MathDB
Problems
Contests
National and Regional Contests
Poland Contests
Poland - Second Round
2016 Poland - Second Round
2016 Poland - Second Round
Part of
Poland - Second Round
Subcontests
(6)
6
1
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Green points and red segments in a data space (a. k. a. cherries)
n
n
n
(
n
≥
4
n \ge 4
n
≥
4
) green points are in a data space and no
4
4
4
green points lie on one plane. Some segments which connect green points have been colored red. Number of red segments is even. Each two green points are connected with polyline which is build from red segments. Show that red segments can be split on pairs, such that segments from one pair have common end.
5
1
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Show that middle point of line segment lies on line
Quadrilateral
A
B
C
D
ABCD
A
BC
D
is inscribed in circle. Points
P
P
P
and
Q
Q
Q
lie respectively on rays
A
B
→
AB^{\rightarrow}
A
B
→
and
A
D
→
AD^{\rightarrow}
A
D
→
such that
A
P
=
C
D
AP = CD
A
P
=
C
D
,
A
Q
=
B
C
AQ = BC
A
Q
=
BC
. Show that middle point of line segment
P
Q
PQ
PQ
lies on the line
A
C
AC
A
C
.
4
1
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Show that exists positive integer
Let
k
k
k
be a positive integer. Show that exists positive integer
n
n
n
, such that sets
A
=
{
1
2
,
2
2
,
3
3
,
.
.
.
}
A = \{ 1^2, 2^2, 3^3, ...\}
A
=
{
1
2
,
2
2
,
3
3
,
...
}
and
B
=
{
1
2
+
n
,
2
2
+
n
,
3
2
+
n
,
.
.
.
}
B = \{1^2 + n, 2^2 + n, 3^2 + n, ... \}
B
=
{
1
2
+
n
,
2
2
+
n
,
3
2
+
n
,
...
}
have exactly
k
k
k
common elements.
3
1
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Function on integers
Determine, whether exists function
f
f
f
, which assigns each integer
k
k
k
, nonnegative integer
f
(
k
)
f(k)
f
(
k
)
and meets the conditions:
f
(
0
)
>
0
f(0) > 0
f
(
0
)
>
0
, for each integer
k
k
k
minimal number of the form
f
(
k
−
l
)
+
f
(
l
)
f(k - l) + f(l)
f
(
k
−
l
)
+
f
(
l
)
, where
l
∈
Z
l \in \mathbb{Z}
l
∈
Z
, equals
f
(
k
)
f(k)
f
(
k
)
.
2
1
Hide problems
Show that circumcircle is tangent
In acute triangle
A
B
C
ABC
A
BC
bisector of angle
B
A
C
BAC
B
A
C
intersects side
B
C
BC
BC
in point
D
D
D
. Bisector of line segment
A
D
AD
A
D
intersects circumcircle of triangle
A
B
C
ABC
A
BC
in points
E
E
E
and
F
F
F
. Show that circumcircle of triangle
D
E
F
DEF
D
EF
is tangent to line
B
C
BC
BC
.
1
1
Hide problems
Rational distances in triangle
Point
P
P
P
lies inside triangle of sides of length
3
,
4
,
5
3, 4, 5
3
,
4
,
5
. Show that if distances between
P
P
P
and vertices of triangle are rational numbers then distances from
P
P
P
to sides of triangle are rational numbers too.