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National and Regional Contests
Hungary Contests
Kürschák Math Competition
1978 Kurschak Competition
1978 Kurschak Competition
Part of
Kürschák Math Competition
Subcontests
(3)
3
1
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H >=r+R where H is max altitude
A triangle has inradius
r
r
r
and circumradius
R
R
R
. Its longest altitude has length
H
H
H
. Show that if the triangle does not have an obtuse angle, then
H
≥
r
+
R
H \ge r+R
H
≥
r
+
R
. When does equality hold?
2
1
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coloring vertices of convex n-gon
The vertices of a convex
n
n
n
-gon are colored so that adjacent vertices have different colors. Prove that if
n
n
n
is odd, then the polygon can be divided into triangles with non-intersecting diagonals such that no diagonal has its endpoints the same color.
1
1
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ax^2 + by^2 = 1 infinite rational solutions if it has one
a
a
a
and
b
b
b
are rationals. Show that if
a
x
2
+
b
y
2
=
1
ax^2 + by^2 = 1
a
x
2
+
b
y
2
=
1
has a rational solution (in
x
x
x
and
y
y
y
), then it must have infinitely many.