Strange Algebra

Algebra Level 4

Let $\color{#D61F06}{'S'}$ be the ordered pairs of $\color{#3D99F6}{(x,y)}$ such that $\color{#20A900}{0<x\le 1}$ and $\color{#69047E}{0<y\le 1}$ and $\color{#624F41}{ [\ log _3(\frac{1}{x})\ ]}$ and $\color{darkred}{[ \ log _5(\frac{1}{y}) \ ]}$ are both even.

While plotting a graph between the values of $\color{darkred}{x,y}$ that satisfy the condition the area of the graph comes out to be of the form $\color{maroon}{\frac{m}{n}}$ where $\color{magenta}{m}$ and $\color{magenta}{n}$ are co-prime integers.

Then what is the value of $\color{#20A900}{m+n ?}$ $\color{#D61F06}{NOTE}$

$\color{magenta}{[ .]}$ represents $\color{magenta}{GIF}$

For eg. - [1.2]=1 , [2.3]=2

The answer is 13.

2 solutions

Shubhendra Singh
Sep 13, 2014

On working a bit we can find that the value of $\color{magenta}{x}$ that satisfy the condition lies between $\color{magenta}{1 and 1/3 , 1/9and1/27 and \ so\ on........}$ similarly the values of $\color{magenta}{y}$ that will satisfy the condition will lie between $\color{magenta}{1 and 1/5 , 1/25 and 1/125 and \ so \ on ..........}$

So the intersection of the values of values of $\color{magenta}{x}$ and $\color{magenta}{y }$ will form rectangles . The area of the rectangles will come out to be

(1 -1/3 + 1/9 -1/27....................).(1-1/5+1/25-1/125...........) {It will form a GP)

By applyin the formula of infinite GP

= (3/4).(5/6)= 5/8

So the ans would be 5+8=13

Aditya Raj
Feb 4, 2015

On working a bit we can find that the value of x that satisfy the condition lies between 1 & 1/3,1/9,1/27.... similarly the values of that will satisfy the condition will lie between 1&1/5,1/125.... So the intersection of the values of values of and will form rectangles . The area of the rectangles will come out to be (1 -1/3 + 1/9 -1/27....................).(1-1/5+1/25-1/125...........) {It will form a GP) By applying the formula of infinite GP = (3/4).(5/6)= 5/8 So the ans would be 5+8=13

Strange Algebra

The answer is 13.

2 solutions

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