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Problems
Contests
National and Regional Contests
Czech Republic Contests
Czech and Slovak Olympiad III A
1996 Czech And Slovak Olympiad IIIA
1996 Czech And Slovak Olympiad IIIA
Part of
Czech and Slovak Olympiad III A
Subcontests
(6)
5
1
Hide problems
f(xy) = f(x)+ f(y)+k f(gcd(x,y)) , f(1995) =1996
For which integers
k
k
k
does there exist a function
f
:
N
→
Z
f : N \to Z
f
:
N
→
Z
such that
f
(
1995
)
=
1996
f(1995) =1996
f
(
1995
)
=
1996
and
f
(
x
y
)
=
f
(
x
)
+
f
(
y
)
+
k
f
(
g
c
d
(
x
,
y
)
)
f(xy) = f(x)+ f(y)+k f(gcd(x,y))
f
(
x
y
)
=
f
(
x
)
+
f
(
y
)
+
k
f
(
g
c
d
(
x
,
y
))
for all
x
,
y
∈
N
x,y \in N
x
,
y
∈
N
?
1
1
Hide problems
G(n) = n-G(G(n)), G(0) = 0 , G(k) \ge G(k -1) , G(k -1) = G(k) = G(k +1)
A sequence
(
G
n
)
n
=
0
∞
(G_n)_{n=0}^{\infty}
(
G
n
)
n
=
0
∞
satisfies
G
(
0
)
=
0
G(0) = 0
G
(
0
)
=
0
and
G
(
n
)
=
n
−
G
(
G
(
n
−
1
)
)
G(n) = n-G(G(n-1))
G
(
n
)
=
n
−
G
(
G
(
n
−
1
))
for each
n
∈
N
n \in N
n
∈
N
. Show that (a)
G
(
k
)
≥
G
(
k
−
1
)
G(k) \ge G(k -1)
G
(
k
)
≥
G
(
k
−
1
)
for every
k
∈
N
k \in N
k
∈
N
; (b) there is no integer
k
k
k
for which
G
(
k
−
1
)
=
G
(
k
)
=
G
(
k
+
1
)
G(k -1) = G(k) = G(k +1)
G
(
k
−
1
)
=
G
(
k
)
=
G
(
k
+
1
)
.
2
1
Hide problems
exists a tetrahedron with max distance
Let
A
P
,
B
Q
AP,BQ
A
P
,
BQ
and
C
R
CR
CR
be altitudes of an acute-angled triangle
A
B
C
ABC
A
BC
. Show that for any point
X
X
X
inside the triangle
P
Q
R
PQR
PQR
there exists a tetrahedron
A
B
C
D
ABCD
A
BC
D
such that
X
X
X
is the point on the face
A
B
C
ABC
A
BC
at the greatest distance from
D
D
D
(measured along the surface of the tetrahedron).
3
1
Hide problems
color the elements of 6 three-element subsets of a finite set X in two colors
Given six three-element subsets of a finite set
X
X
X
, show that it is possible to color the elements of
X
X
X
in two colors so that none of the given subsets is in one color
4
1
Hide problems
KA \cdot YB = XA \cdot MB = LA\cdot LB \ne 0 , geometry
Points
A
A
A
and
B
B
B
on the rays
C
X
CX
CX
and
C
Y
CY
C
Y
respectively of an acute angle
X
C
Y
XCY
XC
Y
are given so that
C
X
<
C
A
=
C
B
<
C
Y
CX < CA = CB < CY
CX
<
C
A
=
CB
<
C
Y
. Construct a line meeting the ray
C
X
CX
CX
and the segments
A
B
,
B
C
AB,BC
A
B
,
BC
at
K
,
L
,
M
K,L,M
K
,
L
,
M
, respectively, such that
K
A
⋅
Y
B
=
X
A
⋅
M
B
=
L
A
⋅
L
B
≠
0
KA \cdot YB = XA \cdot MB = LA\cdot LB \ne 0
K
A
⋅
Y
B
=
X
A
⋅
MB
=
L
A
⋅
L
B
=
0
.
6
1
Hide problems
circumcircles of 3 triangles are equal, so are the incircles
Let
K
,
L
,
M
K,L,M
K
,
L
,
M
be points on sides
A
B
,
B
C
,
C
A
AB,BC,CA
A
B
,
BC
,
C
A
, respectively, of a triangle
A
B
C
ABC
A
BC
such that
A
K
/
A
B
=
B
L
/
B
C
=
C
M
/
C
A
=
1
/
3
AK/AB = BL/BC = CM/CA = 1/3
A
K
/
A
B
=
B
L
/
BC
=
CM
/
C
A
=
1/3
. Show that if the circumcircles of the triangles
A
K
M
,
B
L
K
,
C
M
L
AKM, BLK, CML
A
K
M
,
B
L
K
,
CM
L
are equal, then so are the incircles of these triangles.