Difference between revisions of "2023 AIME II Problems/Problem 1"

Revision as of 05:05, 18 February 2023

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 $990.$ Find the greatest number of apples growing on any of the six trees.

Solution 1

In the arithmetic sequence, let $a$ be the first term and $d$ be the common difference, where $d>0.$ The sum of the first six terms is $a+(a+d)+(a+2d)+(a+3d)+(a+4d)+(a+5d) = 6a+15d.$ We are given that $\begin{align*} 6a+15d &= 990, \\ 2a &= a+5d. \end{align*}$ The second equation implies that $a=5d.$ Substituting this into the first equation, we get $\begin{align*} 6(5d)+15d &=990, \\ 45d &= 990 \\ d &= 22. \end{align*}$ It follows that $a=110.$ Therefore, the greatest number of apples growing on any of the six trees is $a+5d=\boxed{220}.$

~MRENTHUSIASM

Solution 2

Let the terms in the sequence be defined as $a_1, a_2, ..., a_6.$

Since this is an arithmetic sequence, we have $a_1+a_6=a_2+a_5=a_3+a_4.$ So, $\sum_{i=1}^6 a_i=3(a_1+a_6)=990.$ Hence, $(a_1+a_6)=330.$ And, since we are given that $a_6=2a_1,$ we get $3a_1=330\implies a_1=110$ and $a_6=\boxed{220}.$

~Kiran

Solution 3 (ChatGPT)

Let the number of apples growing on the least productive tree be $a$ , and let the common difference between the numbers of apples on the trees be $d$ . Then the number of apples on each of the trees is $a$ , $a + d$ , $a + 2d$ , $a + 3d$ , $a + 4d$ , and $a + 5d$ .

Since the greatest number of apples growing on any tree is double the least number of apples growing on any tree, we have $(a + 5d) = 2a$ . Simplifying gives $a = 5d$ .

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, $d = \frac{990}{45} = 22$ and $a = 5d = 110$ .

The greatest number of apples growing on any of the six trees is $a+5d = 110 + 5\cdot 22 = \boxed{220}$ .

Video Solution 1 by SpreadTheMathLove

https://www.youtube.com/watch?v=nNhfDCX5-bw

@@ Line 28: / Line 28: @@
 ~Kiran
+==Solution 3 (ChatGPT) ==
+Let the number of apples growing on the least productive tree be <math>a</math>, and let the common difference between the numbers of apples on the trees be <math>d</math>. Then the number of apples on each of the trees is <math>a</math>, <math>a + d</math>, <math>a + 2d</math>, <math>a + 3d</math>, <math>a + 4d</math>, and <math>a + 5d</math>.
+Since the greatest number of apples growing on any tree is double the least number of apples growing on any tree, we have <math>(a + 5d) = 2a</math>. Simplifying gives <math>a = 5d</math>.
+The total number of apples on all six trees is \begin{align*}
+&= a + (a+d) + (a+2d) + (a+3d) + (a+4d) + (a+5d) \
+&= 6a + 15d \
+&= 6(5d) + 15d \
+&= 45d.
+\end{align*}Therefore, <math>d = \frac{990}{45} = 22</math> and <math>a = 5d = 110</math>.
+The greatest number of apples growing on any of the six trees is <math>a+5d = 110 + 5\cdot 22 = \boxed{220}</math>.
 ==Video Solution 1 by SpreadTheMathLove==

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