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Problems
Contests
National and Regional Contests
Czech Republic Contests
Czech and Slovak Olympiad III A
2001 Czech And Slovak Olympiad IIIA
2001 Czech And Slovak Olympiad IIIA
Part of
Czech and Slovak Olympiad III A
Subcontests
(5)
6
1
Hide problems
f(x) = 1- f(x-a_1)f(x-a_2)... f(x-a_n) for integers x>=0, f(x)=1 if integer x<0
Let be given natural numbers
a
1
,
a
2
,
.
.
.
,
a
n
a_1,a_2,...,a_n
a
1
,
a
2
,
...
,
a
n
and a function
f
:
Z
→
R
f : Z \to R
f
:
Z
→
R
such that
f
(
x
)
=
1
f(x) = 1
f
(
x
)
=
1
for all integers
x
<
0
x < 0
x
<
0
and
f
(
x
)
=
1
−
f
(
x
−
a
1
)
f
(
x
−
a
2
)
.
.
.
f
(
x
−
a
n
)
f(x) = 1- f(x-a_1)f(x-a_2)... f(x-a_n)
f
(
x
)
=
1
−
f
(
x
−
a
1
)
f
(
x
−
a
2
)
...
f
(
x
−
a
n
)
for all integers
x
≥
0
x \ge 0
x
≥
0
. Prove that there exist natural numbers
s
s
s
and
t
t
t
such that for all integers
x
>
s
x > s
x
>
s
it holds that
f
(
x
+
t
)
=
f
(
x
)
f(x+t) = f(x)
f
(
x
+
t
)
=
f
(
x
)
.
5
1
Hide problems
isosceles trapezoid, sides wanted, area given
A sheet of paper has the shape of an isosceles trapezoid
C
1
A
B
2
C
2
C_1AB_2C_2
C
1
A
B
2
C
2
with the shorter base
B
2
C
2
B_2C_2
B
2
C
2
. The foot of the perpendicular from the midpoint
D
D
D
of
C
1
C
2
C_1C_2
C
1
C
2
to
A
C
1
AC_1
A
C
1
is denoted by
B
1
B_1
B
1
. Suppose that upon folding the paper along
D
B
1
,
A
D
DB_1, AD
D
B
1
,
A
D
and
A
C
1
AC_1
A
C
1
points
C
1
,
C
2
C_1,C_2
C
1
,
C
2
become a single point
C
C
C
and points
B
1
,
B
2
B_1,B_2
B
1
,
B
2
become a point
B
B
B
. The area of the tetrahedron
A
B
C
D
ABCD
A
BC
D
is
64
64
64
cm
3
^3
3
. Find the sides of the initial trapezoid.
3
1
Hide problems
\sqrt{2x^2 +ax+b} > x-c <=> x \in (-\infty,0)\cup(1,\infty)
Find all triples of real numbers
(
a
,
b
,
c
)
(a,b,c)
(
a
,
b
,
c
)
for which the set of solutions
x
x
x
of
2
x
2
+
a
x
+
b
>
x
−
c
\sqrt{2x^2 +ax+b} > x-c
2
x
2
+
a
x
+
b
>
x
−
c
is the set
(
−
∞
,
0
]
∪
(
1
,
∞
)
(-\infty,0]\cup(1,\infty)
(
−
∞
,
0
]
∪
(
1
,
∞
)
.
2
1
Hide problems
right triangle construction
Given a triangle
P
Q
X
PQX
PQX
in the plane, with
P
Q
=
3
,
P
X
=
2.6
PQ = 3, PX = 2.6
PQ
=
3
,
PX
=
2.6
and
Q
X
=
3.8
QX = 3.8
QX
=
3.8
. Construct a right-angled triangle
A
B
C
ABC
A
BC
such that the incircle of
△
A
B
C
\vartriangle ABC
△
A
BC
touches
A
B
AB
A
B
at
P
P
P
and
B
C
BC
BC
at
Q
Q
Q
, and point
X
X
X
lies on the line
A
C
AC
A
C
.
1
1
Hide problems
P(x)^2 +P(-x) = P(x^2)+P(x) , polynomial
Determine all polynomials
P
P
P
such that for every real number
x
x
x
,
P
(
x
)
2
+
P
(
−
x
)
=
P
(
x
2
)
+
P
(
x
)
P(x)^2 +P(-x) = P(x^2)+P(x)
P
(
x
)
2
+
P
(
−
x
)
=
P
(
x
2
)
+
P
(
x
)