Difference between revisions of "2019 AMC 10A Problems/Problem 5"

Latest revision as of 22:16, 19 November 2024

The following problem is from both the 2019 AMC 10A #5 and 2019 AMC 12A #4, so both problems redirect to this page.

Problem

What is the greatest number of consecutive integers whose sum is $45?$

$\textbf{(A) } 9 \qquad\textbf{(B) } 25 \qquad\textbf{(C) } 45 \qquad\textbf{(D) } 90 \qquad\textbf{(E) } 120$

Solution 1

We might at first think that the answer would be $9$ , because $1+2+3 \dots +n = 45$ when $n = 9$ . But note that the problem says that they can be integers, not necessarily positive. Observe also that every term in the sequence $-44, -43, \cdots, 44, 45$ cancels out except $45$ . Thus, the answer is, intuitively, $\boxed{\textbf{(D) } 90 }$ integers.

Though impractical, a proof of maximality can proceed as follows: Let the desired sequence of consecutive integers be $a, a+1, \cdots, a+(N-1)$ , where there are $N$ terms, and we want to maximize $N$ . Then the sum of the terms in this sequence is $aN + \frac{(N-1)(N)}{2}=45$ . Rearranging and factoring, this reduces to $N(2a+N-1) = 90$ . Since $N$ must divide $90$ , and we know that $90$ is an attainable value of the sum, $90$ must be the maximum.

Solution 2

To maximize the number of integers, we need to make the average of them as low as possible while still being positive. The average can be $\frac12$ if the middle two numbers are $0$ and $1$ , so the answer is $\frac{45}{\frac12}=\boxed{\textbf{(D) } 90 }$ .

Solution 2a (more detailed ver. of Solution 2)

Note that the sum of $n$ consecutive integers whose mean (median; the two are equal in this case) is $a$ is equal to $an$ . Here, we want to maximize $n$ such that $an=45$ . The mean/median of a set of consecutive integers is either an integer or half of an integer; clearly, $n$ is maximized when $a=0.5$ and $n=\boxed{\textbf{(D) }90}$ .

~Technodoggo

Solution 3

If we want the answer to have as much terms as possible we need to make as much terms cancel out as much as we can.Because 45 - 1 < $\frac12$ 60 so we can eliminate E. Now lets make every integer cancel to zero except for 45 and then we account for 0 and 45 to get 44*2+1+1= $\boxed{\textbf{(D) }90}$ . Pls help me clarify idk how to explain it clearer.

~Twinotter

Video Solution 1

https://youtu.be/ZhAZ1oPe5Ds?t=665

~ pi_is_3.14

Video Solution 2

https://youtu.be/ivYp-eNOIZA

~savannahsolver

2019 AMC 10A (Problems • Answer Key • Resources)
Preceded by Problem 4	Followed by Problem 6
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AMC 10 Problems and Solutions

2019 AMC 12A (Problems • Answer Key • Resources)
Preceded by Problem 3	Followed by Problem 5
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AMC 12 Problems and Solutions

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.

@@ Line 7: / Line 7: @@
 ==Solution 1==
-We first think that the solution would be 9 because <math>1+2+3 \dots +n = 45</math> (n = 9). But note that the problem says that they can be integers, not necessarily positive. Note that every term in the sequence <math>-44, -43, \cdots, 44, 45</math> cancels out except <math>45</math>.  Thus, the answer is, intuitively, <math>\boxed{\textbf{(D) } 90 }</math> integers.
+We might at first think that the answer would be <math>9</math>, because <math>1+2+3 \dots +n = 45</math> when <math>n = 9</math>. But note that the problem says that they can be integers, not necessarily positive. Observe also that every term in the sequence <math>-44, -43, \cdots, 44, 45</math> cancels out except <math>45</math>.  Thus, the answer is, intuitively, <math>\boxed{\textbf{(D) } 90 }</math> integers.
 Though impractical, a proof of maximality can proceed as follows:  Let the desired sequence of consecutive integers be <math>a, a+1, \cdots, a+(N-1)</math>, where there are <math>N</math> terms, and we want to maximize <math>N</math>.  Then the sum of the terms in this sequence is <math>aN + \frac{(N-1)(N)}{2}=45</math>.  Rearranging and factoring, this reduces to <math>N(2a+N-1) = 90</math>.  Since <math>N</math> must divide <math>90</math>, and we know that <math>90</math> is an attainable value of the sum, <math>90</math> must be the maximum.
---clara32356
 ==Solution 2==
 To maximize the number of integers, we need to make the average of them as low as possible while still being positive. The average can be <math>\frac12</math> if the middle two numbers are <math>0</math> and <math>1</math>, so the answer is <math>\frac{45}{\frac12}=\boxed{\textbf{(D) } 90 }</math>.
+===Solution 2a (more detailed ver. of Solution 2)===
+Note that the sum of <math>n</math> consecutive integers whose mean (median; the two are equal in this case) is <math>a</math> is equal to <math>an</math>. Here, we want to maximize <math>n</math> such that <math>an=45</math>. The mean/median of a set of consecutive integers is either an integer or half of an integer; clearly, <math>n</math> is maximized when <math>a=0.5</math> and <math>n=\boxed{\textbf{(D) }90}</math>.
+~Technodoggo
+==Solution 3==
+If we want the answer to have as much terms as possible we need to make as much terms cancel out as much as we can.Because 45 - 1 < <math>\frac12</math>60 so we can eliminate E. Now lets make every integer cancel to zero except for 45 and then we account for 0 and 45 to get 44*2+1+1=<math>\boxed{\textbf{(D) }90}</math>. Pls help me clarify idk how to explain it clearer.
+~Twinotter
+== Video Solution 1 ==
+https://youtu.be/ZhAZ1oPe5Ds?t=665
+~ pi_is_3.14
+==Video Solution 2==
+https://youtu.be/ivYp-eNOIZA
+~savannahsolver
 ==See Also==

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