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Problems
Contests
National and Regional Contests
Czech Republic Contests
Czech and Slovak Olympiad III A
1988 Czech And Slovak Olympiad IIIA
1988 Czech And Slovak Olympiad IIIA
Part of
Czech and Slovak Olympiad III A
Subcontests
(6)
6
1
Hide problems
triangles P_1A_2A_3, A_1P_2A_3, A_1A_2P_3 cover the triangle A_1A_2A_3
Inside the triangle
A
1
A
2
A
3
A_1A_2A_3
A
1
A
2
A
3
with sides
a
1
a_1
a
1
,
a
2
a_2
a
2
,
a
3
a_3
a
3
, three points are given, which we label
P
1
P_1
P
1
,
P
2
P_2
P
2
,
P
3
P_3
P
3
so that the product of their distances from the corresponding sides
a
1
a_1
a
1
,
a
2
a_2
a
2
,
a
3
a_3
a
3
is as large as possible. Prove that the triangles
P
1
A
2
A
3
P_1A_2A_3
P
1
A
2
A
3
,
A
1
P
2
A
3
A_1P_2A_3
A
1
P
2
A
3
,
A
1
A
2
P
3
A_1A_2P_3
A
1
A
2
P
3
cover the triangle.[hide=original wording]V trojúhelníku A1A2A3 se stranami a1, a2, a3 jsou dány tři body, které označíme Pi, P2, P3 tak, aby součin jejich vzdáleností od odpovídajících stran a1, a2, a3 byl co největší. Dokažte, že trojúhelníky P1A2A3, A1P2A3, A1A2P3 pokrývají trojúhelník.
5
1
Hide problems
x^{154}-ax^{77}+1 is a multiple of x^{14}-ax^{7}+1
Find all numbers
a
∈
(
−
2
,
2
)
a \in (-2, 2)
a
∈
(
−
2
,
2
)
for which the polynomial
x
154
−
a
x
77
+
1
x^{154}-ax^{77}+1
x
154
−
a
x
77
+
1
is a multiple of the polynomial
x
14
−
a
x
7
+
1
x^{14}-ax^{7}+1
x
14
−
a
x
7
+
1
.
4
1
Hide problems
2 colors for numbers 1,2,3,..., 2^n
Prove that each of the numbers
1
,
2
,
3
,
.
.
.
,
2
n
1, 2, 3, ..., 2^n
1
,
2
,
3
,
...
,
2
n
can be written in one of two colors (red and blue) such that no non-constant
2
n
2n
2
n
-term arithmetic sequence chosen from these numbers is monochromatic .
3
1
Hide problems
min |AX|+|BX|+|CX|+|DX| for tetrahedron ABCD
Given a tetrahedron
A
B
C
D
ABCD
A
BC
D
with edges
∣
A
D
∣
=
∣
B
C
∣
=
a
|AD|=|BC|= a
∣
A
D
∣
=
∣
BC
∣
=
a
,
∣
A
C
∣
=
∣
B
D
∣
=
b
|AC|=|BD|=b
∣
A
C
∣
=
∣
B
D
∣
=
b
,
A
B
=
c
AB=c
A
B
=
c
and
∣
C
D
∣
=
d
|CD| = d
∣
C
D
∣
=
d
. Determine the smallest value of the sum
∣
A
X
∣
+
∣
B
X
∣
+
∣
C
X
∣
+
∣
D
X
∣
|AX|+|BX|+|CX|+|DX|
∣
A
X
∣
+
∣
BX
∣
+
∣
CX
∣
+
∣
D
X
∣
, where
X
X
X
is any point in space.
2
1
Hide problems
|a-c|<= 2 if a^2= 2(b+1) and x^3+ax^2+bx+c=0 has all real roots
If for the coefficients of equation
x
3
+
a
x
2
+
b
x
+
c
=
0
x^3+ax^2+bx+c=0
x
3
+
a
x
2
+
b
x
+
c
=
0
whose roots are all real, holds,
a
2
=
2
(
b
+
1
)
a^2= 2(b+1)
a
2
=
2
(
b
+
1
)
then
∣
a
−
c
∣
≤
2
|a-c|\le 2
∣
a
−
c
∣
≤
2
. Prove it.
1
1
Hide problems
x_{2m} = x_m if x_1 = f(1), x_{n+1} = f(x_n)
Let
f
f
f
be a representation of the set
M
=
{
1
,
2
,
.
.
.
,
1988
}
M = \{1, 2,..., 1988\}
M
=
{
1
,
2
,
...
,
1988
}
into
M
M
M
. For any natural
n
n
n
, let
x
1
=
f
(
1
)
x_1 = f(1)
x
1
=
f
(
1
)
,
x
n
+
1
=
f
(
x
n
)
x_{n+1} = f(x_n)
x
n
+
1
=
f
(
x
n
)
. Find out if there exists
m
m
m
such that
x
2
m
=
x
m
x_{2m} = x_m
x
2
m
=
x
m
.