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We have

Germany 1-0 Argentina after extra time. Well done Germany!

**Nehushtan**- Replies: 14

Germany – out of this world!

Brazil – out of this World Cup …

That’s all you can do. The quadratic polynomial

is irreducible over and .

In a way you could write

but this is probably not what you want.

Agnishom wrote:

The series is undefined as the first term is .
Agnishom wrote:

If a,b,c > 0

and abc=1Prove that a^2 + b^2 + c^2 ≥ a+b+c

Jensen’s inequality states that if is a convex function then

The function is convex; therefore

But by AM–GM.

Hence

Consider

This is a G.P. with common ratio

so we haveBut *n* is odd so

Multiplying both sides by

givesSince the LHS is an integer, the RHS must be an integer. QED.

**Nehushtan**- Replies: 2

The periodic table is set to get a new addition say hello to the superheavy element ununseptium

A moth Small Dusty Wave (*Idaea seriata*).

**Nehushtan**- Replies: 2

There are 60 seconds in a minute. There are 60 minutes in an hour. And a day has twenty-four hours 12 hours of daylight and 12 hours night averaged through the year.

What makes the numbers 12 and 60 so special?

Agnishom wrote:

Please Tell me how I can show

Note that is a subset of . In general, if then .

Proof:

We must prove that both

and . is clear since is a subset of .To prove the other inclusion, suppose

and let . . If we are done. If then, since , we have as well. In either case . Hence .Agnishom wrote:

Sorry, it is supposed to be This:

Prove that:

Thats better.

and it is easy to see that

.Something is wrong with your question.

What does mean?

The best site to visit if youre interested in IMO, IMHO, is this one: http://www.artofproblemsolving.com/

Their forum: http://www.artofproblemsolving.com/Forum/index.php?

Lets assume

is 1 or prime. Then it divides 2 or .(i)

divides 2 or or . So .(ii)

divides . For this to be an integer, . So . So which is the same as (i) above.So

if is 1 or prime. There are other solutions if is not 1 and not prime but as you are not told to find them all, I think Ill stop here.JaneFairfax wrote:

Contradictory is perhaps not a good word to use in the title of this thread. Oxymoronic might be better, I believe. It is not contradiction that I wish to explore, but rather oxymoron: http://z8.invisionfree.com/DYK/index.php?showtopic=69

If I said that someone was an **unpleasant kind** of person, I would be uttering an oxymoron.

Done, thanks.

Here is another problem for all the brainy people here.

**Easy puzzle challenge**. Therefore if your solution is messy and complicated, youre on the wrong tack.