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Problems
Contests
National and Regional Contests
Italy Contests
ITAMO
1988 ITAMO
1988 ITAMO
Part of
ITAMO
Subcontests
(7)
2
1
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v^2_1 +v^2_2 +...+v^2_n = p^2_1 +p^2_2 +...+p^2_n, in basketball tournament
In a basketball tournament any two of the
n
n
n
teams
S
1
,
S
2
,
.
.
.
,
S
n
S_1,S_2,...,S_n
S
1
,
S
2
,
...
,
S
n
play one match (no draws). Denote by
v
i
v_i
v
i
and
p
i
p_i
p
i
the number of victories and defeats of team
S
i
S_i
S
i
(
i
=
1
,
2
,
.
.
.
,
n
i = 1,2,...,n
i
=
1
,
2
,
...
,
n
), respectively. Prove that
v
1
2
+
v
2
2
+
.
.
.
+
v
n
2
=
p
1
2
+
p
2
2
+
.
.
.
+
p
n
2
v^2_1 +v^2_2 +...+v^2_n = p^2_1 +p^2_2 +...+p^2_n
v
1
2
+
v
2
2
+
...
+
v
n
2
=
p
1
2
+
p
2
2
+
...
+
p
n
2
4
1
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terms of 1,11,111,1111,... base 9 are triangular
Show that all terms of the sequence
1
,
11
,
111
,
1111
,
.
.
.
1,11,111,1111,...
1
,
11
,
111
,
1111
,
...
in base
9
9
9
are triangular numbers, i.e. of the form
m
(
m
+
1
)
2
\frac{m(m+1)}{2}
2
m
(
m
+
1
)
for an integer
m
m
m
7
1
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3 of n positive integers with same gcd, have also the same gcd
Given
n
≥
3
n \ge 3
n
≥
3
positive integers not exceeding
100
100
100
, let
d
d
d
be their greatest common divisor. Show that there exist three of these numbers whose greatest common divisor is also equal to
d
d
d
.
3
1
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5 equal disks cover a regular pentagon, min radius
A regular pentagon of side length
1
1
1
is given. Determine the smallest
r
r
r
for which the pentagon can be covered by five discs of radius
r
r
r
and justify your answer.
6
1
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x+y+z <= a+b+c+3d , in a tetrahedron
The edge lengths of the base of a tetrahedron are
a
,
b
,
c
a,b,c
a
,
b
,
c
, and the lateral edge lengths are
x
,
y
,
z
x,y,z
x
,
y
,
z
. If
d
d
d
is the distance from the top vertex to the centroid of the base, prove that
x
+
y
+
z
≤
a
+
b
+
c
+
3
d
x+y+z \le a+b+c+3d
x
+
y
+
z
≤
a
+
b
+
c
+
3
d
.
5
1
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projections of 4 points onto a plane are the vertices of a parallelogram
Given four non-coplanar points, is it always possible to find a plane such that the orthogonal projections of the points onto the plane are the vertices of a parallelogram? How many such planes are there in general?
1
1
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A tosses a coin n times, and B does n+1 times, n for fair game
Players
A
A
A
and
B
B
B
play the following game:
A
A
A
tosses a coin
n
n
n
times, and
B
B
B
does
n
+
1
n+1
n
+
1
times. The player who obtains more ”heads” wins; or in the case of equal balances,
A
A
A
is assigned victory. Find the values of
n
n
n
for which this game is fair (i.e. both players have equal chances for victory).