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Problems
Contests
National and Regional Contests
Poland Contests
Polish MO Finals
2001 Polish MO Finals
2001 Polish MO Finals
Part of
Polish MO Finals
Subcontests
(3)
1
2
Hide problems
inequality with powers
Prove the following inequality:
x
1
+
2
x
2
+
3
x
3
+
.
.
.
+
n
x
n
≤
n
(
n
−
1
)
2
+
x
1
+
x
2
2
+
x
3
3
+
.
.
.
+
x
n
n
x_1 + 2x_2 + 3x_3 + ... + nx_n \leq \frac{n(n-1)}{2} + x_1 + x_2 ^2 + x_3 ^3 + ... + x_n ^n
x
1
+
2
x
2
+
3
x
3
+
...
+
n
x
n
≤
2
n
(
n
−
1
)
+
x
1
+
x
2
2
+
x
3
3
+
...
+
x
n
n
where
∀
x
i
x
i
>
0
\forall _{x_i} x_i > 0
∀
x
i
x
i
>
0
2^n a+b being always a square
Assume that
a
,
b
a,b
a
,
b
are integers and
n
n
n
is a natural number.
2
n
a
+
b
2^na+b
2
n
a
+
b
is a perfect square for every
n
n
n
.Prove that
a
=
0
a=0
a
=
0
.
3
2
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Fibonaccis Sequence
A sequence
x
0
=
A
x_0=A
x
0
=
A
and
x
1
=
B
x_1=B
x
1
=
B
and
x
n
+
2
=
x
n
+
1
+
x
n
x_{n+2}=x_{n+1} +x_n
x
n
+
2
=
x
n
+
1
+
x
n
is called a Fibonacci type sequence. Call a number
C
C
C
a repeated value if
x
t
=
x
s
=
c
x_t=x_s=c
x
t
=
x
s
=
c
for
t
t
t
different from
s
s
s
. Prove one can choose
A
A
A
and
B
B
B
to have as many repeated value as one likes but never infinite.
partition of a set
Given positive integers
n
1
<
n
2
<
.
.
.
<
n
2000
<
1
0
100
n_1<n_2<...<n_{2000}<10^{100}
n
1
<
n
2
<
...
<
n
2000
<
1
0
100
. Prove that we can choose from the set
{
n
1
,
.
.
.
,
n
2000
}
\{n_1,...,n_{2000}\}
{
n
1
,
...
,
n
2000
}
nonempty, disjont sets
A
A
A
and
B
B
B
which have the same number of elements, the same sum and the same sum of squares.
2
2
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Polish tetrahedron
Given a regular tetrahedron
A
B
C
D
ABCD
A
BC
D
with edge length
1
1
1
and a point
P
P
P
inside it. What is the maximum value of
∣
P
A
∣
+
∣
P
B
∣
+
∣
P
C
∣
+
∣
P
D
∣
\left|PA\right|+\left|PB\right|+\left|PC\right|+\left|PD\right|
∣
P
A
∣
+
∣
PB
∣
+
∣
PC
∣
+
∣
P
D
∣
.
parallelogram
Let
A
B
C
D
ABCD
A
BC
D
be a parallelogram and let
K
K
K
and
L
L
L
be points on the segments
B
C
BC
BC
and
C
D
CD
C
D
, respectively, such that
B
K
⋅
A
D
=
D
L
⋅
A
B
BK\cdot AD=DL\cdot AB
B
K
⋅
A
D
=
D
L
⋅
A
B
. Let the lines
D
K
DK
DK
and
B
L
BL
B
L
intersect at
P
P
P
. Show that
∡
D
A
P
=
∡
B
A
C
\measuredangle DAP=\measuredangle BAC
∡
D
A
P
=
∡
B
A
C
.