MathDB
Problems
Contests
National and Regional Contests
Italy Contests
ITAMO
1992 ITAMO
1992 ITAMO
Part of
ITAMO
Subcontests
(5)
4
1
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italian jury of n persons , probability a fixed juror is a part of the majority
A jury of
9
9
9
persons should decide whether a verdict is guilty or not. Each juror votes independently with the probability
1
/
2
1/2
1/2
for each of the two possibilities, and noone is allowed to be abstinent. Find the probability that a fixed juror will be a part of the majority. In the case of a jury of
n
n
n
persons, find the values of n for which the probability of being a part of the majority is greater than, equal to, and smaller than
1
/
2
1/2
1/2
, respectively. (For
n
=
2
k
n = 2k
n
=
2
k
,
k
+
1
k +1
k
+
1
votes are needed for a majority.)
3
1
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n distinct divisors d_1,d_2, ...,d_n of n! with n! = d_1 +d_2 +···+d_n
Prove that for each
n
≥
3
n \ge 3
n
≥
3
there exist
n
n
n
distinct positive divisors
d
1
,
d
2
,
.
.
.
,
d
n
d_1,d_2, ...,d_n
d
1
,
d
2
,
...
,
d
n
of
n
!
n!
n
!
such that
n
!
=
d
1
+
d
2
+
.
.
.
+
d
n
n! = d_1 +d_2 +...+d_n
n
!
=
d
1
+
d
2
+
...
+
d
n
.
2
1
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3 points out of 4 in a convex ABCD of area 1 with area >=1/4
A convex quadrilateral of area
1
1
1
is given. Prove that there exist four points in the interior or on the sides of the quadrilateral such that each triangle with the vertices in three of these four points has an area greater than or equal to
1
/
4
1/4
1/4
.
1
1
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max no of 27 cubes of a large cube, a plane intersects
A cube is divided into
27
27
27
equal smaller cubes. A plane intersects the cube. Find the maximum possible number of smaller cubes the plane can intersect.
5
1
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Inequality
a
a
a
,
b
b
b
,
c
c
c
are real numbers. Show that
min
(
(
a
−
b
)
2
,
(
b
−
c
)
2
,
(
c
−
a
)
2
)
≤
a
2
+
b
2
+
c
2
2
\min((a-b)^2,(b-c)^2,(c-a)^2)\leq \frac{a^2+b^2+c^2}{2}
min
((
a
−
b
)
2
,
(
b
−
c
)
2
,
(
c
−
a
)
2
)
≤
2
a
2
+
b
2
+
c
2