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National and Regional Contests
Italy Contests
ITAMO
2002 ITAMO
2002 ITAMO
Part of
ITAMO
Subcontests
(6)
2
1
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Italian mathematical olympiad 2002, problem 2
The plan of a house has the shape of a capital
L
L
L
, obtained by suitably placing side-by-side four squares whose sides are
10
10
10
metres long. The external walls of the house are
10
10
10
metres high. The roof of the house has six faces, starting at the top of the six external walls, and each face forms an angle of
3
0
∘
30^\circ
3
0
∘
with respect to a horizontal plane. Determine the volume of the house (that is, of the solid delimited by the six external walls, the six faces of the roof, and the base of the house).
3
1
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Italian mathematical olympiad 2002, problem 3
Let
A
A
A
and
B
B
B
are two points on a plane, and let
M
M
M
be the midpoint of
A
B
AB
A
B
. Let
r
r
r
be a line and let
R
R
R
and
S
S
S
be the projections of
A
A
A
and
B
B
B
onto
r
r
r
. Assuming that
A
A
A
,
M
M
M
, and
R
R
R
are not collinear, prove that the circumcircle of triangle
A
M
R
AMR
A
MR
has the same radius as the circumcircle of
B
S
M
BSM
BSM
.
4
1
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Italian mathematical olympiad 2002, problem 4
Find all values of
n
n
n
for which all solutions of the equation
x
3
−
3
x
+
n
=
0
x^3-3x+n=0
x
3
−
3
x
+
n
=
0
are integers.
5
1
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Italian mathematical olympiad 2002, problem 5
Prove that if
m
=
5
n
+
3
n
+
1
m=5^n+3^n+1
m
=
5
n
+
3
n
+
1
is a prime, then
12
12
12
divides
n
n
n
.
1
1
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Italian mathematical olympiad 2002, problem 1
Find all
3
3
3
-digit positive integers that are
34
34
34
times the sum of their digits.
6
1
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Italian mathematical olympiad 2002, problem 6
We are given a chessboard with 100 rows and 100 columns. Two squares of the board are said to be adjacent if they have a common side. Initially all squares are white.a) Is it possible to colour an odd number of squares in such a way that each coloured square has an odd number of adjacent coloured squares?b) Is it possible to colour some squares in such a way that an odd number of them have exactly
4
4
4
adjacent coloured squares and all the remaining coloured squares have exactly
2
2
2
adjacent coloured squares?c) Is it possible to colour some squares in such a way that an odd number of them have exactly
2
2
2
adjacent coloured squares and all the remaining coloured squares have exactly
4
4
4
adjacent coloured squares?