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Problems
Contests
National and Regional Contests
Czech Republic Contests
Czech and Slovak Olympiad III A
2004 Czech and Slovak Olympiad III A
2004 Czech and Slovak Olympiad III A
Part of
Czech and Slovak Olympiad III A
Subcontests
(6)
6
1
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a functional equation on positive reals
Find all functions
f
:
R
+
→
R
+
f:\mathbb R^+ \rightarrow \mathbb R^+
f
:
R
+
→
R
+
such that for all positive real numbers
x
,
y
x,y
x
,
y
,
x
2
[
f
(
x
)
+
f
(
y
)
]
=
(
x
+
y
)
f
(
y
f
(
x
)
)
.
x^2[f(x)+f(y)]=(x+y)f(yf(x)).
x
2
[
f
(
x
)
+
f
(
y
)]
=
(
x
+
y
)
f
(
y
f
(
x
))
.
5
1
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Four points are concyclic in a square
Let
L
L
L
be an arbitrary point on the minor arc
C
D
CD
C
D
of the circumcircle of square
A
B
C
D
ABCD
A
BC
D
. Let
K
,
M
,
N
K,M,N
K
,
M
,
N
be the intersection points of
A
L
,
C
D
AL,CD
A
L
,
C
D
;
C
L
,
A
D
CL,AD
C
L
,
A
D
; and
M
K
,
B
C
MK,BC
M
K
,
BC
respectively. Prove that
B
,
M
,
L
,
N
B,M,L,N
B
,
M
,
L
,
N
are concyclic.
4
1
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find all n for a sum to be an integer
Find all positive integers
n
n
n
such that
∑
k
=
1
n
n
k
!
\sum_{k=1}^{n}\frac{n}{k!}
∑
k
=
1
n
k
!
n
is an integer.
3
1
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121 chords on a circle
Given a circle
S
S
S
and its
121
121
121
chords
P
i
(
i
=
1
,
2
,
…
,
121
)
P_i (i=1,2,\ldots,121)
P
i
(
i
=
1
,
2
,
…
,
121
)
, each with a point
A
i
(
i
=
1
,
2
,
…
,
121
)
A_i(i=1,2,\ldots,121)
A
i
(
i
=
1
,
2
,
…
,
121
)
on it. Prove that there exists a point
X
X
X
on the circumference of
S
S
S
such that: there exist
29
29
29
distinct indices
1
≤
k
1
≤
k
2
≤
…
≤
k
29
≤
121
1\le k_1\le k_2\le\ldots\le k_{29}\le 121
1
≤
k
1
≤
k
2
≤
…
≤
k
29
≤
121
, such that the angle formed by
A
k
j
X
{A_{k_j}}X
A
k
j
X
and
P
k
j
{P_{k_j}}
P
k
j
is smaller than
21
21
21
degrees for every
j
=
1
,
2
,
…
,
29
j=1,2,\ldots,29
j
=
1
,
2
,
…
,
29
.
2
1
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n-letter words containing two letters
Consider all words containing only letters
A
A
A
and
B
B
B
. For any positive integer
n
n
n
,
p
(
n
)
p(n)
p
(
n
)
denotes the number of all
n
n
n
-letter words without four consecutive
A
A
A
's or three consecutive
B
B
B
's. Find the value of the expression
p
(
2004
)
−
p
(
2002
)
−
p
(
1999
)
p
(
2001
)
+
p
(
2000
)
.
\frac{p(2004)-p(2002)-p(1999)}{p(2001)+p(2000)}.
p
(
2001
)
+
p
(
2000
)
p
(
2004
)
−
p
(
2002
)
−
p
(
1999
)
.
1
1
Hide problems
find all real triples with an inequality condition
Find all triples
(
x
,
y
,
z
)
(x,y,z)
(
x
,
y
,
z
)
of real numbers such that
x
2
+
y
2
+
z
2
≤
6
+
min
(
x
2
−
8
x
4
,
y
2
−
8
y
4
,
z
2
−
8
z
4
)
.
x^2+y^2+z^2\le 6+\min (x^2-\frac{8}{x^4},y^2-\frac{8}{y^4},z^2-\frac{8}{z^4}).
x
2
+
y
2
+
z
2
≤
6
+
min
(
x
2
−
x
4
8
,
y
2
−
y
4
8
,
z
2
−
z
4
8
)
.