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Problems
Contests
National and Regional Contests
Poland Contests
Poland - Second Round
2009 Poland - Second Round
2009 Poland - Second Round
Part of
Poland - Second Round
Subcontests
(3)
2
2
Hide problems
Some divisibilities imply a/b is less than root 3
Given are two integers
a
>
b
>
1
a>b>1
a
>
b
>
1
such that
a
+
b
∣
a
b
+
1
a+b \mid ab+1
a
+
b
∣
ab
+
1
and
a
−
b
∣
a
b
−
1
a-b \mid ab-1
a
−
b
∣
ab
−
1
. Prove that
a
<
3
b
a<\sqrt{3}b
a
<
3
b
.
Intersection of 2 subsets amongst n has one element
Find all integer numbers
n
≥
4
n\ge 4
n
≥
4
which satisfy the following condition: from every
n
n
n
different
3
3
3
-element subsets of
n
n
n
-element set it is possible to choose
2
2
2
subsets, which have exactly one element in common.
1
2
Hide problems
Inequality with products of a_1,a_2, ... a_n
Let
a
1
≥
a
2
≥
…
≥
a
n
>
0
a_1\ge a_2\ge \ldots \ge a_n>0
a
1
≥
a
2
≥
…
≥
a
n
>
0
be
n
n
n
reals. Prove the inequality
a
1
a
2
…
a
n
−
1
+
(
2
a
2
−
a
1
)
(
2
a
3
−
a
2
)
…
(
2
a
n
−
a
n
−
1
)
≥
2
a
2
a
3
…
a
n
a_1a_2\ldots a_{n-1}+(2a_2-a_1)(2a_3-a_2)\ldots (2a_n-a_{n-1})\ge 2a_2a_3\ldots a_n
a
1
a
2
…
a
n
−
1
+
(
2
a
2
−
a
1
)
(
2
a
3
−
a
2
)
…
(
2
a
n
−
a
n
−
1
)
≥
2
a
2
a
3
…
a
n
If tangents at C,D meet at P then PC=PE
A
B
C
D
ABCD
A
BC
D
is a cyclic quadrilateral inscribed in the circle
Γ
\Gamma
Γ
with
A
B
AB
A
B
as diameter. Let
E
E
E
be the intersection of the diagonals
A
C
AC
A
C
and
B
D
BD
B
D
. The tangents to
Γ
\Gamma
Γ
at the points
C
,
D
C,D
C
,
D
meet at
P
P
P
. Prove that
P
C
=
P
E
PC=PE
PC
=
PE
.
3
2
Hide problems
Two circles and two tangents
Disjoint circles
o
1
,
o
2
o_1, o_2
o
1
,
o
2
, with centers
I
1
,
I
2
I_1, I_2
I
1
,
I
2
respectively, are tangent to the line
k
k
k
at
A
1
,
A
2
A_1, A_2
A
1
,
A
2
respectively and they lie on the same side of this line. Point
C
C
C
lies on segment
I
1
I
2
I_1I_2
I
1
I
2
and \angle A_1CA_2 \equal{} 90^{\circ}. Let
B
1
B_1
B
1
be the second intersection of
A
1
C
A_1C
A
1
C
with
o
1
o_1
o
1
, and let
B
2
B_2
B
2
be the second intersection of
A
2
C
A_2C
A
2
C
with
o
2
o_2
o
2
. Prove that
B
1
B
2
B_1B_2
B
1
B
2
is tangent to the circles
o
1
,
o
2
o_1, o_2
o
1
,
o
2
.
Find sequences of reals with two sums
For every integer
n
≥
3
n\ge 3
n
≥
3
find all sequences of real numbers
(
x
1
,
x
2
,
…
,
x
n
)
(x_1,x_2,\ldots ,x_n)
(
x
1
,
x
2
,
…
,
x
n
)
such that
∑
i
=
1
n
x
i
=
n
\sum_{i=1}^{n}x_i=n
∑
i
=
1
n
x
i
=
n
and
∑
i
=
1
n
(
x
i
−
1
−
x
i
+
x
i
+
1
)
2
=
n
\sum_{i=1}^{n} (x_{i-1}-x_i+x_{i+1})^2=n
∑
i
=
1
n
(
x
i
−
1
−
x
i
+
x
i
+
1
)
2
=
n
, where
x
0
=
x
n
x_0=x_n
x
0
=
x
n
and
x
n
+
1
=
x
1
x_{n+1}=x_1
x
n
+
1
=
x
1
.