2023 AMC 8 Problems/Problem 25

Revision as of 19:43, 24 January 2023 by Eibc (talk | contribs) (Solution 2)

Problem

Fifteen integers $a_1, a_2, a_3, \dots, a_{15}$ are arranged in order on a number line. The integers are equally spaced and have the property that \[1 \le a_1 \le 10, \thickspace 13 \le a_2 \le 20, \thickspace 241 \le a_{15}\le 250.\] What is the sum of digits of $a_{14}$?

\[\textbf{(A) } 8 \qquad \textbf{(B) } 9 \qquad \textbf{(C) } 10 \qquad \textbf{(D) } 11 \qquad \textbf{(E) } ~12\]

Video Solution - Divisibility makes diophantine equation trivial

https://www.youtube.com/watch?v=5LLl26VI-7Y&list=PLT9bNzqjDoMnk_gSjh66yRuDMdGecx5zI&index=8

Animated Video Solution

https://youtu.be/itDH7AgxYFo

~Star League (https://starleague.us)

Solution 1

\[1\leq a_1\leq10\]

\[13\leq a_2\leq20\]

\[241\leq a_{15}\leq250\]


We can find the possible values of the common difference by finding the numbers which satisfy the conditions. To do this, we find the minimum of the last two–$241-20=221$, and the maximum–$250-13=237$. There is a difference of 13 between them, so only $17$ and $18$ work, as $17*13=221$, so $17$ satisfies $221\leq 13x\leq237$. The number $18$ is similarly found. $19$, however, is too much.


Now, we check with the first and last equations using the same method. We know $241-10\leq 14x\leq250-1$. Therefore, $231\leq 14x\leq249$. We test both values we just got, and we can realize that $18$ is too large to satisfy this inequality. On the other hand, we can now find that the difference will be $17$, which satisfies this inequality.


The last step is to find the first term. We know that the first term can only be from $1$ to $3$, since any larger value would render the second inequality invalid. Testing these three, we find that only $a_1=3$ will satisfy all the inequalities. Therefore, $a_{14}=13\cdot17+3=224$. The sum of the digits is therefore $\boxed{\text{(A)} \hspace{0.1 in} 8}$


~apex304, SohumUttamchandani, wuwang2002, TaeKim, Cxrupptedpat

Solution 2

Let the common difference between consecutive $a_i$ be $d$. Then, since $a_{15} - a_1 = 14d$, we find from the first and last inequalities that $231 \le d \le 249$. As $d$ must be an integer, this means $d = 17$. Plugging this into all of the given inequalities so we may extract information about $a_1$ gives \[1 \le a_1 \le 10, \thickspace 13 \le a_1 + 17 \le 20, \thickspace 241 \le a_1 + 238 \le 250.\] The second inequality tells us that $a_1 \le 3$, while the last inequality tells us $3 \le a_1$, so we must have $a_1 = 3$. Finally, to solve for $a_{14}$, we simply have $a_{14} = a_1 + 13d = 3 + 221 = 224$, so our answer is $\boxed{\textbf{(A)}\ 8}$.

~eibc