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 Post subject: Secondary school's competition - marathon.Posted: Sun, 29 Aug 2010 11:14:09 UTC
 S.O.S. Newbie

Joined: Sat, 28 Aug 2010 22:15:41 UTC
Posts: 1
Hello everyone.
I would like to start 'marathon' of the exercises from different competition in the secondary school level. (ages 16-19)

Exercise 1.
Let be the real positive numbers which . Prove that
.

Enjoy!

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 Post subject: Re: Secondary school's competition - marathon.Posted: Sun, 29 Aug 2010 14:24:56 UTC
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Joined: Mon, 29 Dec 2008 17:49:32 UTC
Posts: 7607
Location: NCTS/TPE, Taiwan
ghostbuster9 wrote:
Hello everyone.
I would like to start 'marathon' of the exercises from different competition in the secondary school level. (ages 16-19)

Exercise 1.
Let be the real positive numbers which . Prove that
.

Enjoy!

Generalisation: for all .

Spoiler:
Apply AM-GM on each degree piece.

Exercise 2. (1992 IMO Q5, rephrased)
Let be a finite set, and let be the projection of S along x onto yz coordinates, along y onto zx coordinates, and along z onto xy-coordinates respectively. Prove that .

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