MathDB
Problems
Contests
National and Regional Contests
Italy Contests
ITAMO
1996 ITAMO
1996 ITAMO
Part of
ITAMO
Subcontests
(6)
6
1
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Square grid
What is the minimum number of squares that is necessary to draw on a white sheet to obtain a square grid of side
n
n
n
?
5
1
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Locus with a vertex of a square
Given a circle
C
C
C
and an exterior point
A
A
A
. For every point
P
P
P
on the circle construct the square
A
P
Q
R
APQR
A
PQR
(in counterclock order). Determine the locus of the point
Q
Q
Q
when
P
P
P
moves on the circle
C
C
C
.
4
1
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Even number of draws
There is a list of
n
n
n
football matches. Determine how many possible columns, with an even number of draws, there are.
3
1
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A sphere and a cube
Given a cube of unit side. Let
A
A
A
and
B
B
B
be two opposite vertex. Determine the radius of the sphere, with center inside the cube, tangent to the three faces of the cube with common point
A
A
A
and tangent to the three sides with common point
B
B
B
.
2
1
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Triples on diophantic
Show that the equation
a
2
+
b
2
=
c
2
+
3
a^2 + b^2 = c^2 + 3
a
2
+
b
2
=
c
2
+
3
has infinetely many triples of integers
a
,
b
,
c
a, b, c
a
,
b
,
c
that are solutions.
1
1
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Maximum product of altitudes
Among all the triangles which have a fixed side
l
l
l
and a fixed area
S
S
S
, determine for which triangles the product of the altitudes is maximum.