Difference between revisions of "Random Problem"
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== Medium Problem == | == Medium Problem == | ||
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>. | ||
| + | |||
| + | ==Solution== | ||
| + | ??? | ||
| + | |||
| + | == Hardish Problem == | ||
| + | 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 is <math>\frac{a}{b}\pi</math>? | ||
| + | |||
| + | ==Solution== | ||
| + | ??? | ||
| + | |||
| + | == Hard Problem == | ||
| + | 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>? | ||
==Solution== | ==Solution== | ||
??? | ??? | ||
Revision as of 11:09, 30 January 2023
Contents
Easy Problem
The sum![\[\frac{1}{2!}+\frac{2}{3!}+\frac{3}{4!}+\cdots+\frac{2021}{2022!}\]](http://latex.artofproblemsolving.com/7/2/6/726b88e230d5c0a0303ef6ca7177fa73b7eec900.png)
can be expressed as 
, where 
and 
are positive integers. What is 
?
Solution
???
Medium Problem
Show that there exist no finite decimals 
such that when its digits are rearranged to a different decimal 
, 
.
Solution
???
Hardish Problem
A cylinder is inscribed in a circular cone with base radius of 
and height of 
. What is the maximum possible volume of this cylinder is 
?
Solution
???
Hard Problem
A regular 
-gon is inscribed in a circle with radius 
. Let 
be the set of distances (not necessarily distinct) from the center of the circle to each side of the -gon, and 
be the set of distances (not necessarily distinct) from the center of the circle to each diagonal of the -gon. Let 
be the union of and . What is the sum of the squares of all of the elements in ?
Solution
???
