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Problems
Contests
National and Regional Contests
Hungary Contests
Kürschák Math Competition
2023 Kurschak Competition
2023 Kurschak Competition
Part of
Kürschák Math Competition
Subcontests
(3)
3
1
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Cyclic pentagon
Given is a convex cyclic pentagon
A
B
C
D
E
ABCDE
A
BC
D
E
and a point
P
P
P
inside it, such that
A
B
=
A
E
=
A
P
AB=AE=AP
A
B
=
A
E
=
A
P
and
B
C
=
C
E
BC=CE
BC
=
CE
. The lines
A
D
AD
A
D
and
B
E
BE
BE
intersect in
Q
Q
Q
. Points
R
R
R
and
S
S
S
are on segments
C
P
CP
CP
and
B
P
BP
BP
such that
D
R
=
Q
R
DR=QR
D
R
=
QR
and
S
R
∣
∣
B
C
SR||BC
SR
∣∣
BC
. Show that the circumcircles of
B
E
P
BEP
BEP
and
P
Q
S
PQS
PQS
are tangent to each other.
2
1
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Vital edges
Let
n
≥
2
n\geq 2
n
≥
2
be a positive integer. We call a vertex every point in the coordinate plane, whose
x
x
x
and
y
y
y
coordinates both are from the set
{
1
,
2
,
3
,
.
.
.
,
n
}
\{1,2,3,...,n\}
{
1
,
2
,
3
,
...
,
n
}
. We call a segment between two vertices an edge, if its length if
1
1
1
. We've colored some edges red, such that between any two vertices, there is a unique path of red edges (a path may contain each edge at most once). The red edge
f
f
f
is vital for an edge
e
e
e
, if the path of red edges connecting the two endpoints of
e
e
e
contain
f
f
f
. Prove that there is a red edge, such that it is vital for at least
n
n
n
edges.
1
1
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Self dividing polynomial
Let
f
(
x
)
f(x)
f
(
x
)
be a non-constant polynomial with non-negative integer coefficients. Prove that there are infinitely many positive integers
n
n
n
, for which
f
(
n
)
f(n)
f
(
n
)
is not divisible by any of
f
(
2
)
f(2)
f
(
2
)
,
f
(
3
)
f(3)
f
(
3
)
, ...,
f
(
n
−
1
)
f(n-1)
f
(
n
−
1
)
.