MathDB
Problems
Contests
National and Regional Contests
Italy Contests
ITAMO
2009 ITAMO
2009 ITAMO
Part of
ITAMO
Subcontests
(3)
3
2
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find all nice ones
A natural number
n
n
n
is called nice if it enjoys the following properties: • The expression is made up of
4
4
4
decimal digits; • the first and third digits of
n
n
n
are equal; • the second and fourth digits of
n
n
n
are equal; • the product of the digits of
n
n
n
divides
n
2
n^2
n
2
. Determine all nice numbers.
number of k which are n-squared
A natural number
k
k
k
is said
n
n
n
-squared if by colouring the squares of a
2
n
×
k
2n \times k
2
n
×
k
chessboard, in any manner, with
n
n
n
different colours, we can find
4
4
4
separate unit squares of the same colour, the centers of which are vertices of a rectangle having sides parallel to the sides of the board. Determine, in function of
n
n
n
, the smallest natural
k
k
k
that is
n
n
n
-squared.
1
2
Hide problems
Determine all values of 'e'
Let
a
<
b
<
c
<
d
<
e
a < b < c < d < e
a
<
b
<
c
<
d
<
e
be real numbers. We calculate all possible sums in pairs of these 5 numbers. Of these 10 sums, the three smaller ones are 32, 36, 37, while the two larger ones are 48 and 51. Determine all possible values that
e
e
e
can take.
A flea jumps n times from origin
A flea is initially at the point
(
0
,
0
)
(0, 0)
(
0
,
0
)
in the Cartesian plane. Then it makes
n
n
n
jumps. The direction of the jump is taken in a choice of the four cardinal directions. The first step is of length
1
1
1
, the second of length
2
2
2
, the third of length
4
4
4
, and so on. The
n
t
h
n^{th}
n
t
h
-jump is of length
2
n
−
1
2^{n-1}
2
n
−
1
. Prove that, if you know the final position flea, then it is possible to uniquely determine its position after each of the
n
n
n
jumps.
2
2
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ABCD is a square with centre O
A
B
C
D
ABCD
A
BC
D
is a square with centre
O
O
O
. Two congruent isosceles triangle
B
C
J
BCJ
BC
J
and
C
D
K
CDK
C
DK
with base
B
C
BC
BC
and
C
D
CD
C
D
respectively are constructed outside the square. let
M
M
M
be the midpoint of
C
J
CJ
C
J
. Show that
O
M
OM
OM
and
B
K
BK
B
K
are perpendicular to each other.
T,R,K are collinear
Let
A
B
C
ABC
A
BC
be an acute-angled scalene triangle and
Γ
\Gamma
Γ
be its circumcircle.
K
K
K
is the foot of the internal bisector of
∠
B
A
C
\angle BAC
∠
B
A
C
on
B
C
BC
BC
. Let
M
M
M
be the midpoint of the arc
B
C
BC
BC
containing
A
A
A
.
M
K
MK
M
K
intersect
Γ
\Gamma
Γ
again at
A
′
A'
A
′
.
T
T
T
is the intersection of the tangents at
A
A
A
and
A
′
A'
A
′
.
R
R
R
is the intersection of the perpendicular to
A
K
AK
A
K
at
A
A
A
and perpendicular to
A
′
K
A'K
A
′
K
at
A
′
A'
A
′
. Show that
T
,
R
T, R
T
,
R
and
K
K
K
are collinear.