# If a,b,c are three distinctWatch

#1
If a,b,c are three distinct nonzero real numbers and

a + (1 / b) .=b + (1 / c) .=c + (1 / a) .=t

for some real number t

prove that abc + t=0 .
0
10 years ago
#2
(Original post by drmath)
If a,b,c are three distinct nonzero real numbers and

a + (1 / b) .=b + (1 / c) .=c + (1 / a) .=t

for some real number t

prove that abc + t=0 .
a + (1/b) = t
b + (1/c) = t
c + (1/a) = t

abc = [t-(1/b)][t-(1/c)][t-(1/a)] = [t^2 - (t/b) - (t/c) + (1/ab)][t-(1/a)]
= t^3 - (t^2/b) - (t^2/c) + (t/ab) - (t^2/a) + (t/ab) + (t/ac) - (1/a^2b)
= t^3 - t^2[(1/b) + (1/c) + (1/a)] + t[(1/ab) + (1/ab) + (1/ac)] - (1/a^2b)
= t^3 - t^2[3t-(a+b+c)] + t[...

Argh, I cannae be assed with this. I'm sure there's a very clever trick you can use that I'm not spotting.

I imgaine that by bashing away hard enough like I have you can get t[(1/ab) + (1/ab) + (1/ac)] - (1/a^2b) purely in terms of t, and then you'll get a "cubic" in t where actually all the non-linear terms cancel out with each other.
0
10 years ago
#3
I would begin by noting that is fully determined by and that are not entirely arbitrary - if you have two of them, the third is automatically determined simply by considering one of the three equations. I wouldn't be surprised if there is only one solution set for - there are three variables and three equations:

Also, by considering , and assuming that , we find that .
0
10 years ago
#4
0
X

new posts
Latest
My Feed

### Oops, nobody has postedin the last few hours.

Why not re-start the conversation?

see more

### See more of what you like onThe Student Room

You can personalise what you see on TSR. Tell us a little about yourself to get started.

### University open days

• Cranfield University
Cranfield Forensic MSc Programme Open Day Postgraduate
Thu, 25 Apr '19
• University of the Arts London
Open day: MA Footwear and MA Fashion Artefact Postgraduate
Thu, 25 Apr '19
• Cardiff Metropolitan University
Sat, 27 Apr '19

### Poll

Join the discussion

#### Have you registered to vote?

Yes! (260)
38.92%
No - but I will (45)
6.74%
No - I don't want to (49)
7.34%
No - I can't vote (<18, not in UK, etc) (314)
47.01%