2023 AIME II Problems/Problem 1
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.
Solution 1
In the arithmetic sequence, let 
be the first term and 
be the common difference, where 
The sum of the first six terms is ![\[a+(a+d)+(a+2d)+(a+3d)+(a+4d)+(a+5d) = 6a+15d.\]](http://latex.artofproblemsolving.com/5/f/8/5f8aeca3955d001532722393e1fc38cfd900cbbb.png)
We are given that
The second equation implies that 
Substituting this into the first equation, we get
It follows that 
Therefore, the greatest number of apples growing on any of the six trees is 
~MRENTHUSIASM
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 | ||
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