Find all triples of positive integers a,b and c such that
(1 + 1/a)(1 + 1/b)(1 + 1/c) = 2
Good luck.
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Moderator: Jon O'Neill
BMO1 or BMO2? I sucked fairly hard at the former when I was at school (I got one right and a partial solution to another, which I think was near the median).Conor wrote:It's a BMO question, actually.
BMO2, although it seemed a bit on the easier side for it. I'm pretty bad at Olympiad style questions, so it's always pleasing when I manage to do one. I'll have a look into a geometrical interpretation -- my method was purely algebraic.Paul Howe wrote:BMO1 or BMO2? I sucked fairly hard at the former when I was at school (I got one right and a partial solution to another, which I think was near the median).
I've actually got a vague memory of seeing this one before, possibly got my wires crossed somewhere but remember it having a nice geometrical interpretation. I'll certainly have a stab at it tomorrow.
I did it this far and, having spotted the geometrical significance, couldn't think of anyway to use it to solve the problem (since geometry isn't constrained to integers.) Did I miss something, or was the geometrical bit just for "fun"?Paul Howe wrote: So the answer is (2,4,15),(2,5,9),(2,6,7),(3,3,8),(3,4,5), and all permutations of these. I think the geometric interpretation was to multiply everything by abc so (a+1)(b+1)(c+1) = 2abc. The solutions then describe the set of all cuboids having integer length sides such that increasing the length of each side by 1 doubles the volume. Cute, but not really relevant to solving the problem.
No, not relevant, I saw this in a solution booklet when I was practising for BMO yonks ago and had a fuzzy memory of it having geometrical flavour, but couldn't think of the significance when I posted last night. It would've been a nicer question if they'd given the geometry description and required you to translate into algebra, I think.Charlie Reams wrote:
I did it this far and, having spotted the geometrical significance, couldn't think of anyway to use it to solve the problem (since geometry isn't constrained to integers.) Did I miss something, or was the geometrical bit just for "fun"?
But 1/infinity isn't defined...Dinos Sfyris wrote:Also I know you said integers but there's also: 2,3,infinity and 1,infinity,infinity
But if it were it would be zero, yes? :pMichael Wallace wrote:But 1/infinity isn't defined...Dinos Sfyris wrote:Also I know you said integers but there's also: 2,3,infinity and 1,infinity,infinity
You can say as a tends to infinity, 1/a tends to zero. But 1/infinity isn't defined since infinity isn't a number.Dinos Sfyris wrote: But if it were it would be zero, yes? :p
To be fair, he did acknowledge that bit. I just get unreasonably annoyed when people say things like 1/0 = infinityCharlie Reams wrote:And in any case the question asked for integers, and infinity definitely ain't one of those.
I just went to a very weird visual place involving a raccoon-like Hulk!Michael Wallace wrote:I just get unreasonably annoyed when people say things like 1/0 = infinityCharlie Reams wrote:And in any case the question asked for integers, and infinity definitely ain't one of those.