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Problems
Contests
International Contests
Mediterranean Mathematics Olympiad
2001 Mediterranean Mathematics Olympiad
2001 Mediterranean Mathematics Olympiad
Part of
Mediterranean Mathematics Olympiad
Subcontests
(4)
4
1
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Problem with function on points
Let
S
S
S
be the set of points inside a given equilateral triangle
A
B
C
ABC
A
BC
with side
1
1
1
or on its boundary. For any
M
∈
S
,
a
M
,
b
M
,
c
M
M \in S, a_M, b_M, c_M
M
∈
S
,
a
M
,
b
M
,
c
M
denote the distances from
M
M
M
to
B
C
,
C
A
,
A
B
BC,CA,AB
BC
,
C
A
,
A
B
, respectively. Define
f
(
M
)
=
a
M
3
(
b
M
−
c
M
)
+
b
M
3
(
c
M
−
a
M
)
+
c
M
3
(
a
M
−
b
M
)
.
f(M) = a_M^3 (b_M - c_M) + b_M^3(c_M - a_M) + c_M^3(a_M - b_M).
f
(
M
)
=
a
M
3
(
b
M
−
c
M
)
+
b
M
3
(
c
M
−
a
M
)
+
c
M
3
(
a
M
−
b
M
)
.
(a) Describe the set
{
M
∈
S
∣
f
(
M
)
≥
0
}
\{M \in S | f(M) \geq 0\}
{
M
∈
S
∣
f
(
M
)
≥
0
}
geometrically.(b) Find the minimum and maximum values of
f
(
M
)
f(M)
f
(
M
)
as well as the points in which these are attained.
3
1
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Show that there exists a positive integer N
Show that there exists a positive integer
N
N
N
such that the decimal representation of
200
0
N
2000^N
200
0
N
starts with the digits
200120012001.
200120012001.
200120012001.
2
1
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Find all integers n that polynomial can be represented
Find all integers
n
n
n
for which the polynomial
p
(
x
)
=
x
5
−
n
x
−
n
−
2
p(x) = x^5 -nx -n -2
p
(
x
)
=
x
5
−
n
x
−
n
−
2
can be represented as a product of two non-constant polynomials with integer coefficients.
1
1
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the intersection point depends only on k and P, Q
Let
P
P
P
and
Q
Q
Q
be points on a circle
k
k
k
. A chord
A
C
AC
A
C
of
k
k
k
passes through the midpoint
M
M
M
of
P
Q
PQ
PQ
. Consider a trapezoid
A
B
C
D
ABCD
A
BC
D
inscribed in
k
k
k
with
A
B
∥
P
Q
∥
C
D
AB \parallel PQ \parallel CD
A
B
∥
PQ
∥
C
D
. Prove that the intersection point
X
X
X
of
A
D
AD
A
D
and
B
C
BC
BC
depends only on
k
k
k
and
P
,
Q
.
P,Q.
P
,
Q
.