Difference between revisions of "1998 USAMO Problems/Problem 1"
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== Solution == | == Solution == | ||
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If <math>S=|a_1-b_1|+|a_2-b_2|+\cdots +|a_{999}-b_{999}|</math>, then <math>S \equiv 1+1+\cdots + 1 \equiv 99 \equiv 4 (mod 5)</math>. | If <math>S=|a_1-b_1|+|a_2-b_2|+\cdots +|a_{999}-b_{999}|</math>, then <math>S \equiv 1+1+\cdots + 1 \equiv 99 \equiv 4 (mod 5)</math>. |
Revision as of 10:42, 6 June 2011
Problem
Suppose that the set has been partitioned into disjoint pairs
(
) so that for all
,
equals
or
. Prove that the sum
ends in the digit
.
Solution
If , then
.
For integers M, N we have .
So we also have also, so
.