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National and Regional Contests
Hungary Contests
Kürschák Math Competition
2002 Kurschak Competition
2002 Kurschak Competition
Part of
Kürschák Math Competition
Subcontests
(3)
3
1
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3^n-vertex cliques are partitionable into 3 long cycles
Prove that the edges of a complete graph with
3
n
3^n
3
n
vertices can be partitioned into disjoint cycles of length
3
3
3
.
2
1
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Where Fibonacci meets Farey
The Fibonacci sequence is defined as
f
1
=
f
2
=
1
f_1=f_2=1
f
1
=
f
2
=
1
,
f
n
+
2
=
f
n
+
1
+
f
n
f_{n+2}=f_{n+1}+f_n
f
n
+
2
=
f
n
+
1
+
f
n
(
n
∈
N
n\in\mathbb{N}
n
∈
N
). Suppose that
a
a
a
and
b
b
b
are positive integers such that
a
b
\frac ab
b
a
lies between the two fractions
f
n
f
n
−
1
\frac{f_n}{f_{n-1}}
f
n
−
1
f
n
and
f
n
+
1
f
n
\frac{f_{n+1}}{f_{n}}
f
n
f
n
+
1
. Show that
b
≥
f
n
+
1
b\ge f_{n+1}
b
≥
f
n
+
1
.
1
1
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A vertex of a triangle concyclic with H, O, I
We have an acute-angled triangle which is not isosceles. We denote the orthocenter, the circumcenter and the incenter of it by
H
H
H
,
O
O
O
,
I
I
I
respectively. Prove that if a vertex of the triangle lies on the circle
H
O
I
HOI
H
O
I
, then there must be another vertex on this circle as well.