MathDB
Problems
Contests
National and Regional Contests
Poland Contests
Poland - Second Round
2014 Poland - Second Round
2014 Poland - Second Round
Part of
Poland - Second Round
Subcontests
(6)
6.
1
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99% fine numbers
Call a positive number
n
n
n
fine, if there exists a prime number
p
p
p
such that
p
∣
n
p|n
p
∣
n
and
p
2
∤
n
p^2\nmid n
p
2
∤
n
. Prove that at least 99% of numbers
1
,
2
,
3
,
…
,
1
0
12
1, 2, 3, \ldots, 10^{12}
1
,
2
,
3
,
…
,
1
0
12
are fine numbers.
5.
1
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Tangency and circles
Circles
o
1
o_1
o
1
and
o
2
o_2
o
2
tangent to some line at points
A
A
A
and
B
B
B
, respectively, intersect at points
X
X
X
and
Y
Y
Y
(
X
X
X
is closer to the line
A
B
AB
A
B
). Line
A
X
AX
A
X
intersects
o
2
o_2
o
2
at point
P
≠
X
P\neq X
P
=
X
. Tangent to
o
2
o_2
o
2
at point
P
P
P
intersects line
A
B
AB
A
B
at point
Q
Q
Q
. Prove that
∢
X
Y
B
=
∢
B
Y
Q
\sphericalangle XYB = \sphericalangle BYQ
∢
X
Y
B
=
∢
B
Y
Q
.
4.
1
Hide problems
Traveling teams
2
n
2n
2
n
(
n
≥
2
n\ge 2
n
≥
2
) teams took part in the football league matches and there were
2
n
−
1
2n-1
2
n
−
1
matchweeks. In each matchweek each team played one match. Any two teams met with each other during the matches in exactly one game. Moreover, in each match one team was the host and the second was a guest. Say a team is traveling, if in any two consecutive matchweeks it was once a host and once a guest. Prove that there are at most two traveling teams.
3.
1
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Smallest possible value
For each positive integer
n
n
n
, determine the smallest possible value of the polynomial
W
n
(
x
)
=
x
2
n
+
2
x
2
n
−
1
+
3
x
2
n
−
2
+
…
+
(
2
n
−
1
)
x
2
+
2
n
x
.
W_n(x)=x^{2n}+2x^{2n-1}+3x^{2n-2}+\ldots + (2n-1)x^2+2nx.
W
n
(
x
)
=
x
2
n
+
2
x
2
n
−
1
+
3
x
2
n
−
2
+
…
+
(
2
n
−
1
)
x
2
+
2
n
x
.
2.
1
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Equation with radii
Distinct points
A
A
A
,
B
B
B
and
C
C
C
lie on a line in this order. Point
D
D
D
lies on the perpendicular bisector of the segment
B
C
BC
BC
. Denote by
M
M
M
the midpoint of the segment
B
C
BC
BC
. Let
r
r
r
be the radius of the incircle of the triangle
A
B
D
ABD
A
B
D
and let
R
R
R
be the radius of the circle with center lying outside the triangle
A
C
D
ACD
A
C
D
, tangent to
C
D
CD
C
D
,
A
C
AC
A
C
and
A
D
AD
A
D
. Prove that
D
M
=
r
+
R
DM=r+R
D
M
=
r
+
R
.
1.
1
Hide problems
Prove that y|x^2
Let
x
,
y
x, y
x
,
y
be positive integers such that
x
2
y
+
y
2
x
\frac{x^2}{y}+\frac{y^2}{x}
y
x
2
+
x
y
2
is an integer. Prove that
y
∣
x
2
y|x^2
y
∣
x
2
.