2016 AMC 10A Problems/Problem 14
Problem
How many ways are there to write as the sum of twos and threes, ignoring order? (For example, and are two such ways.)
Solution 1
The amount of twos in our sum ranges from to , with differences of because .
The possible amount of twos is .
Solution 2
You can also see that you can rewrite the word problem into an equation + = . Therefore the question is just how many multiples of subtracted from 2016 will be an even number. We can see that , all the way to , and works, with being incremented by 's.Therefore, between and , the number of multiples of is .
Solution 3
We can utilize the stars-and-bars distribution technique to solve this problem. We have 2 "buckets" in which we will distribute parts of our total sum, 2016. By doing this, we know we will have "total" answers. We want every third x and second y, so we divide our previous total by 6, which will result in . We have to round down to the nearest integer, and we have to add 2 because we did not consider the 2 solutions involving x or y being 0. So, .
Video Solution
https://youtu.be/dHY8gjoYFXU?t=1058
~IceMatrix
~savannahsolver
https://youtu.be/ZhAZ1oPe5Ds?t=2959
~ pi_is_3.14
See Also
2016 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 13 |
Followed by Problem 15 | |
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.