Difference between revisions of "2023 AIME II Problems/Problem 1"
(→Video Solution 1 by the Power of Logic) |
(→Solution 1) |
||
| Line 3: | Line 3: | ||
The numbers of apples growing on each of six apple trees form an arithmetic sequence where the greatest number of apples growing on any of the six trees is double the least number of apples growing on any of the six trees. The total number of apples growing on all six trees is <math>990.</math> Find the greatest number of apples growing on any of the six trees. | The numbers of apples growing on each of six apple trees form an arithmetic sequence where the greatest number of apples growing on any of the six trees is double the least number of apples growing on any of the six trees. The total number of apples growing on all six trees is <math>990.</math> Find the greatest number of apples growing on any of the six trees. | ||
| − | + | \This is cool | |
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | \ | ||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
| − | |||
==Solution 2== | ==Solution 2== | ||
Revision as of 09:10, 29 February 2024
Contents
Problem
The numbers of apples growing on each of six apple trees form an arithmetic sequence where the greatest number of apples growing on any of the six trees is double the least number of apples growing on any of the six trees. The total number of apples growing on all six trees is 
Find the greatest number of apples growing on any of the six trees.
\This is cool
Solution 2
Let the terms in the sequence be defined as ![\[a_1, a_2, ..., a_6.\]](http://latex.artofproblemsolving.com/7/e/6/7e65f57f79b98b0ebbeb36be340162b45a2c3e81.png)
Since this is an arithmetic sequence, we have 
So, ![\[\sum_{i=1}^6 a_i=3(a_1+a_6)=990.\]](http://latex.artofproblemsolving.com/f/5/4/f54a6d7b6ffbceab3e69e3a23a474668f3b48737.png)
Hence, 
And, since we are given that 
we get 
and 
~Kiran
Video Solution 1 by SpreadTheMathLove
https://www.youtube.com/watch?v=nNhfDCX5-bw
Video Solution by the Power of Logic(both #1 and #2)
See also
| 2023 AIME II (Problems • Answer Key • Resources) | ||
| Preceded by First Problem |
Followed by Problem 2 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
| All AIME Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
