Difference between revisions of "2011 IMO Problems/Problem 1"
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ANSWER: <math>\{11a, a, 5a, 7a\}</math>,<math>\{a, 14a, 6a, 9a\}</math>, for any positive integer <math>a</math>. | ANSWER: <math>\{11a, a, 5a, 7a\}</math>,<math>\{a, 14a, 6a, 9a\}</math>, for any positive integer <math>a</math>. | ||
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+ | (Note: The above solution looks generally correct, but the actual answer should be <math>\{11a, a, 5a, 7a\}</math>,<math>\{a, 11a, 19a, 29a\}</math>. You can check that <math>\{a, 14a, 6a, 9a\}</math> doesn't actually work. -Someone who didn't write up the above solution but solved the problem in a similar way) | ||
==See Also== | ==See Also== |
Revision as of 01:39, 3 March 2014
Problem
Given any set of four distinct positive integers, we denote the sum
by
. Let
denote the number of pairs
with
for which
divides
. Find all sets
of four distinct positive integers which achieve the largest possible value of
.
Author: Fernando Campos, Mexico
Solution
Firstly, if we order , we see
, so
isn't a couple that satisfies the conditions of the problem. Also,
, so again
isn't a good couple. We have in total 6 couples. So
.
We now find all sets with
. If
and
are both good couples, and
, we have
.
So WLOG
with
and
. It's easy to see
and since
are bad, all couples containing
must be good. Obviously
and
are good (
). So we have
and
.
Using the second equation, we see that if ,
, for some
a positive integer.
So now we use the first equation to get , for a natural
.
Finally, we obtain 1, 2 or 4. We divide in cases:
CASE I: .
So
and
. But
3, 4,5 or 6.
implies
, impossible.
when
. We easily see
and
, impossible since
. When
,
, and we get
.Uf
,
and we get
.
CASE II and III:2, 4. Left to the reader.
ANSWER: ,
, for any positive integer
.
(Note: The above solution looks generally correct, but the actual answer should be ,
. You can check that
doesn't actually work. -Someone who didn't write up the above solution but solved the problem in a similar way)
See Also
2011 IMO (Problems) • Resources | ||
Preceded by First question |
1 • 2 • 3 • 4 • 5 • 6 | Followed by Problem 2 |
All IMO Problems and Solutions |