Difference between revisions of "2000 AMC 12 Problems/Problem 18"

Latest revision as of 06:46, 24 August 2024

The following problem is from both the 2000 AMC 12 #18 and 2000 AMC 10 #25, so both problems redirect to this page.

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 $100$ ^th day of year $N-1$ occur?

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

Solution 1

There are either $65 + 200 = 265$ or $66 + 200 = 266$ days between the first two dates depending upon whether or not year $N+1$ is a leap year (since the February 29th of the leap year would come between the 300th day of year $N$ and 200th day of year $N + 1$ ). Since $7$ divides into $266$ but not $265$ , for both days to be a Tuesday, year $N$ must be a leap year.

Hence, year $N-1$ is not a leap year, and so since there are $265 + 300 = 565$ days between the date in years $N,\text{ }N-1$ , this leaves a remainder of $5$ upon division by $7$ . Since we are subtracting days, we count 5 days before Tuesday, which gives us $\boxed{\mathbf{(A)} \ \text{Thursday}.}$

Solution 2

The $300-29\cdot 7=97^{\text{th}}$ day of year $N$ and the $200-15\cdot 7=95^{\text{th}}$ day of year $N+1$ are Tuesdays. If there were the same number of days in year $N$ and year $N-1,$ the $99^{\text{th}}$ day of year $N-1$ will be a Tuesday. But year $N$ is a leap year because $97-95 \cong 366 \pmod{7},$ so the $98^{\text{th}}$ day of year $N-1$ is a Tuesday. It follows that the $100^{\text{th}}$ day of year $N-1$ is $\boxed{\mathbf{(A)} \ \text{Thursday}.}$

~dolphin7

Video Solution

https://youtu.be/aFJtbvTAmm4

https://www.youtube.com/watch?v=YfiUn8hLlpA ~David

2000 AMC 12 (Problems • Answer Key • Resources)
Preceded by Problem 17	Followed by Problem 19
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AMC 12 Problems and Solutions

2000 AMC 10 (Problems • Answer Key • Resources)
Preceded by Problem 24	Followed by Last Problem
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AMC 10 Problems and Solutions

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.

@@ Line 6: / Line 6: @@
 <math>\text {(A)}\ \text{Thursday} \qquad \text {(B)}\ \text{Friday}\qquad \text {(C)}\ \text{Saturday}\qquad \text {(D)}\ \text{Sunday}\qquad \text {(E)}\ \text{Monday}</math>
-== Solution ==
+== Solution 1 ==
-There are either <cmath>265 + 300 = 565</cmath> or <cmath>266 + 300 = 566</cmath> days between year <math>N</math> and year <math>N-1</math>, depending on whether year <math>N-1</math> is a leap year.
+There are either <cmath>65 + 200 = 265</cmath> or <cmath>66 + 200 = 266</cmath> days between the first two dates depending upon whether or not year <math>N+1</math> is a leap year (since the February 29th of the leap year would come between the 300th day of year <math>N</math> and 200th day of year <math>N + 1</math>). Since <math>7</math> divides into <math>266</math> but not <math>265</math>, for both days to be a Tuesday, year <math>N</math> must be a leap year.
-In addition, there are either <cmath>65 + 200 = 265</cmath> or <cmath>66 + 200 = 266</cmath> days between the first two dates depending upon whether or not year <math>N</math> is a leap year. Since <math>7</math> divides into <math>266</math> but not <math>265</math>, for both days to be a Tuesday, year <math>N</math> must be a leap year.
+Hence, year <math>N-1</math> is not a leap year, and so since there are <cmath>265 + 300 = 565</cmath> days between the date in years <math>N,\text{ }N-1</math>, this leaves a remainder of <math>5</math> upon division by <math>7</math>. Since we are subtracting days, we count 5 days before Tuesday, which gives us  <math>\boxed{\mathbf{(A)} \ \text{Thursday}.}</math>
-Hence, year <math>N-1</math> is not a leap year, and so since there are <cmath>265 + 300 = 565</cmath> days between the date in years <math>N,\text{ }N-1</math>, this leaves a remainder of <math>5</math> upon division by <math>7</math>. Since we are subtracting days, we count 5 days before Tuesday, which gives us  <math>\mathrm {Thursday} \text{ (A)}</math>.
+== Solution 2 ==
+The <math>300-29\cdot 7=97^{\text{th}}</math> day of year <math>N</math> and the <math>200-15\cdot 7=95^{\text{th}}</math> day of year <math>N+1</math> are Tuesdays. If there were the same number of days in year <math>N</math> and year <math>N-1,</math> the <math>99^{\text{th}}</math> day of year <math>N-1</math> will be a Tuesday. But year <math>N</math> is a leap year because <math>97-95 \cong 366 \pmod{7},</math> so the <math>98^{\text{th}}</math> day of year <math>N-1</math> is a Tuesday. It follows that the <math>100^{\text{th}}</math> day of year <math>N-1</math> is <math>\boxed{\mathbf{(A)} \ \text{Thursday}.}</math>
-Edited by Max0815
+~dolphin7
+==Video Solution ==
+https://youtu.be/aFJtbvTAmm4
+https://www.youtube.com/watch?v=YfiUn8hLlpA   ~David
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