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
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
Since this is an arithmetic sequence, we have So,
Hence,
And, since we are given that
we get
and
~Kiran
Solution 3 (ChatGPT)
Let the number of apples growing on the least productive tree be , and let the common difference between the numbers of apples on the trees be
. Then the number of apples on each of the trees is
,
,
,
,
, and
.
Since the greatest number of apples growing on any tree is double the least number of apples growing on any tree, we have . Simplifying gives
.
The total number of apples on all six trees is
\begin{align*} 990 &= a + (a+d) + (a+2d) + (a+3d) + (a+4d) + (a+5d) \ &= 6a + 15d \ &= 6(5d) + 15d \ &= 45d. \end{align*}
Therefore, and
.
The greatest number of apples growing on any of the six trees is .
Video Solution 1 by SpreadTheMathLove
https://www.youtube.com/watch?v=nNhfDCX5-bw
See also
2023 AIME II (Problems • Answer Key • Resources) | ||
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Followed by Problem 2 | |
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