IMO 2012 #2

by Wolstenholme, Oct 28, 2014, 6:01 PM

Let $n\ge 3$ be an integer, and let $a_2,a_3,\ldots ,a_n$ be positive real numbers such that $a_{2}a_{3}\cdots a_{n}=1$ . Prove that
$(1 + a_2)^2 (1 + a_3)^3 \dotsm (1 + a_n)^n > n^n.$

Proof:

Note that for all integer $2 \le i \le n$ we have by AM - GM that $(1 + a_i)^i = \left(\frac{1}{i - 1} + \frac{1}{i - 1} + \dots + \frac{1}{i - 1} + a_i\right)^i \ge \frac{i^ia_i}{(i - 1)^{i - 1}}.$ Therefore $\prod_{i = 2}^{n}(1 + a_i)^i \ge \prod_{i = 2}^{n}\frac{i^ia_i}{(i - 1)^{i - 1}} = n^na_2a_3a_4\dots{a_n} = n^n$ . Equality holds only if $a_i = \frac{1}{i - 1}$ for all integer $2 \le i \le n$ but as this is clearly impossible, we are done.

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glad to have found a fellow chipotle lover <3
by nukelauncher, Aug 13, 2020, 6:40 AM
the random chinese tst problem is the only thing I read, but I'll assume your blog is nice and give you a shout even though you probably never use aops anymoer
by fukano_2, Jun 14, 2020, 6:24 AM
wolstenholme - op
by AopsUser101, Jan 29, 2020, 8:27 PM
this blog is so hot
by mathleticguyyy, Jun 5, 2019, 8:26 PM
Hi. Nice Blog!
by User360702, Jan 10, 2019, 6:03 PM
helloooooo
by songssari, Jun 12, 2016, 8:21 AM
shouts make blogs happier
by briantix, Mar 18, 2016, 9:57 PM
You were just featured on AoPS's facebook page.
by mishka1980, Sep 12, 2015, 10:33 PM
This is late, but where is the ARML results post?
by donot, Aug 31, 2015, 11:07 PM
"I am Sam"
"Sam I am"
by mathwizard888, Aug 12, 2015, 9:13 PM
HW $\textcolor{white}{}$
by Eugenis, Apr 20, 2015, 10:10 PM
Uh-oh ARML practice is Thursday... I should start the homework.
by nosaj, Apr 20, 2015, 12:34 AM
Yes I am Sam, and Chebyshev polynomials aren't trivial, although they do make some problems trivial
by Wolstenholme, Apr 15, 2015, 10:00 PM
How are Chebyshev Polynomials trivial?
by nosaj, Apr 13, 2015, 4:10 AM
Are you Sam?
by Eugenis, Apr 4, 2015, 2:05 AM
@Brian: yes, yes I did #whoneedsalgskillz?

@gauss1181; hey!
by Wolstenholme, Mar 1, 2015, 11:25 PM
hello!!!
by gauss1181, Nov 27, 2014, 12:19 AM
Hi Wolstenholme did you actually use calc on that tstst problem
by briantix, Aug 2, 2014, 12:25 AM

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IMO 2012 #2

by Wolstenholme, Oct 28, 2014, 6:01 PM

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