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Difference between revisions of "2020 AIME II Problems/Problem 8"

(→‎Solution 2 (Same idea, easier to see))
(→‎Solution 2 (Same idea, easier to see))
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Starting from <math>f_1(x)=|x-1|</math>, we can track the solutions, the number of solutions, and their sum.  
 
Starting from <math>f_1(x)=|x-1|</math>, we can track the solutions, the number of solutions, and their sum.  
  
−
<cmath>\begin{center}
+
<cmath>\begin{array}{c|c|c|c}
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\begin{array}{c|c|c|c}
+
x&Solutions&number&sum\\  
−
x & Solutions & number & sum \\  
+
1&1&1&1\\   
−
1 & 1 & 1 & 1 \\   
+
2&1,3&2&4\\   
−
2 & 1,3 & 2 & 4 \\   
+
3&0,2,4,6&4&12\\
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3 & 0,2,4,6 & 4 & 12 \\
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4&-2,0,2...10&7&28\\
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4 & -2,0,2...10 & 7 & 28 \\
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5&-9,-7,-5...21&11&55\\
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5 & -9,-7,-5...21 & 11 & 55 \\
+
\end{array}</cmath>
−
\end{array}
 
−
\end{center}</cmath>
 
  
 
It is clear that the solutions form arithmetic sequences with a difference of 2, and the negative solutions cancel out all but <math>n</math> of the <math>1+\frac{n(n-1)}{2}</math> solutions. Thus, the sum of the solutions is <math>n \cdot [1+\frac{n(n-1)}{2}]</math>, which is a cubic function.
 
It is clear that the solutions form arithmetic sequences with a difference of 2, and the negative solutions cancel out all but <math>n</math> of the <math>1+\frac{n(n-1)}{2}</math> solutions. Thus, the sum of the solutions is <math>n \cdot [1+\frac{n(n-1)}{2}]</math>, which is a cubic function.

Revision as of 20:48, 6 May 2023

Problem

Define a sequence recursively by $f_1(x)=|x-1|$ and $f_n(x)=f_{n-1}(|x-n|)$ for integers $n>1$. Find the least value of $n$ such that the sum of the zeros of $f_n$ exceeds $500,000$.

Solution 1 (Official MAA)

First it will be shown by induction that the zeros of $f_n$ are the integers $a, {a+2,} {a+4,} \dots, {a + n(n-1)}$, where $a = n - \frac{n(n-1)}2.$

This is certainly true for $n=1$. Suppose that it is true for $n = m-1 \ge 1$, and note that the zeros of $f_m$ are the solutions of $|x - m| = k$, where $k$ is a nonnegative zero of $f_{m-1}$. Because the zeros of $f_{m-1}$ form an arithmetic sequence with common difference $2,$ so do the zeros of $f_m$. The greatest zero of $f_{m-1}$ is\[m-1+\frac{(m-1)(m-2)}2 =\frac{m(m-1)}2,\]so the greatest zero of $f_m$ is $m+\frac{m(m-1)}2$ and the least is $m-\frac{m(m-1)}2$.

It follows that the number of zeros of $f_n$ is $\frac{n(n-1)}2+1=\frac{n^2-n+2}2$, and their average value is $n$. The sum of the zeros of $f_n$ is\[\frac{n^3-n^2+2n}2.\]Let $S(n)=n^3-n^2+2n$, so the sum of the zeros exceeds $500{,}000$ if and only if $S(n) > 1{,}000{,}000 = 100^3\!.$ Because $S(n)$ is increasing for $n > 2$, the values $S(100) = 1{,}000{,}000 - 10{,}000 + 200 = 990{,}200$ and $S(101)=1{,}030{,}301 - 10{,}201 + 202 = 1{,}020{,}302$ show that the requested value of $n$ is $\boxed{101}$.

Solution 2 (Same idea, easier to see)

Starting from $f_1(x)=|x-1|$, we can track the solutions, the number of solutions, and their sum.

\[\begin{array}{c|c|c|c} x&Solutions&number&sum\\ 1&1&1&1\\ 2&1,3&2&4\\ 3&0,2,4,6&4&12\\ 4&-2,0,2...10&7&28\\ 5&-9,-7,-5...21&11&55\\ \end{array}\]

It is clear that the solutions form arithmetic sequences with a difference of 2, and the negative solutions cancel out all but $n$ of the $1+\frac{n(n-1)}{2}$ solutions. Thus, the sum of the solutions is $n \cdot [1+\frac{n(n-1)}{2}]$, which is a cubic function.

$n \cdot [1+\frac{n(n-1)}{2}]>500,000$

Multiplying both sides by $2$,

$n \cdot [2+n(n-1)]>1,000,000$

1 million is $10^6=100^3$, so the solution should be close to $100$.

100 is slightly too small, so $\boxed{101}$ works.

~ dragnin

Video Solution

https://youtu.be/EwGydCuoHNM

~MathProblemSolvingSkills.com

Video Solution

https://youtu.be/g13o0wgj4p0

~IceMatrix

See Also

2020 AIME II (Problems • Answer Key • Resources)
Preceded by
Problem 7 		Followed by
Problem 9
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15
All AIME Problems and Solutions

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