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National and Regional Contests
Hungary Contests
Kürschák Math Competition
1967 Kurschak Competition
1967 Kurschak Competition
Part of
Kürschák Math Competition
Subcontests
(3)
3
1
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ABCD is parallelogram if h(A)=h(B) =h(C)=h(D), sum of distances from 2 sides
For a vertex
X
X
X
of a quadrilateral, let
h
(
X
)
h(X)
h
(
X
)
be the sum of the distances from
X
X
X
to the two sides not containing
X
X
X
. Show that if a convex quadrilateral
A
B
C
D
ABCD
A
BC
D
satisfies
h
(
A
)
=
h
(
B
)
=
h
(
C
)
=
h
(
D
)
h(A) = h(B) = h(C) = h(D)
h
(
A
)
=
h
(
B
)
=
h
(
C
)
=
h
(
D
)
, then it must be a parallelogram.
2
1
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convex n-gon is divided into triangles by diagonal
A convex
n
n
n
-gon is divided into triangles by diagonals which do not intersect except at vertices of the n-gon. Each vertex belongs to an odd number of triangles. Show that
n
n
n
must be a multiple of
3
3
3
.
1
1
Hide problems
difference of any two elements of set of integers A also belongs to A
A
A
A
is a set of integers which is closed under addition and contains both positive and negative numbers. Show that the difference of any two elements of
A
A
A
also belongs to
A
A
A
.