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Problems
Contests
National and Regional Contests
Hungary Contests
Kürschák Math Competition
2003 Kurschak Competition
2003 Kurschak Competition
Part of
Kürschák Math Competition
Subcontests
(3)
3
1
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Lower estimate on Sum gcd(i,j)
Prove that the following inequality holds with the exception of finitely many positive integers
n
n
n
:
∑
i
=
1
n
∑
j
=
1
n
g
c
d
(
i
,
j
)
>
4
n
2
.
\sum_{i=1}^n\sum_{j=1}^n gcd(i,j)>4n^2.
i
=
1
∑
n
j
=
1
∑
n
g
c
d
(
i
,
j
)
>
4
n
2
.
2
1
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Unique 3-coloring of a graph
Prove that if a graph
G
\mathcal{G}
G
on
n
≥
3
n\ge 3
n
≥
3
vertices has a unique
3
3
3
-coloring, then
G
\mathcal{G}
G
has at least
2
n
−
3
2n-3
2
n
−
3
edges.(A graph is
3
3
3
-colorable when there exists a coloring of its vertices with
3
3
3
colors such that no two vertices of the same color are connected by an edge. The graph can be
3
3
3
-colored uniquely if there do not exist vertices
u
u
u
and
v
v
v
of the graph that are painted different colors in one
3
3
3
-coloring, yet are colored the same in another.)
1
1
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Power of a point practise
Draw a circle
k
k
k
with diameter
E
F
‾
\overline{EF}
EF
, and let its tangent in
E
E
E
be
e
e
e
. Consider all possible pairs
A
,
B
∈
e
A,B\in e
A
,
B
∈
e
for which
E
∈
A
B
‾
E\in \overline{AB}
E
∈
A
B
and
A
E
⋅
E
B
AE\cdot EB
A
E
⋅
EB
is a fixed constant. Define
(
A
1
,
B
1
)
=
(
A
F
∩
k
,
B
F
∩
k
)
(A_1,B_1)=(AF\cap k,BF\cap k)
(
A
1
,
B
1
)
=
(
A
F
∩
k
,
BF
∩
k
)
. Prove that the segments
A
1
B
1
‾
\overline{A_1B_1}
A
1
B
1
all concur in one point.