Difference between revisions of "2001 IMO Problems/Problem 2"
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which is the same as | which is the same as | ||
<cmath>\frac{\sqrt{a^3}}{\sqrt{a^3+8abc}}+\frac{\sqrt{b^3}}{\sqrt{b^3+8abc}}+\frac{\sqrt{c^3}}{\sqrt{c^3+8abc}}</cmath> | <cmath>\frac{\sqrt{a^3}}{\sqrt{a^3+8abc}}+\frac{\sqrt{b^3}}{\sqrt{b^3+8abc}}+\frac{\sqrt{c^3}}{\sqrt{c^3+8abc}}</cmath> | ||
− | Now let <math>f(x)=\sqrt{\frac{x^3}{x^3 + | + | Now let <math>f(x)=\sqrt{\frac{x^3}{x^3 + 8abc}}</math>. Then f is convex and f is strictly increasing, so by [[Jensen's inequality]] and [[AM-GM]], |
<cmath>f(a) + f(b) + f(c) \geq 3f((\frac{1}{3})a + (\frac{1}{3})b + (\frac{1}{3})c)) \geq 3f(\sqrt[3]{abc}) = 3(\frac{1}{3}) =1</cmath> | <cmath>f(a) + f(b) + f(c) \geq 3f((\frac{1}{3})a + (\frac{1}{3})b + (\frac{1}{3})c)) \geq 3f(\sqrt[3]{abc}) = 3(\frac{1}{3}) =1</cmath> | ||
Revision as of 21:03, 15 August 2024
Problem
Let be positive real numbers. Prove that
.
Contents
Solution
Firstly, (where
) and its cyclic variations.
Next note that
and
are similarly oriented sequences. Thus
Hence the inequality has been established.
Equality holds if
.
Notation: : AM-GM inequality,
: AM-HM inequality,
: Chebyshev's inequality,
: QM-AM inequality / RMS inequality
Alternate Solution using Hölder's
By Hölder's inequality,
Thus we need only show that
Which is obviously true since
.
Alternate Solution using Jensen's
This inequality is homogeneous so we can assume without loss of generality and apply Jensen's inequality for
, so we get:
but
by AM-GM, and thus the inequality is proven.
Alternate Solution 2 using Jensen's
We can rewrite
as
which is the same as
Now let
. Then f is convex and f is strictly increasing, so by Jensen's inequality and AM-GM,
Alternate Solution 3 using Jensen's
Let ,
,
and
f is convex so we can write:
let
, by substitustion:
we multiply both sides by t
QED
Alternate Solution using Isolated Fudging
We claim that
Cross-multiplying, squaring both sides and expanding, we have
After cancelling the
term, we apply AM-GM to RHS and obtain
as desired, completing the proof of the claim.
Similarly and
.
Summing the three inequalities, we obtain the original inequality.
Alternate Solution using Cauchy
We want to prove
Note that since this inequality is homogenous, assume .
By Cauchy,
Dividing both sides by , we see that we want to prove
or equivalently
Squaring both sides, we have
Now use Cauchy again to obtain
Since , the inequality becomes
after some simplifying.
But this equals and since
we just want to prove
after some simplifying.
But that is true by AM-GM or Muirhead. Thus, proved.
Alternate Solution using Carlson
By Carlson's Inequality, we can know that
Then,
On the other hand, and
Then,
Therefore,
Thus,
-- Haozhe Yang
See also
2001 IMO (Problems) • Resources | ||
Preceded by Problem 1 |
1 • 2 • 3 • 4 • 5 • 6 | Followed by Problem 3 |
All IMO Problems and Solutions |