Difference between revisions of "2000 AIME I Problems/Problem 10"

Revision as of 17:20, 20 November 2022

Problem

A sequence of numbers $x_{1},x_{2},x_{3},\ldots,x_{100}$ has the property that, for every integer $k$ between $1$ and $100,$ inclusive, the number $x_{k}$ is $k$ less than the sum of the other $99$ numbers. Given that $x_{50} = m/n,$ where $m$ and $n$ are relatively prime positive integers, find $m + n$ .

Solution

Let the sum of all of the terms in the sequence be $\mathbb{S}$ . Then for each integer $k$ , $x_k = \mathbb{S}-x_k-k \Longrightarrow \mathbb{S} - 2x_k = k$ . Summing this up for all $k$ from $1, 2, \ldots, 100$ ,

$\begin{align*}100\mathbb{S}-2(x_1 + x_2 + \cdots + x_{100}) &= 1 + 2 + \cdots + 100\\ 100\mathbb{S} - 2\mathbb{S} &= \frac{100 \cdot 101}{2} = 5050\\ \mathbb{S}&=\frac{2525}{49}\end{align*}$

Now, substituting for $x_{50}$ , we get $2x_{50}=\frac{2525}{49}-50=\frac{75}{49} \Longrightarrow x_{50}=\frac{75}{98}$ , and the answer is $75+98=\boxed{173}$ .

Solution 2

Consider $x_k$ and $x_{k+1}$ . Let $S$ be the sum of the rest 98 terms. Then $x_k+k=S+x_{k+1}$ and $x_{k+1}+(k+1)=S+x_k.$ Eliminating $S$ we have $x_{k+1}-x_k=-\dfrac{1}{2}.$ So the sequence is arithmetic with common difference $-\dfrac{1}{2}.$

In terms of $x_{50},$ the sequence is $x_{50}+\dfrac{49}{2}, x_{50}+\dfrac{48}{2},\cdots,x_{50}+\dfrac{1}{2}, x_{50}, x_{50}-\dfrac{1}{2}, \cdots, x_{50}-\dfrac{49}{2}, x_{50}-\dfrac{50}{2}.$ Therefore $x_{50}+50=99x_{50}-\dfrac{50}{2}$ . Solving, we get $x_{50}=\dfrac{75}{98}. The answer is$ 75+98=\boxed{173.}$

-JZ -minor edit by erinb28lms (final solution was not given)

Video solution

https://www.youtube.com/watch?v=TdvxgrSZTQw

@@ Line 14: / Line 14: @@
 Consider <math>x_k</math> and <math>x_{k+1}</math>. Let <math>S</math> be the sum of the rest 98 terms. Then <math>x_k+k=S+x_{k+1}</math> and <math>x_{k+1}+(k+1)=S+x_k.</math> Eliminating <math>S</math> we have <math>x_{k+1}-x_k=-\dfrac{1}{2}.</math> So the sequence is arithmetic with common difference <math>-\dfrac{1}{2}.</math>
-In terms of <math>x_{50},</math> the sequence is <math>x_{50}+\dfrac{49}{2}, x_{50}+\dfrac{48}{2},\cdots,x_{50}+\dfrac{1}{2}, x_{50}, x_{50}-\dfrac{1}{2}, \cdots, x_{50}-\dfrac{49}{2}, x_{50}-\dfrac{50}{2}.</math> Therefore <math>x_{50}+50=99x_{50}-\dfrac{50}{2}</math>. We are done by solving for <math>x_{50}</math>.
+In terms of <math>x_{50},</math> the sequence is <math>x_{50}+\dfrac{49}{2}, x_{50}+\dfrac{48}{2},\cdots,x_{50}+\dfrac{1}{2}, x_{50}, x_{50}-\dfrac{1}{2}, \cdots, x_{50}-\dfrac{49}{2}, x_{50}-\dfrac{50}{2}.</math> Therefore <math>x_{50}+50=99x_{50}-\dfrac{50}{2}</math>. Solving, we get <math>x_{50}=\dfrac{75}{98}. The answer is </math>75+98=\boxed{173.}$
 -JZ
+-minor edit by erinb28lms (final solution was not given)
 ==Video solution==

2000 AIME I (Problems • Answer Key • Resources)
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