2000 AMC 10 Problems/Problem 25

Problem

In year $N$ , the $300^\text{th}$ day of the year is a Tuesday. In year $N+1$ , the $200^\text{th}$ day is also a Tuesday. On what day of the week did the $100^\text{th}$ day of year $N-1$ occur?

$\mathrm{(A)}\ \text{Thursday} \qquad\mathrm{(B)}\ \text{Friday} \qquad\mathrm{(C)}\ \text{Saturday} \qquad\mathrm{(D)}\ \text{Sunday} \qquad\mathrm{(E)}\ \text{Monday}$

Solution

Clearly, identifying what of these years may/must/may not be a leap year will be key in solving the problem.

Let $A$ be the $300^\text{th}$ day of year $N$ , $B$ the $200^\text{th}$ day of year $N+1$ and $C$ the $100^\text{th}$ day of year $N-1$ .

If year $N$ is not a leap year, the day $B$ will be $(365-300) + 200 = 265$ days after $A$ . As $265 \bmod 7 = 6$ , that would be a Monday.

Therefore year $N$ must be a leap year. (Then $B$ is $266$ days after $A$ .)

As there can not be two leap years after each other, $N-1$ is not a leap year. Therefore day $A$ is $265 + 300 = 565$ days after $C$ . We have $565\bmod 7 = 5$ . Therefore $C$ is $5$ weekdays before $A$ , i.e., $C$ is a $\boxed{\text{Thursday}}$ .

(Note that the situation described by the problem statement indeed occurs in our calendar. For example, for $N=2004$ we have $A$ =Tuesday, October 26th 2004, $B$ =Tuesday, July 19th, 2005 and $C$ =Thursday, April 10th 2003.)

2000 AMC 10 (Problems • Answer Key • Resources)
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2000 AMC 10 Problems/Problem 25

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