MathDB
Problems
Contests
National and Regional Contests
Hungary Contests
Kürschák Math Competition
1997 Kurschak Competition
1997 Kurschak Competition
Part of
Kürschák Math Competition
Subcontests
(3)
3
1
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3-coloring, no one-color cycle
Prove that the vertices of any planar graph can be colored with
3
3
3
colors such that there is no monochromatic cycle.
2
1
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O, I and orthocenter of intouch triangle collinear
The center of the circumcircle of
△
A
B
C
\triangle ABC
△
A
BC
is
O
O
O
. The incenter of the triangle is
I
I
I
, and the intouch triangle is
A
1
B
1
C
1
A_1B_1C_1
A
1
B
1
C
1
. Let
H
1
H_1
H
1
be the orthocenter of
△
A
1
B
1
C
1
\triangle A_1B_1C_1
△
A
1
B
1
C
1
. Prove that
O
O
O
,
I
I
I
, and
H
1
H_1
H
1
are collinear.
1
1
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Lattice points, no three collinear
Let
p
>
2
p>2
p
>
2
be a prime number and let
L
=
{
0
,
1
,
…
,
p
−
1
}
2
L=\{0,1,\dots,p-1\}^2
L
=
{
0
,
1
,
…
,
p
−
1
}
2
. Prove that we can find
p
p
p
points in
L
L
L
with no three of them collinear.