Difference between revisions of "1997 AHSME Problems/Problem 29"
Talkinaway (talk | contribs) (Created page with "== See also == {{AHSME box|year=1997|num-b=28|num-a=30}}") |
(→Solution) |
||
| (2 intermediate revisions by 2 users not shown) | |||
| Line 1: | Line 1: | ||
| + | ==Problem== | ||
| + | |||
| + | Call a positive real number special if it has a decimal representation that consists entirely of digits <math>0</math> and <math>7</math>. For example, <math> \frac{700}{99}= 7.\overline{07}= 7.070707\cdots </math> and <math> 77.007 </math> are special numbers. What is the smallest <math>n</math> such that <math>1</math> can be written as a sum of <math>n</math> special numbers? | ||
| + | |||
| + | <math> \textbf{(A)}\ 7\qquad\textbf{(B)}\ 8\qquad\textbf{(C)}\ 9\qquad\textbf{(D)}\ 10\qquad\\ \textbf{(E)}\ \text{The number 1 cannot be represented as a sum of finitely many special numbers.} </math> | ||
| + | |||
| + | ==Solution== | ||
| + | Define a super-special number to be a number whose decimal expansion only consists of <math>0</math>'s and <math>1</math>'s. The problem is equivalent to finding the number of super-special numbers necessary to add up to <math>\frac{1}{7}=0.142857142857\hdots</math>. This can be done in <math>8</math> numbers if we take | ||
| + | <cmath> | ||
| + | 0.111111\hdots, 0.011111\hdots, 0.010111\hdots, 0.010111\hdots, 0.000111\hdots, 0.000101\hdots, 0.000101\hdots, 0.000100\hdots | ||
| + | </cmath> | ||
| + | Now assume for sake of contradiction that we can do this with strictly less than <math>8</math> super-special numbers (in particular, less than <math>10</math>.) Then the result of the addition won't have any carry over, so each digit is simply the number of super-special numbers which had a <math>1</math> in that place. This means that in order to obtain the <math>8</math> in <math>0.1428\hdots</math>, there must be <math>8</math> super-special numbers, so the answer is <math>\boxed{\textbf{(B)}\ 8}</math>. | ||
| + | |||
== See also == | == See also == | ||
{{AHSME box|year=1997|num-b=28|num-a=30}} | {{AHSME box|year=1997|num-b=28|num-a=30}} | ||
| + | {{MAA Notice}} | ||
Latest revision as of 12:11, 9 August 2013
Problem
Call a positive real number special if it has a decimal representation that consists entirely of digits 
and 
. For example, 
and 
are special numbers. What is the smallest 
such that 
can be written as a sum of special numbers?
Solution
Define a super-special number to be a number whose decimal expansion only consists of 's and 's. The problem is equivalent to finding the number of super-special numbers necessary to add up to 
. This can be done in 
numbers if we take
![\[0.111111\hdots, 0.011111\hdots, 0.010111\hdots, 0.010111\hdots, 0.000111\hdots, 0.000101\hdots, 0.000101\hdots, 0.000100\hdots\]](http://latex.artofproblemsolving.com/a/0/3/a037c51326b46658ce3e3616873731a255efd051.png)
Now assume for sake of contradiction that we can do this with strictly less than super-special numbers (in particular, less than 
.) Then the result of the addition won't have any carry over, so each digit is simply the number of super-special numbers which had a in that place. This means that in order to obtain the in 
, there must be super-special numbers, so the answer is 
.
See also
| 1997 AHSME (Problems • Answer Key • Resources) | ||
| Preceded by Problem 28 |
Followed by Problem 30 | |
| 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 • 26 • 27 • 28 • 29 • 30 | ||
| All AHSME Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
