Difference between revisions of "2021 AMC 12A Problems/Problem 16"

(Solution 1)
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Note that <math>\sqrt{20100} \approx 142</math>. Plugging this value in as <math>k</math> gives
 
Note that <math>\sqrt{20100} \approx 142</math>. Plugging this value in as <math>k</math> gives
 
<cmath>\frac{1}{2}(142)(143)=10153.</cmath>
 
<cmath>\frac{1}{2}(142)(143)=10153.</cmath>
<math>10153-142<10050</math>, so <math>142</math> is the <math>152</math>nd and <math>153</math>rd numbers, and hence, our desired answer. <math>\fbox{(C) 142}.</math>.
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<math>10153-142<10050</math>, so <math>142</math> is the <math>152</math>nd and <math>153</math>rd numbers, and hence, our desired answer. <math>\fbox{(C) 142}</math>.
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 +
Note that we can derive <math>\sqrt{20100} \approx 142</math> through the formula <cmath>\sqrt{n} = \sqrt{a+b} \approx \sqrt{a} + \frac{b}{2\sqrt{a} + 1},</cmath>
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where <math>a</math> is a perfect square less than or equal to <math>n</math>. We set <math>a</math> to <math>19600</math>, so <math>\sqrt{a} = 140</math>, and <math>b = 500</math>. We then have <math>n \approx 140 + \frac{500}{2(140)+1} \approx 142</math>. ~approximation by ciceronii
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==Solution 2==
 
==Solution 2==
 
The <math>x</math>th number of this sequence is <math>\left\lceil\frac{-1\pm\sqrt{1+8x}}{2}\right\rceil</math> via the quadratic formula. We can see that if we halve <math>x</math> we end up getting <math>\left\lceil\frac{-1\pm\sqrt{1+4x}}{2}\right\rceil</math>. This is approximately the number divided by <math>\sqrt{2}</math>. <math>\frac{200}{\sqrt{2}} = 141.4</math> and since <math>142</math> looks like the only number close to it, it is answer <math>\boxed{(C) 142}</math> ~Lopkiloinm
 
The <math>x</math>th number of this sequence is <math>\left\lceil\frac{-1\pm\sqrt{1+8x}}{2}\right\rceil</math> via the quadratic formula. We can see that if we halve <math>x</math> we end up getting <math>\left\lceil\frac{-1\pm\sqrt{1+4x}}{2}\right\rceil</math>. This is approximately the number divided by <math>\sqrt{2}</math>. <math>\frac{200}{\sqrt{2}} = 141.4</math> and since <math>142</math> looks like the only number close to it, it is answer <math>\boxed{(C) 142}</math> ~Lopkiloinm

Revision as of 15:07, 12 February 2021

Problem

In the following list of numbers, the integer $n$ appears $n$ times in the list for $1 \leq n \leq 200$.\[1, 2, 2, 3, 3, 3, 4, 4, 4, 4, \ldots, 200, 200, \ldots , 200\]What is the median of the numbers in this list?

Solution 1

There are $1+2+..+199+200=\frac{(200)(201)}{2}=20100$ numbers in total. Let the median be $k$. We want to find the median $k$ such that \[\frac{k(k+1)}{2}=20100/2,\] or \[k(k+1)=20100.\] Note that $\sqrt{20100} \approx 142$. Plugging this value in as $k$ gives \[\frac{1}{2}(142)(143)=10153.\] $10153-142<10050$, so $142$ is the $152$nd and $153$rd numbers, and hence, our desired answer. $\fbox{(C) 142}$.

Note that we can derive $\sqrt{20100} \approx 142$ through the formula \[\sqrt{n} = \sqrt{a+b} \approx \sqrt{a} + \frac{b}{2\sqrt{a} + 1},\] where $a$ is a perfect square less than or equal to $n$. We set $a$ to $19600$, so $\sqrt{a} = 140$, and $b = 500$. We then have $n \approx 140 + \frac{500}{2(140)+1} \approx 142$. ~approximation by ciceronii

Solution 2

The $x$th number of this sequence is $\left\lceil\frac{-1\pm\sqrt{1+8x}}{2}\right\rceil$ via the quadratic formula. We can see that if we halve $x$ we end up getting $\left\lceil\frac{-1\pm\sqrt{1+4x}}{2}\right\rceil$. This is approximately the number divided by $\sqrt{2}$. $\frac{200}{\sqrt{2}} = 141.4$ and since $142$ looks like the only number close to it, it is answer $\boxed{(C) 142}$ ~Lopkiloinm

Video Solution by Hawk Math

https://www.youtube.com/watch?v=AjQARBvdZ20

Video Solution by OmegaLearn (Using Algebra)

https://youtu.be/HkwgH9Lc1hE

See also

2021 AMC 10A (ProblemsAnswer KeyResources)
Preceded by
Problem 15
Followed by
Problem 17
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
All AMC 10 Problems and Solutions
2021 AMC 12A (ProblemsAnswer KeyResources)
Preceded by
Problem 15
Followed by
Problem 17
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25
All AMC 12 Problems and Solutions

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