Difference between revisions of "1993 USAMO Problems/Problem 5"
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each <math>n > 1</math>, | each <math>n > 1</math>, | ||
− | < | + | <cmath>\frac{a_0+\cdots+a_n}{n+1}\cdot\frac{a_1+\cdots+a_{n-1}}{n-1}\ge\frac{a_0+\cdots+a_{n-1}}{n}\cdot\frac{a_1+\cdots+a_{n}}{n}.</cmath> |
==Solution== | ==Solution== | ||
− | {{ | + | Notice that because |
+ | <cmath>(a_0 + a_1 + \dots + a_{n-1})(a_1 + a_2 + \dots + a_n) = (a_0 + (a_1 + \dots + a_{n-1})(-a_0 + (a_0 + a_1 + \dots + a_n))</cmath> | ||
+ | <cmath> = -a_0^2 + a_0 ((a_0 + a_1 + \dots + a_n) - (a_1 + a_2 + \dots + a_{n-1})) + (a_1 + a_2 + \dots + a_{n-1})(a_0 + a_1 + \dots + a_n)</cmath> | ||
+ | <cmath> = a_0 a_n + (a_1 + a_2 + \dots + a_{n-1})(a_0 + a_1 + \dots + a_n),</cmath> | ||
+ | we may subtract <math>\dfrac{(a_1 + a_2 + \dots + a_{n-1})(a_0 + a_1 + \dots + a_n)}{n^2}</math> from both sides of the inequality and observe that it is sufficient to prove that | ||
+ | <cmath>\frac{(a_1 + a_2 + \dots + a_{n-1})(a_0 + a_1 + \dots + a_n)}{(n^2 - 1)n^2} \ge \frac{a_0 a_n}{n^2},</cmath> | ||
+ | or | ||
+ | <cmath>(a_1 + a_2 + \dots + a_{n-1})(a_0 + a_1 + \dots + a_n) \ge (n^2 - 1) a_0 a_n.</cmath> | ||
+ | |||
+ | Fortunately, this is an easy inequality. Indeed, from AM-GM applied on each group of terms we have | ||
+ | <cmath>(a_1 + a_2 + \dots + a_{n-1})(a_0 + a_1 + \dots + a_n) \ge (n^2 - 1) \sqrt[n-1]{a_1 a_2 \dots a_{n-1}} \sqrt[n+1]{a_0 a_1 \dots a_n},</cmath> | ||
+ | and so it suffices to prove | ||
+ | <cmath>\sqrt[n-1]{a_1 a_2 \dots a_{n-1}} \sqrt[n+1]{a_0 a_1 \dots a_n} \ge a_0 a_n,</cmath> | ||
+ | or, after taking both sides to the <math>(n^2 - 1)</math> power, simplifying, and taking the <math>n</math>-th root of both sides, to prove | ||
+ | <cmath>a_1^2 a_2^2 a_3^2 \dots a_{n-1}^2 \ge a_0^{n-1} a_n^{n-1}.</cmath> | ||
+ | This easily follows from the Fact that <math>a_0 a_n \le a_i a_{n-i}</math> for <math>1 \le i \le n-1</math>. Indeed, we are given that | ||
+ | <cmath>a_0 a_2 \le a_1^2</cmath> | ||
+ | <cmath>a_1 a_3 \le a_2^2</cmath> | ||
+ | <cmath>a_2 a_4 \le a_3^2</cmath> | ||
+ | <cmath>\dots</cmath> | ||
+ | <cmath>a_{n-3} a_{n-1} \le a_{n-2}^2</cmath> | ||
+ | <cmath>a_{n-2} a_n \le a_{n-1}^2.</cmath> | ||
+ | Multiply all inequalities together and cancel <math>a_1, a_2^2, a_3^2, \dots, a_{n-2}^2, a_{n-1}</math> to give <math>a_0 a_n \le a_1 a_{n-1}</math>. Similarly, by multiplying all inequalities except the first and the last, we deduce that <math>a_2 a_{n-2} \ge a_1 a_{n-1} \ge a_0 a_n</math>, and a simple induction argument proves the verity of the Fact for <math>1 \le i \le \frac{n}{2}</math>, and so by the Commutative Property the Fact is true for all <math>1 \le i \le n-1</math>, as desired. Now multiply each inequality of the Fact for <math>i = 1, 2, 3, \dots, n-1</math> to give the desired result. | ||
+ | |||
+ | Note: The Fact can be generalized into a Lemma: <math>a_x a_w \le a_y a_z</math> whenever <math>x \le y \le z \le w</math> and <math>x + w = y + z</math>. The proof is similar to that of the Fact and is left as an exercise to the reader. | ||
+ | |||
+ | --User suli, April 15, 2015. | ||
== See Also == | == See Also == |
Latest revision as of 17:08, 13 August 2023
Problem 5
Let be a sequence of positive real numbers satisfying
for
. (Such a sequence is said to be log concave.) Show that for
each
,
Solution
Notice that because
we may subtract
from both sides of the inequality and observe that it is sufficient to prove that
or
Fortunately, this is an easy inequality. Indeed, from AM-GM applied on each group of terms we have
and so it suffices to prove
or, after taking both sides to the
power, simplifying, and taking the
-th root of both sides, to prove
This easily follows from the Fact that
for
. Indeed, we are given that
Multiply all inequalities together and cancel
to give
. Similarly, by multiplying all inequalities except the first and the last, we deduce that
, and a simple induction argument proves the verity of the Fact for
, and so by the Commutative Property the Fact is true for all
, as desired. Now multiply each inequality of the Fact for
to give the desired result.
Note: The Fact can be generalized into a Lemma: whenever
and
. The proof is similar to that of the Fact and is left as an exercise to the reader.
--User suli, April 15, 2015.
See Also
1993 USAMO (Problems • Resources) | ||
Preceded by Problem 4 |
Followed by Last Problem | |
1 • 2 • 3 • 4 • 5 | ||
All USAMO Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.