B
AB
A
B
be a chord that is not a diameter, and let
C
C
C
be a point on the longer arc
A
B
AB
A
B
. Let
K
K
K
and
L
L
L
denote the reflections of
A
A
A
and
B
B
B
, respectively, about lines
B
C
BC
BC
and
A
C
AC
A
C
, respectively. Prove that the distance between the midpoint of
A
B
AB
A
B
and the midpoint of
K
L
KL
K
L
is independent of the choice of
C
C
C
.