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National and Regional Contests
Saudi Arabia Contests
Saudi Arabia GMO TST
2014 Saudi Arabia GMO TST
4
a_1a_2(a_1 - a_2) + a_2a_3(a_2 - a_3) +...+ a_{n-1}a_n(a_{n-1} - a_n) \ge a_1a_n
a_1a_2(a_1 - a_2) + a_2a_3(a_2 - a_3) +...+ a_{n-1}a_n(a_{n-1} - a_n) \ge a_1a_n
Source: 2014 Saudi Arabia GMO TST II p4
July 26, 2020
algebra
inequalities
Problem Statement
Let
a
1
≥
a
2
≥
.
.
.
≥
a
n
>
0
a_1 \ge a_2 \ge ... \ge a_n > 0
a
1
≥
a
2
≥
...
≥
a
n
>
0
be real numbers. Prove that
a
1
a
2
(
a
1
−
a
2
)
+
a
2
a
3
(
a
2
−
a
3
)
+
.
.
.
+
a
n
−
1
a
n
(
a
n
−
1
−
a
n
)
≥
a
1
a
n
(
a
1
−
a
n
)
a_1a_2(a_1 - a_2) + a_2a_3(a_2 - a_3) +...+ a_{n-1}a_n(a_{n-1} - a_n) \ge a_1a_n(a_1 - a_n)
a
1
a
2
(
a
1
−
a
2
)
+
a
2
a
3
(
a
2
−
a
3
)
+
...
+
a
n
−
1
a
n
(
a
n
−
1
−
a
n
)
≥
a
1
a
n
(
a
1
−
a
n
)
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