Difference between revisions of "2001 IMO Problems/Problem 4"
(New page: Let n1, n2, ... , nm be integers where m is odd. Let x = (x1, ... , xm) denote a permutation of the integers 1, 2, ... , m. Let f(x) = x1n1 + x2n2 + ... + xmnm. Show that for some distinct...) |
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| − | Let | + | =Problem== |
| + | Let <math>n_1, n_2, \dots , n_m</math> be integers where <math>m</math> is odd. Let <math>x = (x_1, \dots , x_m)</math> denote a permutation of the integers <math>1, 2, \cdots , m</math>. Let <math>f(x) = x_1n_1 + x_2n_2 + ... + x_mn_m</math>. Show that for some distinct permutations <math>a</math>, <math>b</math> the difference <math>f(a) - f(b)</math> is a multiple of <math>m!</math>. | ||
| + | |||
| + | ==Solution== | ||
| + | {{solution}} | ||
| + | |||
| + | ==See also== | ||
| + | [[Category:Olympiad Combinatorics Problems]] | ||
Revision as of 09:28, 15 October 2008
Problem=
Let 
be integers where 
is odd. Let 
denote a permutation of the integers 
. Let 
. Show that for some distinct permutations 
, 
the difference 
is a multiple of 
.
Solution
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