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National and Regional Contests
Hungary Contests
Kürschák Math Competition
2014 Kurschak Competition
2014 Kurschak Competition
Part of
Kürschák Math Competition
Subcontests
(3)
3
1
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Reflecting on the sides of a convex polygon
Let
K
K
K
be a closed convex polygonal region, and let
X
X
X
be a point in the plane of
K
K
K
. Show that there exists a finite sequence of reflections in the sides of
K
K
K
, such that
K
K
K
contains the image of
X
X
X
after these reflections.
2
1
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Three concurrent circles
We are given an acute triangle
A
B
C
ABC
A
BC
, and inside it a point
P
P
P
, which is not on any of the heights
A
A
1
AA_1
A
A
1
,
B
B
1
BB_1
B
B
1
,
C
C
1
CC_1
C
C
1
. The rays
A
P
AP
A
P
,
B
P
BP
BP
,
C
P
CP
CP
intersect the circumcircle of
A
B
C
ABC
A
BC
at points
A
2
A_2
A
2
,
B
2
B_2
B
2
,
C
2
C_2
C
2
. Prove that the circles
A
A
1
A
2
AA_1A_2
A
A
1
A
2
,
B
B
1
B
2
BB_1B_2
B
B
1
B
2
and
C
C
1
C
2
CC_1C_2
C
C
1
C
2
are concurrent.
1
1
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Easy graph theory
Consider a company of
n
≥
4
n\ge 4
n
≥
4
people, where everyone knows at least one other person, but everyone knows at most
n
−
2
n-2
n
−
2
of the others. Prove that we can sit four of these people at a round table such that all four of them know exactly one of their two neighbors. (Knowledge is mutual.)