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Difference between revisions of "2014 AIME II Problems/Problem 15"

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Multiplication for the sequence <math>x_i</math> will translate to addition for the sequence <math>y_i</math>. Thus, we see that <math>x_{n+1}X(x_n) = x_np(x_n)</math> translates into <math>y_{n+1} = y_n+1</math>. Since <math>x_0=1</math>, and <math>y_0=0</math>, <math>x_i</math> corresponds to <math>y_i</math>, which is <math>i</math> in binary. Since <math>x_{10010101_2} = 2 \cdot 5 \cdot 11 \cdot 19 = 2090</math>, <math>t = 10010101_2</math> = <math>\boxed{149}</math>.
 
Multiplication for the sequence <math>x_i</math> will translate to addition for the sequence <math>y_i</math>. Thus, we see that <math>x_{n+1}X(x_n) = x_np(x_n)</math> translates into <math>y_{n+1} = y_n+1</math>. Since <math>x_0=1</math>, and <math>y_0=0</math>, <math>x_i</math> corresponds to <math>y_i</math>, which is <math>i</math> in binary. Since <math>x_{10010101_2} = 2 \cdot 5 \cdot 11 \cdot 19 = 2090</math>, <math>t = 10010101_2</math> = <math>\boxed{149}</math>.
  
 +
==Solution 2 (Painful Bash)==
 +
We go through the terms and look for a pattern. We find that
 +
 +
<math>x_0 = 1</math>  <math>x_8 = 7</math>
 +
 +
<math>x_1 = 2</math>  <math>x_9 = 14</math>
 +
 +
<math>x_2 = 3</math>  <math>x_{10} = 21</math>
 +
 +
<math>x_3 = 6</math>  <math>x_{11} = 42</math>
 +
 +
<math>x_4 = 5</math>  <math>x_{12} = 35</math>
 +
 +
<math>x_5 = 10</math>  <math>x_{13} = 70</math>
 +
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<math>x_6 = 15</math>  <math>x_{14} = 105</math>
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<math>x_7 = 30</math>  <math>x_{15} = 210</math>
 +
 +
Commit to the bash. Eventually, you will recieve that <math>x_{149} = 2090</math>, so <math>\boxed{149}</math> is the answer. Trust me, this is worth the 10 index points.
 +
 +
<math>\textbf{-RootThreeOverTwo}</math>
 
== See also ==
 
== See also ==
 
{{AIME box|year=2014|n=II|num-b=14|after=Last Problem}}
 
{{AIME box|year=2014|n=II|num-b=14|after=Last Problem}}
 
{{MAA Notice}}
 
{{MAA Notice}}

Revision as of 01:41, 26 January 2019

Problem

For any integer $k\geq 1$, let $p(k)$ be the smallest prime which does not divide $k.$ Define the integer function $X(k)$ to be the product of all primes less than $p(k)$ if $p(k)>2$, and $X(k)=1$ if $p(k)=2.$ Let $\{x_n\}$ be the sequence defined by $x_0=1$, and $x_{n+1}X(x_n)=x_np(x_n)$ for $n\geq 0.$ Find the smallest positive integer $t$ such that $x_t=2090.$

Solution

Note that for any $x_i$, for any prime $p$, $p^2 \nmid x_i$. This provides motivation to translate $x_i$ into a binary sequence $y_i$.

Let the prime factorization of $x_i$ be written as $p_{a_1} \cdot p_{a_2} \cdot p_{a_3} \cdots$, where $p_i$ is the $i$th prime number. Then, for every $p_{a_k}$ in the prime factorization of $x_i$, place a $1$ in the $a_k$th digit of $y_i$. This will result in the conversion $x_1 = 2, x_{2} = 3, x_{3} = 2 * 3 = 6, \cdots$.

Multiplication for the sequence $x_i$ will translate to addition for the sequence $y_i$. Thus, we see that $x_{n+1}X(x_n) = x_np(x_n)$ translates into $y_{n+1} = y_n+1$. Since $x_0=1$, and $y_0=0$, $x_i$ corresponds to $y_i$, which is $i$ in binary. Since $x_{10010101_2} = 2 \cdot 5 \cdot 11 \cdot 19 = 2090$, $t = 10010101_2$ = $\boxed{149}$.

Solution 2 (Painful Bash)

We go through the terms and look for a pattern. We find that

$x_0 = 1$ $x_8 = 7$

$x_1 = 2$ $x_9 = 14$

$x_2 = 3$ $x_{10} = 21$

$x_3 = 6$ $x_{11} = 42$

$x_4 = 5$ $x_{12} = 35$

$x_5 = 10$ $x_{13} = 70$

$x_6 = 15$ $x_{14} = 105$

$x_7 = 30$ $x_{15} = 210$

Commit to the bash. Eventually, you will recieve that $x_{149} = 2090$, so $\boxed{149}$ is the answer. Trust me, this is worth the 10 index points.

$\textbf{-RootThreeOverTwo}$

See also

2014 AIME II (Problems • Answer Key • Resources)
Preceded by
Problem 14 		Followed by
Last Problem
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15
All AIME Problems and Solutions

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