MathDB
Problems
Contests
National and Regional Contests
Poland Contests
Polish MO Finals
1993 Polish MO Finals
1993 Polish MO Finals
Part of
Polish MO Finals
Subcontests
(3)
3
2
Hide problems
Sequence and the greatest odd divisor
Denote
g
(
k
)
g(k)
g
(
k
)
as the greatest odd divisor of
k
k
k
. Put
f
(
k
)
=
k
2
+
k
g
(
k
)
f(k) = \dfrac{k}{2} + \dfrac{k}{g(k)}
f
(
k
)
=
2
k
+
g
(
k
)
k
for
k
k
k
even, and
2
(
k
+
1
)
/
2
2^{(k+1)/2}
2
(
k
+
1
)
/2
for
k
k
k
odd. Define the sequence
x
1
,
x
2
,
x
3
,
.
.
.
x_1, x_2, x_3, ...
x
1
,
x
2
,
x
3
,
...
by
x
1
=
1
x_1 = 1
x
1
=
1
,
x
n
+
1
=
f
(
x
n
)
x_{n+1} = f(x_n)
x
n
+
1
=
f
(
x
n
)
. Find
n
n
n
such that
x
n
=
800
x_n = 800
x
n
=
800
.
Volume of tetrahedron
Find out whether it is possible to determine the volume of a tetrahedron knowing the areas of its faces and its circumradius.
2
1
Hide problems
Circle inscribed in the quadriteral
A circle center
O
O
O
is inscribed in the quadrilateral
A
B
C
D
ABCD
A
BC
D
.
A
B
AB
A
B
is parallel to and longer than
C
D
CD
C
D
and has midpoint
M
M
M
. The line
O
M
OM
OM
meets
C
D
CD
C
D
at
F
F
F
.
C
D
CD
C
D
touches the circle at
E
E
E
. Show that
D
E
=
C
F
DE = CF
D
E
=
CF
iff
A
B
=
2
C
D
AB = 2CD
A
B
=
2
C
D
.
1
2
Hide problems
System of equations in Q
Find all rational solutions to: \begin{eqnarray*} t^2 - w^2 + z^2 &=& 2xy \\ t^2 - y^2 + w^2 &=& 2xz \\ t^2 - w^2 + x^2 &=& 2yz . \end{eqnarray*}
Colored convex polyhedron
Let be given a convex polyhedron whose all faces are triangular. The vertices of the polyhedron are colored using three colors. Prove that the number of faces with vertices in all the three colors is even.