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Contests
International Contests
Mediterranean Mathematics Olympiad
2008 Mediterranean Mathematics Olympiad
2008 Mediterranean Mathematics Olympiad
Part of
Mediterranean Mathematics Olympiad
Subcontests
(4)
4
1
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A Sequence of Polynomials
The sequence of polynomials
(
a
n
)
(a_n)
(
a
n
)
is defined by
a
0
=
0
a_0=0
a
0
=
0
,
a
1
=
x
+
2
a_1=x+2
a
1
=
x
+
2
and
a
n
=
a
n
−
1
+
3
a
n
−
1
a
n
−
2
+
a
n
−
2
a_n=a_{n-1}+3a_{n-1}a_{n-2} +a_{n-2}
a
n
=
a
n
−
1
+
3
a
n
−
1
a
n
−
2
+
a
n
−
2
for
n
>
1
n>1
n
>
1
. (a) Show for all positive integers
k
,
m
k,m
k
,
m
: if
k
k
k
divides
m
m
m
then
a
k
a_k
a
k
divides
a
m
a_m
a
m
. (b) Find all positive integers
n
n
n
such that the sum of the roots of polynomial
a
n
a_n
a
n
is an integer.
2
1
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Points on a Plane
Determine whether there exist two infinite point sequences
A
1
,
A
2
,
…
A_1,A_2,\ldots
A
1
,
A
2
,
…
and
B
1
,
B
2
,
…
B_1,B_2,\ldots
B
1
,
B
2
,
…
in the plane, such that for all
i
,
j
,
k
i,j,k
i
,
j
,
k
with
1
≤
i
<
j
<
k
1\le i < j < k
1
≤
i
<
j
<
k
, (i)
B
k
B_k
B
k
is on the line that passes through
A
i
A_i
A
i
and
A
j
A_j
A
j
if and only if
k
=
i
+
j
k=i+j
k
=
i
+
j
. (ii)
A
k
A_k
A
k
is on the line that passes through
B
i
B_i
B
i
and
B
j
B_j
B
j
if and only if
k
=
i
+
j
k=i+j
k
=
i
+
j
. (Proposed by Gerhard Woeginger, Austria)
3
1
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Calculate the Sum
Let
n
n
n
be a positive integer. Calculate the sum
∑
k
=
1
n
∑
1
≤
i
1
<
…
<
i
k
≤
n
2
k
(
i
1
+
1
)
(
i
2
+
1
)
…
(
i
k
+
1
)
\sum_{k=1}^n\ \ {\sum_{1\le i_1 < \ldots < i_k\le n}^{}{\frac {2^k}{(i_1 + 1)(i_2 + 1)\ldots (i_k + 1)}}}
∑
k
=
1
n
∑
1
≤
i
1
<
…
<
i
k
≤
n
(
i
1
+
1
)
(
i
2
+
1
)
…
(
i
k
+
1
)
2
k
1
1
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Convex Hexagon
Let
A
B
C
D
E
F
ABCDEF
A
BC
D
EF
be a convex hexagon such that all of its vertices are on a circle. Prove that
A
D
AD
A
D
,
B
E
BE
BE
and
C
F
CF
CF
are concurrent if and only if
A
B
B
C
⋅
C
D
D
E
⋅
E
F
F
A
=
1
\frac {AB}{BC}\cdot\frac {CD}{DE}\cdot\frac {EF}{FA}= 1
BC
A
B
⋅
D
E
C
D
⋅
F
A
EF
=
1
.