MathDB
Problems
Contests
National and Regional Contests
Poland Contests
Poland - Second Round
1985 Poland - Second Round
1985 Poland - Second Round
Part of
Poland - Second Round
Subcontests
(6)
6
1
Hide problems
right angle and coplanar points wanted
There are various points in space
A
,
B
,
C
0
,
C
1
,
C
2
A, B, C_0, C_1, C_2
A
,
B
,
C
0
,
C
1
,
C
2
, with
∣
A
C
i
∣
=
2
∣
B
C
i
∣
|AC_i| = 2 |BC_i|
∣
A
C
i
∣
=
2∣
B
C
i
∣
for
i
=
0
,
1
,
2
i = 0,1,2
i
=
0
,
1
,
2
and
∣
C
1
C
2
∣
=
4
3
∣
A
B
∣
|C_1C_2|=\frac{4}{3}|AB|
∣
C
1
C
2
∣
=
3
4
∣
A
B
∣
. Prove that the angle
C
1
C
0
C
2
C_1C_0C_2
C
1
C
0
C
2
is right and the points
A
,
B
,
C
1
,
C
2
A, B, C_1, C_2
A
,
B
,
C
1
,
C
2
lie on one plane.
5
1
Hide problems
equivalent with n being even
Prove that for a natural number
n
n
n
greater than 1, the following conditions are equivalent:a)
n
n
n
is an even number,b) there is a permutation
(
a
0
,
a
1
,
a
2
,
…
,
a
n
−
1
)
(a_0, a_1, a_2, \ldots, a_{n-1})
(
a
0
,
a
1
,
a
2
,
…
,
a
n
−
1
)
of the set
{
0
,
1
,
2
,
…
,
n
—
1
}
\{0,1,2,\ldots,n—1\}
{
0
,
1
,
2
,
…
,
n
—1
}
with the property that the sequence of residues from dividing by
n
n
n
the numbers
a
0
,
a
0
+
a
1
,
a
0
+
a
1
+
a
2
,
…
,
a
0
+
a
1
+
a
2
+
…
a
n
−
1
a_0, a_0 + a_1, a_0 + a_1 + a_2, \ldots, a_0 + a_1 + a_2 + \ldots a_{n-1}
a
0
,
a
0
+
a
1
,
a
0
+
a
1
+
a
2
,
…
,
a
0
+
a
1
+
a
2
+
…
a
n
−
1
is also a permutation of this set.
4
1
Hide problems
\sqrt[3]{a} + \sqrt[3]{b} is rational
Prove that if for natural numbers
a
,
b
a, b
a
,
b
the number
a
3
+
b
3
\sqrt[3]{a} + \sqrt[3]{b}
3
a
+
3
b
is rational, then
a
,
b
a, b
a
,
b
are cubes of natural numbers.
3
1
Hide problems
side of a convex quadrilateral, from vertices of regualr 195-gon
Let
L
L
L
be the set of all polylines
A
B
C
D
A
ABCDA
A
BC
D
A
, where
A
,
B
,
C
,
D
A, B, C, D
A
,
B
,
C
,
D
are different vertices of a fixed regular
1985
1985
1985
-gon. We randomly select a polyline from the set
L
L
L
. Calculate the probability that it is the side of a convex quadrilateral.
2
1
Hide problems
n! is the sum of its n$ various divisors.
Prove that for a natural number
n
>
2
n > 2
n
>
2
the number
n
!
n!
n
!
is the sum of its
n
n
n
various divisors.
1
1
Hide problems
x^2 + xy + y^2 = a^2 , y^2 + yz + z^2 = b^2, z^2 + zx + x^2 = c^2
Inside the triangle
A
B
C
ABC
A
BC
, the point
P
P
P
is chosen. Let
a
,
b
,
c
a, b, c
a
,
b
,
c
be the lengths of the sides
B
C
BC
BC
,
C
A
CA
C
A
,
A
B
AB
A
B
, respectively, and
x
,
y
,
z
x, y, z
x
,
y
,
z
the distances of the point
P
P
P
from the vertices
B
,
C
,
A
B, C, A
B
,
C
,
A
. Prove that if
x
2
+
x
y
+
y
2
=
a
2
x^2 + xy + y^2 = a^2
x
2
+
x
y
+
y
2
=
a
2
y
2
+
y
z
+
z
2
=
b
2
y^2 + yz + z^2 = b^2
y
2
+
yz
+
z
2
=
b
2
z
2
+
z
x
+
x
2
=
c
2
z^2 + zx + x^2 = c^2
z
2
+
z
x
+
x
2
=
c
2
this
a
2
+
a
b
+
b
2
>
c
2
.
a^2 + ab + b^2 > c^2.
a
2
+
ab
+
b
2
>
c
2
.