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Problems
Contests
National and Regional Contests
Poland Contests
Polish MO Finals
1997 Polish MO Finals
1997 Polish MO Finals
Part of
Polish MO Finals
Subcontests
(3)
2
2
Hide problems
system of equations
Find all real solutions to: \begin{eqnarray*} 3(x^2 + y^2 + z^2) &=& 1 \\ x^2y^2 + y^2z^2 + z^2x^2 &=& xyz(x + y + z)^3. \end{eqnarray*}
Convex pentagon - equality of angles
A
B
C
D
E
ABCDE
A
BC
D
E
is a convex pentagon such that
D
C
=
D
E
DC = DE
D
C
=
D
E
and
∠
C
=
∠
E
=
9
0
⋅
\angle C = \angle E = 90^{\cdot}
∠
C
=
∠
E
=
9
0
⋅
.
F
F
F
is a point on the side
A
B
AB
A
B
such that
A
F
B
F
=
A
E
B
C
\frac{AF}{BF}= \frac{AE}{BC}
BF
A
F
=
BC
A
E
. Show that
∠
F
C
E
=
∠
A
D
E
\angle FCE = \angle ADE
∠
FCE
=
∠
A
D
E
and
∠
F
E
C
=
∠
B
D
C
\angle FEC = \angle BDC
∠
FEC
=
∠
B
D
C
.
1
2
Hide problems
find x_7
The positive integers
x
1
,
x
2
,
.
.
.
,
x
7
x_1, x_2, ... , x_7
x
1
,
x
2
,
...
,
x
7
satisfy
x
6
=
144
x_6 = 144
x
6
=
144
,
x
n
+
3
=
x
n
+
2
(
x
n
+
1
+
x
n
)
x_{n+3} = x_{n+2}(x_{n+1}+x_n)
x
n
+
3
=
x
n
+
2
(
x
n
+
1
+
x
n
)
for
n
=
1
,
2
,
3
,
4
n = 1, 2, 3, 4
n
=
1
,
2
,
3
,
4
. Find
x
7
x_7
x
7
.
sequence and a_n=0
The sequence
a
1
,
a
2
,
a
3
,
.
.
.
a_1, a_2, a_3, ...
a
1
,
a
2
,
a
3
,
...
is defined by
a
1
=
0
a_1 = 0
a
1
=
0
,
a
n
=
a
[
n
/
2
]
+
(
−
1
)
n
(
n
+
1
)
/
2
a_n = a_{[n/2]} + (-1)^{n(n+1)/2}
a
n
=
a
[
n
/2
]
+
(
−
1
)
n
(
n
+
1
)
/2
. Show that for any positive integer
k
k
k
we can find
n
n
n
in the range
2
k
≤
n
<
2
k
+
1
2^k \leq n < 2^{k+1}
2
k
≤
n
<
2
k
+
1
such that
a
n
=
0
a_n = 0
a
n
=
0
.
3
2
Hide problems
medians, tetrahedron and inequality
In a tetrahedron
A
B
C
D
ABCD
A
BC
D
, the medians of the faces
A
B
D
ABD
A
B
D
,
A
C
D
ACD
A
C
D
,
B
C
D
BCD
BC
D
from
D
D
D
make equal angles with the corresponding edges
A
B
AB
A
B
,
A
C
AC
A
C
,
B
C
BC
BC
. Prove that each of these faces has area less than or equal to the sum of the areas of the other two faces. [hide="Comment"]Equivalent version of the problem:
A
B
C
D
ABCD
A
BC
D
is a tetrahedron.
D
E
DE
D
E
,
D
F
DF
D
F
,
D
G
DG
D
G
are medians of triangles
D
B
C
DBC
D
BC
,
D
C
A
DCA
D
C
A
,
D
A
B
DAB
D
A
B
. The angles between
D
E
DE
D
E
and
B
C
BC
BC
, between
D
F
DF
D
F
and
C
A
CA
C
A
, and between
D
G
DG
D
G
and
A
B
AB
A
B
are equal. Show that: area
D
B
C
DBC
D
BC
≤
\leq
≤
area
D
C
A
DCA
D
C
A
+ area
D
A
B
DAB
D
A
B
.
n points on a unit circle
Given any
n
n
n
points on a unit circle show that at most
n
2
3
\frac{n^2}{3}
3
n
2
of the segments joining two points have length
>
2
> \sqrt{2}
>
2
.