Difference between revisions of "2024 AMC 10B Problems/Problem 25"
(→Solution 1) |
|||
Line 1: | Line 1: | ||
+ | ==Problem== | ||
+ | You made it! If you're good with asymptote, please insert an image along with the rest of the problem! | ||
+ | |||
==Solution 1== | ==Solution 1== | ||
The <math>3</math>x<math>3</math>x<math>3</math> block has side lengths of <math>3a, 3b, 3c</math>. The <math>2</math>x<math>2</math>x<math>7</math> block has side lengths of <math>2b, 2c, 7a</math>. | The <math>3</math>x<math>3</math>x<math>3</math> block has side lengths of <math>3a, 3b, 3c</math>. The <math>2</math>x<math>2</math>x<math>7</math> block has side lengths of <math>2b, 2c, 7a</math>. | ||
Line 9: | Line 12: | ||
Adding all the equations together, we get <math>b+c+3 = 4a</math>. Adding <math>a-3</math> to both sides, we get <math>a+b+c = 5a-3</math>. The question states that <math>a,b,c</math> are all relatively prime positive integers. Therefore, our answer must be congruent to <math>2 \pmod{5}</math>. The only answer choice satisfying this is <math>\boxed{92}</math>. | Adding all the equations together, we get <math>b+c+3 = 4a</math>. Adding <math>a-3</math> to both sides, we get <math>a+b+c = 5a-3</math>. The question states that <math>a,b,c</math> are all relatively prime positive integers. Therefore, our answer must be congruent to <math>2 \pmod{5}</math>. The only answer choice satisfying this is <math>\boxed{92}</math>. | ||
~lprado | ~lprado | ||
+ | |||
+ | ==See also== | ||
+ | {{AMC10 box|year=2024|ab=B|num-b=20|after=Last Problem}} | ||
+ | {{MAA Notice}} |
Revision as of 01:35, 14 November 2024
Problem
You made it! If you're good with asymptote, please insert an image along with the rest of the problem!
Solution 1
The xx block has side lengths of . The xx block has side lengths of .
We can create the following system of equations, knowing that the new block has unit taller, deeper, and wider than the original:
Adding all the equations together, we get . Adding to both sides, we get . The question states that are all relatively prime positive integers. Therefore, our answer must be congruent to . The only answer choice satisfying this is . ~lprado
See also
2024 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 20 |
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.