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Problems
Contests
National and Regional Contests
Italy Contests
ITAMO
2000 ITAMO
2000 ITAMO
Part of
ITAMO
Subcontests
(6)
2
1
Hide problems
(DB+BC)^2=AD^2+AC^2
Let
A
B
C
D
ABCD
A
BC
D
be a convex quadrilateral, and write
α
=
∠
D
A
B
\alpha=\angle DAB
α
=
∠
D
A
B
,
β
=
∠
A
D
B
\beta=\angle ADB
β
=
∠
A
D
B
,
γ
=
∠
A
C
B
\gamma=\angle ACB
γ
=
∠
A
CB
,
δ
=
∠
D
B
C
\delta= \angle DBC
δ
=
∠
D
BC
and
ϵ
=
∠
D
B
A
\epsilon=\angle DBA
ϵ
=
∠
D
B
A
. Assuming that
α
<
π
/
2
\alpha<\pi/2
α
<
π
/2
,
β
+
γ
=
π
/
2
\beta+\gamma=\pi /2
β
+
γ
=
π
/2
, and
δ
+
2
ϵ
=
π
\delta+2\epsilon=\pi
δ
+
2
ϵ
=
π
, prove that
(
D
B
+
B
C
)
2
=
A
D
2
+
A
C
2
(DB+BC)^2=AD^2+AC^2
(
D
B
+
BC
)
2
=
A
D
2
+
A
C
2
.
6
1
Hide problems
p(0) = 0, 0 \le p(1) \le 10^7, p(a) = 1999, p(b) = 2001, p(1) ?
Let
p
(
x
)
p(x)
p
(
x
)
be a polynomial with integer coefficients such that
p
(
0
)
=
0
p(0) = 0
p
(
0
)
=
0
and
0
≤
p
(
1
)
≤
1
0
7
0 \le p(1) \le 10^7
0
≤
p
(
1
)
≤
1
0
7
. Suppose that there exist positive integers
a
,
b
a,b
a
,
b
such that
p
(
a
)
=
1999
p(a) = 1999
p
(
a
)
=
1999
and
p
(
b
)
=
2001
p(b) = 2001
p
(
b
)
=
2001
. Determine all possible values of
p
(
1
)
p(1)
p
(
1
)
. (Note:
1999
1999
1999
is a prime number.)
5
1
Hide problems
a grill of the shape of an n x n unit net by many metal bars of length 2
A man disposes of sufficiently many metal bars of length
2
2
2
and wants to construct a grill of the shape of an
n
×
n
n \times n
n
×
n
unit net. He is allowed to fold up two bars at an endpoint or to cut a bar into two equal pieces, but two bars may not overlap or intersect. What is the minimum number of pieces he must use?
4
1
Hide problems
number theory game of two, winning strategy in at most 50moves
Let
n
>
1
n > 1
n
>
1
be a fixed integer. Alberto and Barbara play the following game: (i) Alberto chooses a positive integer, (ii) Barbara chooses an integer greater than
1
1
1
which is a multiple or submultiple of the number Alberto chose (including itself), (iii) Alberto increases or decreases the Barbara’s number by
1
1
1
. Steps (ii) and (iii) are alternatively repeated. Barbara wins if she succeeds to reach the number
n
n
n
in at most
50
50
50
moves. For which values of
n
n
n
can she win, no matter how Alberto plays?
3
1
Hide problems
ratio of volumes of pyramids, pyramid inscribed in a sphere
A pyramid with the base
A
B
C
D
ABCD
A
BC
D
and the top
V
V
V
is inscribed in a sphere. Let
A
D
=
2
B
C
AD = 2BC
A
D
=
2
BC
and let the rays
A
B
AB
A
B
and
D
C
DC
D
C
intersect in point
E
E
E
. Compute the ratio of the volume of the pyramid
V
A
E
D
VAED
V
A
E
D
to the volume of the pyramid
V
A
B
C
D
VABCD
V
A
BC
D
.
1
1
Hide problems
no with equal digits and sum of squares of 3 consecutive odd integers
A possitive integer is called special if all its decimal digits are equal and it can be represented as the sum of squares of three consecutive odd integers. (a) Find all
4
4
4
-digit special numbers (b) Are there
2000
2000
2000
-digit special numbers?