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Contests
National and Regional Contests
Hungary Contests
Kürschák Math Competition
2004 Kurschak Competition
2004 Kurschak Competition
Part of
Kürschák Math Competition
Subcontests
(3)
3
1
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Red and blue points on a circle
We have placed some red and blue points along a circle. The following operations are permitted:(a) we may add a red point somewhere and switch the color of its neighbors,(b) we may take off a red point from somewhere and switch the color of its neighbors (if there are at least
3
3
3
points on the circle and there is a red one too).Initially, there are two blue points on the circle. Using a number of these operations, can we reach a state with exactly two red point?
2
1
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Number of roots of f(x)-n
Find the smallest positive integer
n
≠
2004
n\neq 2004
n
=
2004
for which there exists a polynomial
f
∈
Z
[
x
]
f\in\mathbb{Z}[x]
f
∈
Z
[
x
]
such that the equation
f
(
x
)
=
2004
f(x)=2004
f
(
x
)
=
2004
has at least one, and the equation
f
(
x
)
=
n
f(x)=n
f
(
x
)
=
n
has at least
2004
2004
2004
different integer solutions.
1
1
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Mixtilinear circle
Given is a triangle
A
B
C
ABC
A
BC
, its circumcircle
ω
\omega
ω
, and a circle
k
k
k
that touches
ω
\omega
ω
from the outside, and also touches rays
A
B
AB
A
B
and
A
C
AC
A
C
in
P
P
P
and
Q
Q
Q
, respectively. Prove that the
A
A
A
-excenter of
△
A
B
C
\triangle ABC
△
A
BC
is the midpoint of
P
Q
‾
\overline{PQ}
PQ
.