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Difference between revisions of "2024 AIME II Problems/Problem 6"

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==Solution 1==
 
==Solution 1==
 
Let <math>k</math> be one of the elements in Alices set <math>A</math> of positive integers. The number of sets that Bob lists with the property that their maximum element is k is <math>2^{k-1}</math>, since every positive integer less than k can be in the set or out. Thus, fore the number of sets bob have listed to be 2024, we want to find a sum of unique powers of two that can achieve this. 2024 is equal to <math>2^{10}+2^9+2^8+2^7+2^6+2^5+2^3</math>. We must increase each power by 1 to find the elements in set <math>A</math>, which are <math>(11,10,9,8,7,6,4)</math>. Add these up to get <math>\boxed{055}</math>. -westwoodmonster
 
Let <math>k</math> be one of the elements in Alices set <math>A</math> of positive integers. The number of sets that Bob lists with the property that their maximum element is k is <math>2^{k-1}</math>, since every positive integer less than k can be in the set or out. Thus, fore the number of sets bob have listed to be 2024, we want to find a sum of unique powers of two that can achieve this. 2024 is equal to <math>2^{10}+2^9+2^8+2^7+2^6+2^5+2^3</math>. We must increase each power by 1 to find the elements in set <math>A</math>, which are <math>(11,10,9,8,7,6,4)</math>. Add these up to get <math>\boxed{055}</math>. -westwoodmonster
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==Solution 2==
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Let <math>A = \left\{ a_1, a_2, \cdots, a_n \right\}</math> with <math>a_1 < a_2 < \cdots < a_n</math>.
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If the maximum element of <math>B</math> is <math>a_i</math> for some <math>i \in \left\{ 1, 2, \cdots , n \right\}</math>, then each element in <math>\left\{ 1, 2, \cdots, a_i- 1 \right\}</math> can be either in <math>B</math> or not in <math>B</math>.
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Therefore, the number of such sets <math>B</math> is <math>2^{a_i - 1}</math>.
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Therefore, the total number of sets <math>B</math> is
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\begin{align*}
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\sum_{i=1}^n 2^{a_i - 1} & = 2024 .
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\end{align*}
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Thus
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\begin{align*}
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\sum_{i=1}^n 2^{a_i} & = 4048 .
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\end{align*}
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Now, the problem becomes writing 4048 in base 2, say, <math>4048 = \left( \cdots b_2b_1b_0 \right)_2</math>.
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We have <math>A = \left\{ j \geq 1: b_j = 1 \right\}</math>.
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We have <math>4048 = \left( 111,111,010,000 \right)_2</math>.
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Therefore, <math>A = \left\{ 4, 6, 7, 8, 9, 10, 11 \right\}</math>.
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Therefore, the sum of all elements in <math>A</math> is <math>\boxed{\textbf{(55) }}</math>.
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~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)
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==Video Solution==
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https://youtu.be/vdj1kCgjHXk
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 +
~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)
  
 
==See also==
 
==See also==

Revision as of 15:08, 9 February 2024

Alice chooses a set $A$ of positive integers. Then Bob lists all finite nonempty sets $B$ of positive integers with the property that the maximum element of $B$ belongs to $A$. Bob's list has 2024 sets. Find the sum of the elements of A.

Solution 1

Let $k$ be one of the elements in Alices set $A$ of positive integers. The number of sets that Bob lists with the property that their maximum element is k is $2^{k-1}$, since every positive integer less than k can be in the set or out. Thus, fore the number of sets bob have listed to be 2024, we want to find a sum of unique powers of two that can achieve this. 2024 is equal to $2^{10}+2^9+2^8+2^7+2^6+2^5+2^3$. We must increase each power by 1 to find the elements in set $A$, which are $(11,10,9,8,7,6,4)$. Add these up to get $\boxed{055}$. -westwoodmonster

Solution 2

Let $A = \left\{ a_1, a_2, \cdots, a_n \right\}$ with $a_1 < a_2 < \cdots < a_n$.

If the maximum element of $B$ is $a_i$ for some $i \in \left\{ 1, 2, \cdots , n \right\}$, then each element in $\left\{ 1, 2, \cdots, a_i- 1 \right\}$ can be either in $B$ or not in $B$. Therefore, the number of such sets $B$ is $2^{a_i - 1}$.

Therefore, the total number of sets $B$ is \begin{align*} \sum_{i=1}^n 2^{a_i - 1} & = 2024 . \end{align*}

Thus \begin{align*} \sum_{i=1}^n 2^{a_i} & = 4048 . \end{align*}

Now, the problem becomes writing 4048 in base 2, say, $4048 = \left( \cdots b_2b_1b_0 \right)_2$. We have $A = \left\{ j \geq 1: b_j = 1 \right\}$.

We have $4048 = \left( 111,111,010,000 \right)_2$. Therefore, $A = \left\{ 4, 6, 7, 8, 9, 10, 11 \right\}$. Therefore, the sum of all elements in $A$ is $\boxed{\textbf{(55) }}$.

~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)

Video Solution

https://youtu.be/vdj1kCgjHXk

~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)

See also

2024 AIME II (Problems • Answer Key • Resources)
Preceded by
Problem 5 		Followed by
Problem 7
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15
All AIME Problems and Solutions

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