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National and Regional Contests
Hungary Contests
Eotvos Mathematical Competition (Hungary)
1934 Eotvos Mathematical Competition
1934 Eotvos Mathematical Competition
Part of
Eotvos Mathematical Competition (Hungary)
Subcontests
(3)
1
1
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at least one term of the sequence A, 2A,4A,8A,...,2^kA, ... is an integer
Let
n
n
n
be a given positive integer and
A
=
1
⋅
3
⋅
5
⋅
.
.
.
⋅
(
2
n
−
1
)
2
⋅
4
⋅
6
⋅
.
.
.
⋅
2
n
A =\frac{1 \cdot 3 \cdot 5 \cdot ... \cdot (2n- 1)}{2 \cdot 4 \cdot 6 \cdot ... \cdot 2n}
A
=
2
⋅
4
⋅
6
⋅
...
⋅
2
n
1
⋅
3
⋅
5
⋅
...
⋅
(
2
n
−
1
)
Prove that at least one term of the sequence
A
,
2
A
,
4
A
,
8
A
,
.
.
.
,
2
k
A
,
.
.
.
A, 2A,4A,8A,...,2^kA, ...
A
,
2
A
,
4
A
,
8
A
,
...
,
2
k
A
,
...
is an integer.
3
1
Hide problems
exist two rectangles in the set such that one contains the other
We are given an infinite set of rectangles in the plane, each with vertices of the form
(
0
,
0
)
(0, 0)
(
0
,
0
)
,
(
0
,
m
)
(0,m)
(
0
,
m
)
,
(
n
,
0
)
(n, 0)
(
n
,
0
)
and
(
n
,
m
)
(n,m)
(
n
,
m
)
, where
m
m
m
and
n
n
n
are positive integers. Prove that there exist two rectangles in the set such that one contains the other.
2
1
Hide problems
max sum of squares of sidelenghts for cyclic polygon
Which polygon inscribed in a given circle has the property that the sum of the squares of the lengths of its sides is maximum?