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Contests
International Contests
Middle European Mathematical Olympiad
2007 Middle European Mathematical Olympiad
2007 Middle European Mathematical Olympiad
Part of
Middle European Mathematical Olympiad
Subcontests
(4)
4
2
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x!+y!=x^y
Determine all pairs
(
x
,
y
)
(x,y)
(
x
,
y
)
of positive integers satisfying the equation x!\plus{}y!\equal{}x^{y}.
2007|(a+k)^3-a^3
Find all positive integers
k
k
k
with the following property: There exists an integer
a
a
a
so that (a\plus{}k)^{3}\minus{}a^{3} is a multiple of
2007
2007
2007
.
3
2
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5 Circles and a Rectangle (nice but well known)
Let
k
k
k
be a circle and
k
1
,
k
2
,
k
3
,
k
4
k_{1},k_{2},k_{3},k_{4}
k
1
,
k
2
,
k
3
,
k
4
four smaller circles with their centres
O
1
,
O
2
,
O
3
,
O
4
O_{1},O_{2},O_{3},O_{4}
O
1
,
O
2
,
O
3
,
O
4
respectively, on
k
k
k
. For i \equal{} 1,2,3,4 and k_{5}\equal{} k_{1} the circles
k
i
k_{i}
k
i
and k_{i\plus{}1} meet at
A
i
A_{i}
A
i
and
B
i
B_{i}
B
i
such that
A
i
A_{i}
A
i
lies on
k
k
k
. The points
O
1
,
A
1
,
O
2
,
A
2
,
O
3
,
A
3
,
O
4
,
A
4
O_{1},A_{1},O_{2},A_{2},O_{3},A_{3},O_{4},A_{4}
O
1
,
A
1
,
O
2
,
A
2
,
O
3
,
A
3
,
O
4
,
A
4
lie in that order on
k
k
k
and are pairwise different. Prove that
B
1
B
2
B
3
B
4
B_{1}B_{2}B_{3}B_{4}
B
1
B
2
B
3
B
4
is a rectangle.
Tetrahedron with Integer Sidelengths
A tetrahedron is called a MEMO-tetrahedron if all six sidelengths are different positive integers where one of them is
2
2
2
and one of them is
3
3
3
. Let
l
(
T
)
l(T)
l
(
T
)
be the sum of the sidelengths of the tetrahedron
T
T
T
. (a) Find all positive integers
n
n
n
so that there exists a MEMO-Tetrahedron
T
T
T
with l(T)\equal{}n. (b) How many pairwise non-congruent MEMO-tetrahedrons
T
T
T
satisfying l(T)\equal{}2007 exist? Two tetrahedrons are said to be non-congruent if one cannot be obtained from the other by a composition of reflections in planes, translations and rotations. (It is not neccessary to prove that the tetrahedrons are not degenerate, i.e. that they have a positive volume).
2
2
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(Sub-)Sets of balls labelled with numbers
A set of balls contains
n
n
n
balls which are labeled with numbers
1
,
2
,
3
,
…
,
n
.
1,2,3,\ldots,n.
1
,
2
,
3
,
…
,
n
.
We are given
k
>
1
k > 1
k
>
1
such sets. We want to colour the balls with two colours, black and white in such a way, that (a) the balls labeled with the same number are of the same colour, (b) any subset of k\plus{}1 balls with (not necessarily different) labels a_{1},a_{2},\ldots,a_{k\plus{}1} satisfying the condition a_{1}\plus{}a_{2}\plus{}\ldots\plus{}a_{k}\equal{} a_{k\plus{}1}, contains at least one ball of each colour. Find, depending on
k
k
k
the greatest possible number
n
n
n
which admits such a colouring.
Set of 5 Points and Numbers of Acute Triangles
For a set
P
P
P
of five points in the plane, no three of them being collinear, let
s
(
P
)
s(P)
s
(
P
)
be the numbers of acute triangles formed by vertices in
P
P
P
. Find the maximum value of
s
(
P
)
s(P)
s
(
P
)
over all such sets
P
P
P
.
1
2
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Cyclic Inequality in 4 Variables with a+b+c+d=4
Let
a
,
b
,
c
,
d
a,b,c,d
a
,
b
,
c
,
d
be positive real numbers with a\plus{}b\plus{}c\plus{}d \equal{} 4. Prove that a^{2}bc\plus{}b^{2}cd\plus{}c^{2}da\plus{}d^{2}ab\leq 4.
Cyclic Inequality in 4 variables with abcd=1
Let
a
,
b
,
c
,
d
a,b,c,d
a
,
b
,
c
,
d
be real numbers which satisfy
1
2
≤
a
,
b
,
c
,
d
≤
2
\frac{1}{2}\leq a,b,c,d\leq 2
2
1
≤
a
,
b
,
c
,
d
≤
2
and abcd\equal{}1. Find the maximum value of \left(a\plus{}\frac{1}{b}\right)\left(b\plus{}\frac{1}{c}\right)\left(c\plus{}\frac{1}{d}\right)\left(d\plus{}\frac{1}{a}\right).