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Contests
National and Regional Contests
Hungary Contests
Kürschák Math Competition
1958 Kurschak Competition
1958 Kurschak Competition
Part of
Kürschák Math Competition
Subcontests
(3)
3
1
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(ACE)=(BDF) if ABCDEF is convex with oppoite sides //
The hexagon
A
B
C
D
E
F
ABCDEF
A
BC
D
EF
is convex and opposite sides are parallel. Show that the triangles
A
C
E
ACE
A
CE
and
B
D
F
BDF
B
D
F
have equal area
2
1
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m, n divisible by 3 if m^2 + mn + n^2 is divisible by 9
Show that if
m
m
m
and
n
n
n
are integers such that
m
2
+
m
n
+
n
2
m^2 + mn + n^2
m
2
+
mn
+
n
2
is divisible by
9
9
9
, then they must both be divisible by
3
3
3
.
1
1
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ontruse triangle with angles 120^o anomg 6 points
Given any six points in the plane, no three collinear, show that we can always find three which form an obtuse-angled triangle with one angle at least
12
0
o
120^o
12
0
o
.