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Problems
Contests
International Contests
Czech-Polish-Slovak Match
2001 Czech-Polish-Slovak Match
2001 Czech-Polish-Slovak Match
Part of
Czech-Polish-Slovak Match
Subcontests
(5)
6
1
Hide problems
Which parellelepied covers all the coloured lattice points?
Points with integer coordinates in cartesian space are called lattice points. We color
2000
2000
2000
lattice points blue and
2000
2000
2000
other lattice points red in such a way that no two blue-red segments have a common interior point (a segment is blue-red if its two endpoints are colored blue and red). Consider the smallest rectangular parallelepiped that covers all the colored points. (a) Prove that this rectangular parallelepiped covers at least
500
,
000
500,000
500
,
000
lattice points. (b) Give an example of a coloring for which the considered rectangular paralellepiped covers at most
8
,
000
,
000
8,000,000
8
,
000
,
000
lattice points.
5
1
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Functional: f(x^2+y)+f(f(x)-y)=2f(f(x))+2y^2
Find all functions
f
:
R
→
R
f : \mathbb{R} \to \mathbb{R}
f
:
R
→
R
that satisfy f(x^2 + y) + f(f(x) - y) = 2f(f(x)) + 2y^2 \text{ for all }x, y \in \mathbb{R}.
4
1
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The locus of intersection points of two lines
Distinct points
A
A
A
and
B
B
B
are given on the plane. Consider all triangles
A
B
C
ABC
A
BC
in this plane on whose sides
B
C
,
C
A
BC,CA
BC
,
C
A
points
D
,
E
D,E
D
,
E
respectively can be taken so that (i)
B
D
B
C
=
C
E
C
A
=
1
3
\frac{BD}{BC}=\frac{CE}{CA}=\frac{1}{3}
BC
B
D
=
C
A
CE
=
3
1
; (ii) points
A
,
B
,
D
,
E
A,B,D,E
A
,
B
,
D
,
E
lie on a circle in this order.Find the locus of the intersection points of lines
A
D
AD
A
D
and
B
E
BE
BE
.
2
1
Hide problems
Isosceles triangles constructed on the sides of a triangle
A triangle
A
B
C
ABC
A
BC
has acute angles at
A
A
A
and
B
B
B
. Isosceles triangles
A
C
D
ACD
A
C
D
and
B
C
E
BCE
BCE
with bases
A
C
AC
A
C
and
B
C
BC
BC
are constructed externally to triangle
A
B
C
ABC
A
BC
such that
∠
A
D
C
=
∠
A
B
C
\angle ADC = \angle ABC
∠
A
D
C
=
∠
A
BC
and
∠
B
E
C
=
∠
B
A
C
\angle BEC = \angle BAC
∠
BEC
=
∠
B
A
C
. Let
S
S
S
be the circumcenter of
△
A
B
C
\triangle ABC
△
A
BC
. Prove that the length of the polygonal line
D
S
E
DSE
D
SE
equals the perimeter of triangle
A
B
C
ABC
A
BC
if and only if
∠
A
C
B
\angle ACB
∠
A
CB
is right.
1
1
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Czech-Slovak-Polish match 2001
Let
n
≥
2
n\ge2
n
≥
2
be a natural number, and
a
i
a_i
a
i
be positive numbers, where
i
=
1
,
2
,
⋯
,
n
.
i=1,2,\cdots,n.
i
=
1
,
2
,
⋯
,
n
.
Show that
(
a
1
3
+
1
)
(
a
2
3
+
1
)
⋯
(
a
n
3
+
1
)
≥
(
a
1
2
a
2
+
1
)
(
a
2
2
a
3
+
1
)
⋯
(
a
n
2
a
1
+
1
)
\left(a_1^3+1\right)\left(a_2^3+1\right)\cdots\left(a_n^3+1\right) \geq \left(a_1^2a_2+1\right)\left(a_2^2a_3+1\right)\cdots\left(a_n^2a_1+1\right)
(
a
1
3
+
1
)
(
a
2
3
+
1
)
⋯
(
a
n
3
+
1
)
≥
(
a
1
2
a
2
+
1
)
(
a
2
2
a
3
+
1
)
⋯
(
a
n
2
a
1
+
1
)