1998 USAMO Problems/Problem 1
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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
Notice that , so .
Also, for integers we have .
Thus, we also have also, so by the Chinese Remainder Theorem . Thus, ends in the digit 9, as desired.
See Also
FASTEST SOLVE ON STREAM from v_Enhance (:omighty:) https://www.youtube.com/watch?v=jsw3c3yAn7o
1998 USAMO (Problems • Resources) | ||
Preceded by First Question |
Followed by Problem 2 | |
1 • 2 • 3 • 4 • 5 • 6 | ||
All USAMO Problems and Solutions |
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