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

Latest revision as of 14:13, 17 July 2025

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

- edited by erinb28lms

Solution 3 (Sum of equations)

Like Solution 1, let the sum of all of the terms in this sequence be $\mathbb{S}$ . By definition:

$x_1 + 1 = x_2+x_3+x_4+...+x_{100}$ $x_2 + 2 = x_1+x_3+x_4+...+x_{100}$ $x_3 + 3 = x_1+x_2+x_4+...+x_{100}$ $...$ $x_{99} + 99 = x_1+x_2+x_3+...+x_{98}+x_{100}$ $x_{100} + 100 = x_1+x_2+x_3+...+x_{98}+x_{99}$ .

Adding up all of these equations yields:

$\mathbb{S} + TR(100) = 99\mathbb{S}$

Here $TR(100)$ represents the $100$ th triangular number, which is $5050$ . Solving for $\mathbb{S}$ yields:

$\mathbb{S} = \frac{2525}{49}$ .

$\mathbb{S}$ can also be written as $x_{50} + (x_{50} + 50)$ . Solving for $x_{50}$ ,

$2x_{50} = \frac{2525-2450}{49}$ $2x_{50} = \frac{75}{49}$ $x_{50} = \frac{75}{98}$

The requested sum is therefore $75+98 = \boxed{173}$ .

~mathwizard123123

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>.
--JZ
+Solving, we get <math>x_{50}=\dfrac{75}{98}.</math> The answer is <math>75+98=\boxed{173}.</math>
+- JZ
+- edited by erinb28lms
+==Solution 3 (Sum of equations)==
+Like Solution 1, let the sum of all of the terms in this sequence be <math>\mathbb{S}</math>. By definition:
+<cmath>x_1 + 1 = x_2+x_3+x_4+...+x_{100}</cmath>
+<cmath>x_2 + 2 = x_1+x_3+x_4+...+x_{100}</cmath>
+<cmath>x_3 + 3 = x_1+x_2+x_4+...+x_{100}</cmath>
+<cmath>...</cmath>
+<cmath>x_{99} + 99 = x_1+x_2+x_3+...+x_{98}+x_{100}</cmath>
+<cmath>x_{100} + 100 = x_1+x_2+x_3+...+x_{98}+x_{99}</cmath>.
+Adding up all of these equations yields:
+<cmath>\mathbb{S} + TR(100) = 99\mathbb{S}</cmath>
+Here <math>TR(100)</math> represents the <math>100</math>th triangular number, which is <math>5050</math>. Solving for <math>\mathbb{S}</math> yields:
+<cmath>\mathbb{S} = \frac{2525}{49}</cmath>.
+<math>\mathbb{S}</math> can also be written as <math>x_{50} + (x_{50} + 50)</math>. Solving for <math>x_{50}</math>,
+<cmath>2x_{50} = \frac{2525-2450}{49}</cmath>
+<cmath>2x_{50} = \frac{75}{49}</cmath>
+<cmath>x_{50} = \frac{75}{98}</cmath>
+The requested sum is therefore <math>75+98 = \boxed{173}</math>.
+~mathwizard123123
+==Video solution==
+https://www.youtube.com/watch?v=TdvxgrSZTQw
 == See also ==

2000 AIME I (Problems • Answer Key • Resources)
Preceded by Problem 9	Followed by Problem 11
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15
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