1990 USAMO Problems/Problem 1
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[hide]Problem
A certain state issues license plates consisting of six digits (from 0 through 9). The state requires that any two plates differ in at least two places. (Thus the plates and cannot both be used.) Determine, with proof, the maximum number of distinct license plates that the state can use.
Solution 1
Consider license plates of digits, for some fixed , issued with the same criteria.
We first note that by the pigeonhole principle, we may have at most distinct plates. Indeed, if we have more, then there must be two plates which agree on the first digits; these plates thus differ only on one digit, the last one.
We now show that it is possible to issue distinct license plates which satisfy the problem's criteria. Indeed, we issue plates with all possible combinations for the first digit, and for each plate, we let the last digit be the sum of the preceding digits taken mod 10. This way, if two plates agree on the first digits, they agree on the last digit and are thus the same plate, and if two plates differ in only one of the first digits, they must differ as well in the last digit.
It then follows that is the greatest number of license plates the state can issue. For , as in the problem, this number is .
Solution 2
Suppose for , we have a set that satisfies the condition. We need to construct such that it does not contain , denoted as . Construct: If and have exactly one difference: 1. Case 1: 2. Case 2: If for , Contradiction! If , contradiction with the same . At least two positions are different, i.e., there cannot be exactly one difference. Therefore, in the set (hypercube), there are no collinear points (parallel to the coordinate axes).
See Also
1990 USAMO (Problems • Resources) | ||
Preceded by First question |
Followed by Problem 2 | |
1 • 2 • 3 • 4 • 5 | ||
All USAMO Problems and Solutions |
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