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Contests
National and Regional Contests
Hungary Contests
Kürschák Math Competition
1951 Kurschak Competition
1951 Kurschak Competition
Part of
Kürschák Math Competition
Subcontests
(3)
1
1
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lines AE and BF intersect on the circumcircle of the square ABCD
A
B
C
D
ABCD
A
BC
D
is a square.
E
E
E
is a point on the side
B
C
BC
BC
such that
B
E
=
1
/
3
B
C
BE =1/3 BC
BE
=
1/3
BC
, and
F
F
F
is a point on the ray
D
C
DC
D
C
such that
C
F
=
1
/
2
D
C
CF =1/2 DC
CF
=
1/2
D
C
. Prove that the lines
A
E
AE
A
E
and
B
F
BF
BF
intersect on the circumcircle of the square. https://cdn.artofproblemsolving.com/attachments/e/d/09a8235d0748ce4479e21a3bb09b0359de54b5.png
3
1
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4 open half-planes cover the plane
An open half-plane is the set of all points lying to one side of a line, but excluding the points on the line itself. If four open half-planes cover the plane, show that one can select three of them which still cover the plane.
2
1
Hide problems
(m -1)! divisible by m
For which
m
>
1
m > 1
m
>
1
is
(
m
−
1
)
!
(m -1)!
(
m
−
1
)!
divisible by
m
m
m
?