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Problems
Contests
National and Regional Contests
Czech Republic Contests
Czech and Slovak Olympiad III A
2007 Czech and Slovak Olympiad III A
2007 Czech and Slovak Olympiad III A
Part of
Czech and Slovak Olympiad III A
Subcontests
(6)
6
1
Hide problems
find x,y,z for which two sets are equal
Find all pariwise distinct real numbers
x
,
y
,
z
x,y,z
x
,
y
,
z
such that
{
x
−
y
y
−
z
,
y
−
z
z
−
x
,
z
−
x
x
−
y
}
=
{
x
,
y
,
z
}
\left\{\frac{x-y}{y-z},\frac{y-z}{z-x},\frac{z-x}{x-y} \right\} = \{x,y,z\}
{
y
−
z
x
−
y
,
z
−
x
y
−
z
,
x
−
y
z
−
x
}
=
{
x
,
y
,
z
}
. (It means, those three fractions make a permutation of
x
,
y
x, y
x
,
y
, and
z
z
z
.)
5
1
Hide problems
the orthocenter of a triangle
In an acute-angled triangle
A
B
C
ABC
A
BC
(
A
C
≠
B
C
AC\ne BC
A
C
=
BC
), let
D
D
D
and
E
E
E
be points on
B
C
BC
BC
and
A
C
AC
A
C
, respectively, such that the points
A
,
B
,
D
,
E
A,B,D,E
A
,
B
,
D
,
E
are concyclic and
A
D
AD
A
D
intersects
B
E
BE
BE
at
P
P
P
. Knowing that
C
P
⊥
A
B
CP\bot AB
CP
⊥
A
B
, prove that
P
P
P
is the orthocenter of triangle
A
B
C
ABC
A
BC
.
4
1
Hide problems
the existence of an integer m
The set
M
=
{
1
,
2
,
…
,
2007
}
M=\{1,2,\ldots,2007\}
M
=
{
1
,
2
,
…
,
2007
}
has the following property: If
n
n
n
is an element of
M
M
M
, then all terms in the arithmetic progression with its first term
n
n
n
and common difference
n
+
1
n+1
n
+
1
, are in
M
M
M
. Does there exist an integer
m
m
m
such that all integers greater than
m
m
m
are elements of
M
M
M
?
3
1
Hide problems
the smallest possible value for f(2007)
Consider a function
f
:
N
→
N
f:\mathbb N\rightarrow \mathbb N
f
:
N
→
N
such that for any two positive integers
x
,
y
x,y
x
,
y
, the equation
f
(
x
f
(
y
)
)
=
y
f
(
x
)
f(xf(y))=yf(x)
f
(
x
f
(
y
))
=
y
f
(
x
)
holds. Find the smallest possible value of
f
(
2007
)
f(2007)
f
(
2007
)
.
2
1
Hide problems
prove a triangle is isoceles
In a cyclic quadrilateral
A
B
C
D
ABCD
A
BC
D
, let
L
L
L
and
M
M
M
be the incenters of
A
B
C
ABC
A
BC
and
B
C
D
BCD
BC
D
respectively. Let
R
R
R
be a point on the plane such that
L
R
⊥
A
C
LR\bot AC
L
R
⊥
A
C
and
M
R
⊥
B
D
MR\bot BD
MR
⊥
B
D
.Prove that triangle
L
M
R
LMR
L
MR
is isosceles.
1
1
Hide problems
a stone go through Hamilton path in a chessboard
A stone is placed in a square of a chessboard with
n
n
n
rows and
n
n
n
columns. We can alternately undertake two operations: (a) move the stone to a square that shares a common side with the square in which it stands; (b) move it to a square sharing only one common vertex with the square in which it stands.In addition, we are required that the first step must be (b). Find all integers
n
n
n
such that the stone can go through a certain path visiting every square exactly once.