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Difference between revisions of "Random Problem"

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<math>\textbf{(A)}\ 2020 \qquad\textbf{(B)}\ 2021 \qquad\textbf{(C)}\ 2022 \qquad\textbf{(D)}\ 2023 \qquad\textbf{(E)}\ 2024</math>
 
<math>\textbf{(A)}\ 2020 \qquad\textbf{(B)}\ 2021 \qquad\textbf{(C)}\ 2022 \qquad\textbf{(D)}\ 2023 \qquad\textbf{(E)}\ 2024</math>
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==Solution==
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===Solution===
 
'''Submitted by BinouTheGuineaPig''' | ''A step-by-step solution''
 
'''Submitted by BinouTheGuineaPig''' | ''A step-by-step solution''
  
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Show that there exist no finite decimals <math>a = 0.\overline{a_1a_2a_3\ldots a_n}</math> such that when its digits are rearranged to a different decimal <math>b = 0.\overline{a_{b_1}a_{b_2}a_{b_3}\ldots a_{b_n}}</math>, <math>a + b = 1</math>.
 
Show that there exist no finite decimals <math>a = 0.\overline{a_1a_2a_3\ldots a_n}</math> such that when its digits are rearranged to a different decimal <math>b = 0.\overline{a_{b_1}a_{b_2}a_{b_3}\ldots a_{b_n}}</math>, <math>a + b = 1</math>.
  
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==Solution==
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===Solution===
 
???
 
???
  
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A cylinder is inscribed in a circular cone with base radius of <math>7</math> and height of <math>14</math>. What is the maximum possible volume of this cylinder in the form of <math>\frac{a}{b}\pi</math>?
 
A cylinder is inscribed in a circular cone with base radius of <math>7</math> and height of <math>14</math>. What is the maximum possible volume of this cylinder in the form of <math>\frac{a}{b}\pi</math>?
  
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==Solution==
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===Solution===
 
'''Submitted by BinouTheGuineaPig''' | ''A step-by-step solution''
 
'''Submitted by BinouTheGuineaPig''' | ''A step-by-step solution''
  
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A regular <math>48</math>-gon is inscribed in a circle with radius <math>1</math>. Let <math>X</math> be the set of distances (not necessarily distinct) from the center of the circle to each side of the <math>48</math>-gon, and <math>Y</math> be the set of distances (not necessarily distinct) from the center of the circle to each diagonal of the <math>48</math>-gon. Let <math>S</math> be the union of <math>X</math> and <math>Y</math>. What is the sum of the squares of all of the elements in <math>S</math>?
 
A regular <math>48</math>-gon is inscribed in a circle with radius <math>1</math>. Let <math>X</math> be the set of distances (not necessarily distinct) from the center of the circle to each side of the <math>48</math>-gon, and <math>Y</math> be the set of distances (not necessarily distinct) from the center of the circle to each diagonal of the <math>48</math>-gon. Let <math>S</math> be the union of <math>X</math> and <math>Y</math>. What is the sum of the squares of all of the elements in <math>S</math>?
  
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==Solution==
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===Solution===
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??[[Random_Problem/Solutions |?]]
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???

Revision as of 16:40, 24 March 2023

Easy Problem

The sum\[\frac{1}{2!}+\frac{2}{3!}+\frac{3}{4!}+\cdots+\frac{2021}{2022!}\]can be expressed as $a-\frac{1}{b!}$, where $a$ and $b$ are positive integers. What is $a+b$?

$\textbf{(A)}\ 2020 \qquad\textbf{(B)}\ 2021 \qquad\textbf{(C)}\ 2022 \qquad\textbf{(D)}\ 2023 \qquad\textbf{(E)}\ 2024$

Solution

Submitted by BinouTheGuineaPig | A step-by-step solution

We see that the general form for each term can be expressed in terms of $n$ as follows.

$\frac{n}{(n+1)!} =\frac{(n+1)-1}{(n+1)!}$

$\qquad\qquad = \frac{n+1}{(n+1)(n!)} - \frac{1}{(n+1)!}$

$\qquad\qquad = \frac{1}{n!} - \frac{1}{(n+1)!}$

Now, to find the entire sum, it can be expressed as follows.

$\sum_{n=1}^{2022} \frac{n}{(n+1)!}$

$=\sum_{n=1}^{2022} \frac{1}{n!} - \frac{1}{(n+1)!}$

$=(\frac{1}{1!}+\cancel{\frac{1}{2!}+\frac{1}{3!}+\frac{1}{4!}+\cdots+\frac{1}{2022!}}) - (\cancel{\frac{1}{2!}+\frac{1}{3!}+\frac{1}{4!}+\cdots+\frac{1}{2022!}}+\frac{1}{2023!})$

Here, we see that a whole chunk of terms cancel each other out, leaving us with

$=1-\frac{1}{2023!} \equiv a-\frac{1}{b!}$

This means $a+b=1+2023=2024$.

Therefore, the answer is $\boxed{\textbf{(E)}}$.

Medium Problem

Show that there exist no finite decimals $a = 0.\overline{a_1a_2a_3\ldots a_n}$ such that when its digits are rearranged to a different decimal $b = 0.\overline{a_{b_1}a_{b_2}a_{b_3}\ldots a_{b_n}}$, $a + b = 1$.

Solution

???

Hard-ish Problem

A cylinder is inscribed in a circular cone with base radius of $7$ and height of $14$. What is the maximum possible volume of this cylinder in the form of $\frac{a}{b}\pi$?

Solution

Submitted by BinouTheGuineaPig | A step-by-step solution

We create a cross-section of the cylinder in the cone, "slicing down" from the apex of the cone as follows.

https://imgur.com/a/Xag3fSE

Volume of the cylinder

$V=\pi{r^2}h$

$V=\pi{r^2}(14-2r)$

$V=14\pi{r^2}-2\pi{r^3}$

Now, we find all inflection points in the graph of this equation, by finding the values of $r$ where the gradient is $0$, in other words where $\frac{dV}{dr}=0$.

$\frac{dV}{dr}=28\pi{r}-6\pi{r^2}$

$28\pi{r}-6\pi{r^2}=0$

$3r^2-14r=0$

$r(3r-14)=0$

$r=0$ or $r=\frac{14}{3}$

We will obviously use the larger value of $r$ to find the maximum volume of the cylinder, as follows.

$V_{max}=14\pi(\frac{14}{3})^2-2\pi(\frac{14}{3})^3$

$V_{max}=\frac{2744}{27}\pi \equiv \frac{a}{b}\pi$

Therefore, the maximum cylinder volume can be expressed as $\boxed{\frac{2744}{27}\pi}$.

Hard Problem

A regular $48$-gon is inscribed in a circle with radius $1$. Let $X$ be the set of distances (not necessarily distinct) from the center of the circle to each side of the $48$-gon, and $Y$ be the set of distances (not necessarily distinct) from the center of the circle to each diagonal of the $48$-gon. Let $S$ be the union of $X$ and $Y$. What is the sum of the squares of all of the elements in $S$?

Solution

???

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