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Problems
Contests
National and Regional Contests
Hungary Contests
Kürschák Math Competition
1994 Kurschak Competition
1994 Kurschak Competition
Part of
Kürschák Math Competition
Subcontests
(3)
3
1
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Sets of intervals
Consider the sets
A
1
,
A
2
,
…
,
A
n
A_1,A_2,\dots,A_n
A
1
,
A
2
,
…
,
A
n
. Set
A
k
A_k
A
k
is composed of
k
k
k
disjoint intervals on the real axis (
k
=
1
,
2
,
…
,
n
k=1,2,\dots,n
k
=
1
,
2
,
…
,
n
). Prove that from the intervals contained by these sets, one can choose
⌊
n
+
1
2
⌋
\left\lfloor\frac{n+1}2\right\rfloor
⌊
2
n
+
1
⌋
intervals such that they belong to pairwise different sets
A
k
A_k
A
k
, and no two of these intervals have a common point.
2
1
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Erasing diagonals in an n-gon
Prove that if we erase
n
−
3
n-3
n
−
3
diagonals of a regular
n
n
n
-gon, then we may still choose
n
−
3
n-3
n
−
3
of the remaining diagonals such that they don't intersect inside the
n
n
n
-gon; but it is possible to erase
n
−
2
n-2
n
−
2
diagonals such that this statement doesn't hold.
1
1
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Parallelogram, maximum of diagonals' angle
The ratio of the sides of a parallelogram is
λ
>
1
\lambda>1
λ
>
1
. Given
λ
\lambda
λ
, determine the maximum of the acute angle subtended by the diagonals of the parallelogram.