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triples (x,y,z) with particular properties

Posted: Wed May 15, 2013 1:52 pm UTC
by alessandro95
Hello everyone, this problem is taken from the entrance test of an Italian university famous for being extremely hard, this is the first problem of the 13 pages long entrance test and I already have no idea on what to do with it (no, I'm not planning to even try this test!):

the original text for Italian speakers is:
"Si trovino le terne di numeri reali (x,y,z) con la proprieta' che la quarta potenza di ciascuno di essi e' uguale alla somma degli altri due."

That translates as:
"Find the triples of real numers (x,y,z) with the property that the fourth power of each of them is equal to the sum of the other two"

which is equivalent to the system of 3 equations:
x^4=y+z
y^4=x+z
z^4=y+x

but I cannot do anything better than pointing out the trivial solution for (0,0,0), any hint on how to solve this problem?
Thank in advance for any help!

Re: triples (x,y,z) with particular properties

Posted: Wed May 15, 2013 1:59 pm UTC
by vbkid
alessandro95 wrote:Hello everyone, this problem is taken from the entrance test of an Italian university famous for being extremely hard, this is the first problem of the 13 pages long entrance test and I already have no idea on what to do with it (no, I'm not planning to even try this test!):

the original text for Italian speakers is:
"Si trovino le terne di numeri reali (x,y,z) con la proprieta' che la quarta potenza di ciascuno di essi e' uguale alla somma degli altri due."

That translates as:
"Find the triples of real numers (x,y,z) with the property that the fourth power of each of them is equal to the sum of the other two"

which is equivalent to the system of 3 equations:
x^4=y+z
y^4=x+z
z^4=y+x

but I cannot do anything better than pointing out the trivial solution for (0,0,0), any hint on how to solve this problem?
Thank in advance for any help!


I just observed that x=y=z So:
x^4=2x
x^3=2
x=y=z=2^(1/3)

Re: triples (x,y,z) with particular properties

Posted: Wed May 15, 2013 2:03 pm UTC
by alessandro95
yes, I also found 2^(1/3) as solution just after posting this thread, how do you know that x,y and z must all be equal to each other?

Re: triples (x,y,z) with particular properties

Posted: Wed May 15, 2013 2:09 pm UTC
by faldion
If x ≠ y, we can do the following:

x^4 = y + z
y^4 = x + z
Subtract the lower equation from the upper one to get x^4 - y^4 = y - x.

x^4 - y^4 = y - x
(x^2 + y^2)(x^2 - y^2) = y - x
(x^2 + y^2)(x - y)(x + y) = -(x - y), divide by (x - y) to get
(x^2 + y^2)(x + y) = -1

(x^2 + y^2) is positive for real x and y, so (x + y) is necessarily negative because their product is negative. But (x + y) = z^4, which is positive for real z, and we have a contradiction.

Therefore x = y. A similar argument works for y = z, and we're done.

Re: triples (x,y,z) with particular properties

Posted: Wed May 15, 2013 2:14 pm UTC
by alessandro95
you guys are awesome, thank a lot!
It looked harder to me, well it still is the first of a lot of exercise after all...!

I may want to post some other exercise from the same test, after thinking a bit longer, should I open a new thread or post here?

Re: triples (x,y,z) with particular properties

Posted: Wed May 15, 2013 2:15 pm UTC
by vbkid
Mine was intuition (whether it was correct or not is another question!) but the algebra above brought me to the same conclusion.
Thanks for this post, it was a good, quick stretch of the brain!

Re: triples (x,y,z) with particular properties

Posted: Wed May 15, 2013 2:17 pm UTC
by vbkid
I'd say just post here!

Re: triples (x,y,z) with particular properties

Posted: Wed May 15, 2013 7:32 pm UTC
by jestingrabbit
vbkid wrote:I just observed that x=y=z So:
x^4=2x
x^3=2
x=y=z=2^(1/3)


You're forgetting the rather tedious x = y = z = 0 solution.

Re: triples (x,y,z) with particular properties

Posted: Wed May 15, 2013 8:27 pm UTC
by vbkid
jestingrabbit wrote:
vbkid wrote:I just observed that x=y=z So:
x^4=2x
x^3=2
x=y=z=2^(1/3)


You're forgetting the rather tedious x = y = z = 0 solution.


Thw OP actually covered it already.

Re: triples (x,y,z) with particular properties

Posted: Thu May 16, 2013 1:09 am UTC
by alessandro95
vbkid wrote:Mine was intuition (whether it was correct or not is another question!) but the algebra above brought me to the same conclusion.
Thanks for this post, it was a good, quick stretch of the brain!


The problem seems to ask only for the values, not for a demonstration that these are the only solutions, so your answer would have probably been enough for the test :)


ok, next problem, I already did some work on it (spoilered):
"Consider an integer n and the number Sn of increasing sequences of integer starting from 0 and ending at n, alternatively even and odd, for example S3=2 since the only 2 such sequences are 0,1,2,3 and 0,3.'
Consider the Fibonacci sequence:
f1=f2=1
fn=fn-1+fn-2, n>2
Prove that Sn=fn

Spoiler:
I started writing by hand the first values of Sn
f1=1 (0,1)
f2=1 (0,1,2)
f3=2 (0,1,2,3)(0,3)
f4=3 (0,1,2,3,4)(0,1,4)(0,3,4)
f5=5 (0,1,2,3,4,5)(0,1,4,5)(0,3,4,5)(0,1,2,5)(0,5)
f6=8 (0,1,2,3,4,5,6)(0,1,4,5,6)(0,3,4,5,6)(0,1,2,5,6)(0,5,6)(0,1,6)(0,1,2,3,6)(0,3,6)

To make the sequences for an even n we only need to append n to all the sequences with an odd index smaller than n, for example to make S4 we append 4 to the sequences for S1 and S3.
In a similar way to make the sequences for an odd Sn we append n to the sequences with an even smaller index and then we add the sequence (0,n) that cannot be formed in this way, for example to make S5 we append 5 to the sequences for S2 and S4 and then we also add the sequence (0,5).

Which is equivalent to say that:
Image

and
Image

which, I guess, is a well known property of the Fibonacci sequence, tough I have no idea on how to demonstrate it, any hints?


Please use spoilers if you post a whole solution!

p.s.
how do I use the [imath] tag? I had to use and external LaTeX implementation

Re: triples (x,y,z) with particular properties

Posted: Thu May 16, 2013 1:50 am UTC
by DeGuerre
A comment on your spoiler, which I am posting here not in a spoiler, because it doesn't give anything away:

alessandro95 wrote:which, I guess, is a well known property of the Fibonacci sequence, tough I have no idea on how to demonstrate it, any hints?


You're going to be embarrassed by how obvious this hint is.

f(2n) = f(2n-1) + f(2n-2) = f(2n-1) + f(2(n-1))

Re: triples (x,y,z) with particular properties

Posted: Thu May 16, 2013 9:03 am UTC
by jedelmania
alessandro95 wrote:To make the sequences for an even n we only need to append n to all the sequences with an odd index smaller than n, for example to make S4 we append 4 to the sequences for S1 and S3.


This is correct, and you get there from the definition of the Fibonacci sequence

Hint
Spoiler:
write out the definition of the series f(k) = f(k-1) + f(k-2) + use induction


There is a quicker way to the answer
Spoiler:
To make the sequences for n we only need to append n to all the sequences from Sn-1 and replace n-2 with n from all the sequences of Sn-2. Proof follows immediately from this observation.