Difference between revisions of "2008 IMO Problems/Problem 2"
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'''(ii)''' Prove that equality is achieved for infinitely many triples of rational numbers <math>x</math>, <math>y</math> and <math>z</math>. | '''(ii)''' Prove that equality is achieved for infinitely many triples of rational numbers <math>x</math>, <math>y</math> and <math>z</math>. | ||
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== Solution == | == Solution == | ||
Consider the transormation <math>f:\mathbb{R}/\{1\} \rightarrow \mathbb{R}/\{-1\}</math> defined by <math>f(u) = \frac{u}{1-u}</math> and put <math>\alpha = f(x), \beta = f(y), \gamma = f(z)</math>. Since <math>f</math> is also one-to one from <math>\mathbb{Q}/\{1\}</math> to <math>\mathbb{Q}/\{-1\}</math>, the problem is equivalent to showing that | Consider the transormation <math>f:\mathbb{R}/\{1\} \rightarrow \mathbb{R}/\{-1\}</math> defined by <math>f(u) = \frac{u}{1-u}</math> and put <math>\alpha = f(x), \beta = f(y), \gamma = f(z)</math>. Since <math>f</math> is also one-to one from <math>\mathbb{Q}/\{1\}</math> to <math>\mathbb{Q}/\{-1\}</math>, the problem is equivalent to showing that |
Revision as of 21:57, 4 September 2008
Problem 2
(i) If ,
and
are three real numbers, all different from
, such that
, then prove that
.
(With the
sign for cyclic summation, this inequality could be rewritten as
.)
(ii) Prove that equality is achieved for infinitely many triples of rational numbers ,
and
.
Solution
Consider the transormation defined by
and put
. Since
is also one-to one from
to
, the problem is equivalent to showing that
subject to
and that equallity holds for infinitely many triplets of rational
.
Now, rewrite (2) as and express it as
where
and
. Notice that (1) can be written as
But from
, we get
with equality holding iff
. That proves part (i) and points us in the direction of looking for rational
for which
and (hence)
, that is:
Expressing
from the first equation and substituting into the second, we get
as the sole equation we need to satisfy in rational numbers.