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Problems
Contests
National and Regional Contests
Canada Contests
Canada National Olympiad
1972 Canada National Olympiad
2
2
Part of
1972 Canada National Olympiad
Problems
(1)
Sum of all products of pairs
Source: Canada 1972, Problem 2
6/24/2006
Let
a
1
,
a
2
,
…
,
a
n
a_1,a_2,\ldots,a_n
a
1
,
a
2
,
…
,
a
n
be non-negative real numbers. Define
M
M
M
to be the sum of all products of pairs
a
i
a
j
a_ia_j
a
i
a
j
(
i
<
j
)
(i<j)
(
i
<
j
)
,
<
s
p
a
n
c
l
a
s
s
=
′
l
a
t
e
x
−
i
t
a
l
i
c
′
>
i
.
e
.
<
/
s
p
a
n
>
<span class='latex-italic'>i.e.</span>
<
s
p
an
c
l
a
ss
=
′
l
a
t
e
x
−
i
t
a
l
i
c
′
>
i
.
e
.
<
/
s
p
an
>
,
M
=
a
1
(
a
2
+
a
3
+
⋯
+
a
n
)
+
a
2
(
a
3
+
a
4
+
⋯
+
a
n
)
+
⋯
+
a
n
−
1
a
n
.
M = a_1(a_2+a_3+\cdots+a_n)+a_2(a_3+a_4+\cdots+a_n)+\cdots+a_{n-1}a_n.
M
=
a
1
(
a
2
+
a
3
+
⋯
+
a
n
)
+
a
2
(
a
3
+
a
4
+
⋯
+
a
n
)
+
⋯
+
a
n
−
1
a
n
.
Prove that the square of at least one of the numbers
a
1
,
a
2
,
…
,
a
n
a_1,a_2,\ldots,a_n
a
1
,
a
2
,
…
,
a
n
does not exceed
2
M
/
n
(
n
−
1
)
2M/n(n-1)
2
M
/
n
(
n
−
1
)
.