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Contests
National and Regional Contests
Italy Contests
Italy TST
1996 Italy TST
1996 Italy TST
Part of
Italy TST
Subcontests
(4)
4
1
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italy tst
4.4. Prove that there exists a set X of 1996 positive integers with the following properties: (i) the elements of X are pairwise coprime; (ii) all elements of X and all sums of two or more distinct elements of X are composite numbers
3
1
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italy TST
3.Let ABCD be a parallelogram with side AB longer than AD and acute angle
∠
D
A
B
\angle DAB
∠
D
A
B
. The bisector of ∠DAB meets side CD at L and line BC at K. If O is the circumcenter of triangle LCK, prove that the points B,C,O,D lie on a circle.
2
1
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italy TST
2. Let
A
1
,
A
2
,
.
.
.
,
A
n
A_1,A_2,...,A_n
A
1
,
A
2
,
...
,
A
n
be distinct subsets of an n-element set
X
X
X
(
n
≥
2
n \geq 2
n
≥
2
). Show that there exists an element
x
x
x
of
X
X
X
such that the sets
A
1
∖
{
x
}
A_1\setminus \{x\}
A
1
∖
{
x
}
,:.......,
A
n
∖
{
x
}
A_n\setminus \{x\}
A
n
∖
{
x
}
are all distinct.
1
1
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italy TST
1-Let
A
A
A
and
B
B
B
be two diametrically opposite points on a circle with radius
1
1
1
. Points
P
1
,
P
2
,
.
.
.
,
P
n
P_1,P_2,...,P_n
P
1
,
P
2
,
...
,
P
n
are arbitrarily chosen on the circle. Let a and b be the geometric means of the distances of
P
1
,
P
2
,
.
.
.
,
P
n
P_1,P_2,...,P_n
P
1
,
P
2
,
...
,
P
n
from
A
A
A
and
B
B
B
, respectively. Show that at least one of the numbers
a
a
a
and
b
b
b
does not exceed
2
\sqrt{2}
2