Difference between revisions of "2024 AIME II Problems/Problem 3"
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Find the number of ways to place a digit in each cell of a 2x3 grid so that the sum of the two numbers formed by reading left to right is <math>999</math>, and the sum of the three numbers formed by reading top to bottom is <math>99</math>. The grid below is an example of such an arrangement because <math>8+991=999</math> and <math>9+9+81=99</math>. | Find the number of ways to place a digit in each cell of a 2x3 grid so that the sum of the two numbers formed by reading left to right is <math>999</math>, and the sum of the three numbers formed by reading top to bottom is <math>99</math>. The grid below is an example of such an arrangement because <math>8+991=999</math> and <math>9+9+81=99</math>. | ||
| − | < | + | <cmath> |
\begin{array}{|c|c|c|} \hline | \begin{array}{|c|c|c|} \hline | ||
0 & 0 & 8 \\ \hline | 0 & 0 & 8 \\ \hline | ||
| − | 9 & 9 & 1\\ \hline | + | 9 & 9 & 1 \\ \hline |
\end{array} | \end{array} | ||
| − | </ | + | </cmath> |
==Solution 1== | ==Solution 1== | ||
Revision as of 13:52, 10 February 2024
Contents
Problem
Find the number of ways to place a digit in each cell of a 2x3 grid so that the sum of the two numbers formed by reading left to right is 
, and the sum of the three numbers formed by reading top to bottom is 
. The grid below is an example of such an arrangement because 
and 
.
![\[\begin{array}{|c|c|c|} \hline 0 & 0 & 8 \\ \hline 9 & 9 & 1 \\ \hline \end{array}\]](http://latex.artofproblemsolving.com/3/7/8/3789264ba7fedec68fa8488d65f286df5cdbefbc.png)
Solution 1
Consider this table:
We note that 
, because 
, meaning it never achieves a unit's digit sum of 
otherwise. Since no values are carried onto the next digit, this implies 
and 
. We can then simplify our table into this:
We want 
, or 
, or 
. Since zeroes are allowed, we just need to apply stars and bars on 
, to get 
. ~akliu
Video Solution
~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)
See also
| 2024 AIME II (Problems • Answer Key • Resources) | ||
| Preceded by Problem 2 |
Followed by Problem 4 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
| All AIME Problems and Solutions | ||
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The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
