Hello everyone.

I would like to start 'marathon' of the exercises from different competition in the secondary school level. (ages 15-18)

let's start with inequality :)

Exercise 1.

Let be positive real numbers. Prove that

where is natural.

Enjoy!

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- Aug 29th 2010, 03:13 AMghostbuster9Secondary school's competition - marathon.
Hello everyone.

I would like to start 'marathon' of the exercises from different competition in the secondary school level. (ages 15-18)

let's start with inequality :)

Exercise 1.

Let be positive real numbers. Prove that

where is natural.

Enjoy! - Aug 29th 2010, 03:31 PMBobP
If we have the equality so assume, wlog, that and that

We then have to prove that

Now,

so,

If then and the result follows, so assume that and that with

Then,

,

and again the result follows. - Aug 30th 2010, 12:49 AMghostbuster9
You could a bit easier: AM-GM two times. But your solution is also OK. Now is your turn - send your exercise.

- Aug 30th 2010, 03:08 AMBobP
O.K.

Exercise 2.

Prove that there do not exist positive integers such that

- Aug 30th 2010, 04:45 AMghostbuster9

what is impossible.

Exercise 3.

Calculate