Cool problem of the day! (3 Viewers)

[Carrotsticks]

Excellent!

Consider the following diagram:

The pattern of circles is repeated infinitely and the triangle is equilateral.

What percentage of the whole triangle is covered by circles?

 

[barbernator]

82.7% correct to 1dp?

this is wrong, i made a mistake

 

[seanieg89]

I got:

 

[barbernator]

I got:

did it again and got this. I will post up a solution if people want a bit later.

 

[Carrotsticks]

Correct!

Given that x and y are positive integers, solve:

 

[barbernator]

Correct!

Given that x and y are positive integers, solve:

y=x? lol

 

[RealiseNothing]

y=x? lol

There's more solutions than that.

For example:

 

[Carrotsticks]

y=x? lol

There is more to it than that =p

 

[barbernator]

would putting it in the form of a infinite product help? ive got no idea...

 

[Carrotsticks]

would putting it in the form of a infinite product help? ive got no idea...

Not quite...

Clue: Begin by dividing both sides by x^x.

 

[AAEldar]

Correct!

Given that x and y are positive integers, solve:

I like this one! Hahah.

Not quite...


Clue: Begin by dividing both sides by x^x.

Different to how I started.

 

[RealiseNothing]

Is it just?

 

[AAEldar]

Is it just?

Nope.

If it helps the only integer solutions are for 2 and 4.

EDIT: Integer solutions requested so yes >.<

 

[Carrotsticks]

Is it just?

Yep! But how would you prove that it's the ONLY solution apart from the trivial one x=y=1 ?

 

[seanieg89]

Only nontrivial solns are the pairs (4,2), (2,4). Proof:



so we cannot have equality if x and y are not equal. Here I have assumed a commonly known limit expression for e (and the fact that this expression is increasing...an easy thing to check).

 

[Carrotsticks]

barbernator,

 

[seanieg89]

Here's one I really liked in high school:

Find all solutions in positive integers to:

 

[AAEldar]

barbernator,

Oh I didn't see that either, never mind haha!

 

[barbernator]

Here's one I really liked in high school:

Find all solutions in positive integers to:

does z have any restrictions?

 

[seanieg89]

All three unknowns are positive integers, that is the only restriction.