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Contests
National and Regional Contests
Italy Contests
ITAMO
1999 ITAMO
1999 ITAMO
Part of
ITAMO
Subcontests
(6)
6
1
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3^k -1 = x^n, 3^k -1 = x^3, diophantine
(a) Find all pairs
(
x
,
k
)
(x,k)
(
x
,
k
)
of positive integers such that
3
k
−
1
=
x
3
3^k -1 = x^3
3
k
−
1
=
x
3
. (b) Prove that if
n
>
1
n > 1
n
>
1
is an integer,
n
≠
3
n \ne 3
n
=
3
, then there are no pairs
(
x
,
k
)
(x,k)
(
x
,
k
)
of positive integers such that
3
k
−
1
=
x
n
3^k -1 = x^n
3
k
−
1
=
x
n
.
5
1
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rectangular grid , village of pile-built dwellings on a lake
There is a village of pile-built dwellings on a lake, set on the gridpoints of an
m
×
n
m \times n
m
×
n
rectangular grid. Each dwelling is connected by exactly
p
p
p
bridges to some of the neighboring dwellings (diagonal connections are not allowed, two dwellings can be connected by more than one bridge). Determine for which values
m
,
n
,
p
m,n, p
m
,
n
,
p
it is possible to place the bridges so that from any dwelling one can reach any other dwelling.
3
1
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AB contains intersection point of 2 tangent circles iff r_1 +r_2 = r
Let
r
1
,
r
2
,
r
r_1,r_2,r
r
1
,
r
2
,
r
, with
r
1
<
r
2
<
r
r_1 < r_2 < r
r
1
<
r
2
<
r
, be the radii of three circles
Γ
1
,
Γ
2
,
Γ
\Gamma_1,\Gamma_2,\Gamma
Γ
1
,
Γ
2
,
Γ
, respectively. The circles
Γ
1
,
Γ
2
\Gamma_1,\Gamma_2
Γ
1
,
Γ
2
are internally tangent to
Γ
\Gamma
Γ
at two distinct points
A
,
B
A,B
A
,
B
and intersect in two distinct points. Prove that the segment
A
B
AB
A
B
contains an intersection point of
Γ
1
\Gamma_1
Γ
1
and
Γ
2
\Gamma_2
Γ
2
if and only if
r
1
+
r
2
=
r
r_1 +r_2 = r
r
1
+
r
2
=
r
.
4
1
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Albert and Barbara paly a game with 1999 sticks, winning strategy wanted
Albert and Barbara play the following game. On a table there are
1999
1999
1999
sticks, and each player in turn removes some of them: at least one stick, but at most half of the currently remaining sticks. The player who leaves just one stick on the table loses the game. Barbara moves first. Decide which player has a winning strategy and describe that strategy.
1
1
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a rectangular sheet is fold along a diagonal, overlapping triangle area
A rectangular sheet with sides
a
a
a
and
b
b
b
is fold along a diagonal. Compute the area of the overlapping triangle.
2
1
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"Balanced Integers"
An integer is balance if the number of digit in its decimal representation is equal to the number of its distinct prime factors (For example, 15 is balanced, but not 49). Prove that there are finite balanced number.