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Contests
National and Regional Contests
Italy Contests
Italy TST
2001 Italy TST
2001 Italy TST
Part of
Italy TST
Subcontests
(3)
2
1
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0 ≤ a ≤ b ≤ c product inequality
Let
0
≤
a
≤
b
≤
c
0\le a\le b\le c
0
≤
a
≤
b
≤
c
be real numbers. Prove that
(
a
+
3
b
)
(
b
+
4
c
)
(
c
+
2
a
)
≥
60
a
b
c
(a+3b)(b+4c)(c+2a)\ge 60abc
(
a
+
3
b
)
(
b
+
4
c
)
(
c
+
2
a
)
≥
60
ab
c
4
1
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equaalize the balloons
We are given
2001
2001
2001
balloons and a positive integer
k
k
k
. Each balloon has been blown up to a certain size (not necessarily the same for each balloon). In each step it is allowed to choose at most
k
k
k
balloons and equalize their sizes to their arithmetic mean. Determine the smallest value of
k
k
k
such that, whatever the initial sizes are, it is possible to make all the balloons have equal size after a finite number of steps.
3
1
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p divides 5^q+1
Find all pairs
(
p
,
q
)
(p, q)
(
p
,
q
)
of prime numbers such that
p
p
p
divides 5^q \plus{} 1 and
q
q
q
divides 5^p \plus{} 1.