Kaleidoscope Octagon

Geometry Level 5

A regular octagon A B C D E F G H ABCDEFGH , which shall be named O 1 O_1 , has squares A C E G ACEG and B D F H BDFH inscribed in it. These squares form a smaller octagon, O 2 O_2 .

O 2 O_2 has squares inscribed in it in the same manner, and the resulting hexagon formed is named O 3 O_3 . This pattern is repeated until infinity.

Let the area of octagon O i O_i be denoted by A i A_i . Given that A 1 = 1 A_1=1 , for some integers a a and b b , where a a is square-free, i = 1 A i = a + b \large \sum _{ i=1 }^{ \infty }{ { A }_{ i } } =\sqrt { a } +b Find a + b a+b .


Hint: 200 follower special (Part 1) .


The answer is 3.

This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try refreshing the page, (b) enabling javascript if it is disabled on your browser and, finally, (c) loading the non-javascript version of this page . We're sorry about the hassle.

1 solution

Agil Saelan
Feb 19, 2016

Using the hint, we know that the geometric progression A 1 , A 2 , A 3 , . . . A_1,A_2,A_3,... has the ratio r = 2 2 r= 2-\sqrt2 Therefore, the sum S = 1 1 ( 2 2 ) = 2 + 1 S=\frac{1}{1-(2-\sqrt2)} =\sqrt2+1

Hence a = 2 , b = 1 a=2 , b=1 and a + b = 3 a+b=3

The hint was almost an answer giveaway !

Venkata Karthik Bandaru - 5 years, 3 months ago

Log in to reply

True. I've never posted a solution before, and since this looks quite simple to write, I decided to give it a try

Agil Saelan - 5 years, 3 months ago

Log in to reply

Yeah, I too am attracted to posting short and simple solutions. I usually skip the tough ones ( Not very patient to type it all down :P ).

Venkata Karthik Bandaru - 5 years, 3 months ago

Where does the hint come from? God or Lenin?

Andreas Wendler - 5 years, 3 months ago

Log in to reply

There's a link below the problem

Agil Saelan - 5 years, 3 months ago

Same way. I too had solved the problem in hint.

Niranjan Khanderia - 5 years, 3 months ago

0 pending reports

×

Problem Loading...

Note Loading...

Set Loading...