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Problems
Contests
National and Regional Contests
Poland Contests
Polish MO Finals
1992 Polish MO Finals
1992 Polish MO Finals
Part of
Polish MO Finals
Subcontests
(3)
3
2
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divisibility of fractionals
Show that
(
k
3
)
!
(k^3)!
(
k
3
)!
is divisible by
(
k
!
)
k
2
+
k
+
1
(k!)^{k^2+k+1}
(
k
!
)
k
2
+
k
+
1
.
inequality for real numbers and 2 sums
Show that for real numbers
x
1
,
x
2
,
.
.
.
,
x
n
x_1, x_2, ... , x_n
x
1
,
x
2
,
...
,
x
n
we have:
∑
i
=
1
n
∑
j
=
1
n
x
i
x
j
i
+
j
≥
0
\sum\limits_{i=1}^n \sum\limits_{j=1}^n \dfrac{x_ix_j}{i+j} \geq 0
i
=
1
∑
n
j
=
1
∑
n
i
+
j
x
i
x
j
≥
0
When do we have equality?
2
2
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functional equation in Q+
Find all functions
f
:
Q
+
→
Q
+
f : \mathbb{Q}^{+} \rightarrow \mathbb{Q}^{+}
f
:
Q
+
→
Q
+
, where
Q
+
\mathbb{Q}^{+}
Q
+
is the set of positive rationals, such that
f
(
x
+
1
)
=
f
(
x
)
+
1
f(x+1) = f(x) + 1
f
(
x
+
1
)
=
f
(
x
)
+
1
and
f
(
x
3
)
=
f
(
x
)
3
f(x^3) = f(x)^3
f
(
x
3
)
=
f
(
x
)
3
for all
x
x
x
.
regular pyramid and 2n-gon as a base
The base of a regular pyramid is a regular
2
n
2n
2
n
-gon
A
1
A
2
.
.
.
A
2
n
A_1A_2...A_{2n}
A
1
A
2
...
A
2
n
. A sphere passing through the top vertex
S
S
S
of the pyramid cuts the edge
S
A
i
SA_i
S
A
i
at
B
i
B_i
B
i
(for
i
=
1
,
2
,
.
.
.
,
2
n
i = 1, 2, ... , 2n
i
=
1
,
2
,
...
,
2
n
). Show that
∑
i
=
1
n
S
B
2
i
−
1
=
∑
i
=
1
n
S
B
2
i
\sum\limits_{i=1}^n SB_{2i-1} = \sum\limits_{i=1}^n SB_{2i}
i
=
1
∑
n
S
B
2
i
−
1
=
i
=
1
∑
n
S
B
2
i
.
1
2
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perpendicular segments
Segments
A
C
AC
A
C
and
B
D
BD
B
D
meet at
P
P
P
, and
∣
P
A
∣
=
∣
P
D
∣
|PA| = |PD|
∣
P
A
∣
=
∣
P
D
∣
,
∣
P
B
∣
=
∣
P
C
∣
|PB| = |PC|
∣
PB
∣
=
∣
PC
∣
.
O
O
O
is the circumcenter of the triangle
P
A
B
PAB
P
A
B
. Show that
O
P
OP
OP
and
C
D
CD
C
D
are perpendicular.
Sequence of functions
The functions
f
0
,
f
1
,
f
2
,
.
.
.
f_0, f_1, f_2, ...
f
0
,
f
1
,
f
2
,
...
are defined on the reals by
f
0
(
x
)
=
8
f_0(x) = 8
f
0
(
x
)
=
8
for all
x
x
x
,
f
n
+
1
(
x
)
=
x
2
+
6
f
n
(
x
)
f_{n+1}(x) = \sqrt{x^2 + 6f_n(x)}
f
n
+
1
(
x
)
=
x
2
+
6
f
n
(
x
)
. For all
n
n
n
solve the equation
f
n
(
x
)
=
2
x
f_n(x) = 2x
f
n
(
x
)
=
2
x
.