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Problems
Contests
National and Regional Contests
Portugal Contests
Portugal MO
2008 Portugal MO
2008 Portugal MO
Part of
Portugal MO
Subcontests
(6)
6
1
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Rock disorganization - Portuguese MO, Problem 6, 2008
Let
n
n
n
be a natural number larger than
2
2
2
. Vanessa has
n
n
n
piles of jade stones, and all the piles have a different number of stones. Vanessa can distribute the stones from any pile by the other piles and stay with
n
−
1
n-1
n
−
1
piles with the same number of stones. She also can distribute the stones from any two piles by the other piles and stay with
n
−
2
n-2
n
−
2
piles with the same number of stones. Find the smallest possible number of jade's stones that the pile with the largest number of stones can have.
5
1
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Mexican geometry - Portuguese MO, Problem 5, 2008
Let
A
B
C
ABC
A
BC
be a right-angled triangle in
A
A
A
such that
A
B
<
A
C
AB<AC
A
B
<
A
C
. Let
M
M
M
be the midpoint of
B
C
BC
BC
and let
D
D
D
be the intersection of
A
C
AC
A
C
with the perpendicular line to
B
C
BC
BC
which passes through
M
M
M
. Let
E
E
E
be the intersection point of the parallel line to
A
C
AC
A
C
which passes through
M
M
M
with the perpendicular line to
B
D
BD
B
D
which passes through
B
B
B
. Prove that triangles
A
E
M
AEM
A
EM
and
M
C
A
MCA
MC
A
are similar if and only if
∠
A
B
C
=
6
0
∘
\angle ABC=60^{\circ}
∠
A
BC
=
6
0
∘
.
4
1
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Italian nice game - Portuguese MO, Problem 4, 2008
Nelson challenges Telma for the following game: First Telma takes
2
9
2^9
2
9
numbers from the set
{
0
,
1
,
2
,
3
,
⋯
,
1024
}
\left\{0,1,2,3,\cdots,1024\right\}
{
0
,
1
,
2
,
3
,
⋯
,
1024
}
, then Nelson takes
2
8
2^8
2
8
of the remaining numbers. Then Telma takes
2
7
2^7
2
7
numbers and successively, until only two numbers remain. Nelson will have to give Telma the difference between these two numbers in euros. What is the largest amount Telma can win, whatever Nelson's strategy is?
3
1
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Friendly numbers - Portuguese MO, Problem 3, 2008
Let
d
d
d
be a natural number. Given two natural numbers
M
M
M
and
N
N
N
with
d
d
d
digits,
M
M
M
is a friend of
N
N
N
if and only if the
d
d
d
numbers obtained substituting each one of the digits of
M
M
M
by the digit of
N
N
N
which is on the same position are all multiples of
7
7
7
. Find all the values of
d
d
d
for which the following condition is valid: For any two numbers
M
M
M
and
N
N
N
with
d
d
d
digits,
M
M
M
is a friend of
N
N
N
if and only if
N
N
N
is a friend of
M
M
M
.
2
1
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Equivalence between sides - Portuguese MO, Problem 2, 2008
Let
A
E
B
C
AEBC
A
EBC
be a cyclic quadrilateral. Let
D
D
D
be a point on the ray
A
E
AE
A
E
which is outside the circumscribed circumference of
A
E
B
C
AEBC
A
EBC
. Suppose that
∠
C
A
B
=
∠
B
A
E
\angle CAB=\angle BAE
∠
C
A
B
=
∠
B
A
E
. Prove that
A
B
=
B
D
AB=BD
A
B
=
B
D
if and only if
D
E
=
A
C
DE=AC
D
E
=
A
C
.
1
1
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Trivial high school count - Portuguese MO, Problem 1, 2008
What is the maximum number of triangles with vertices on the points of the fork/graph which is possible to construct?