Let $ABCD$ be a unit square. Draw a quadrant of the a circle with $A$ as centre and $B,D$ as end points of the arc. Similarly, draw a quadrant of a circle with $B$ as centre and $A,C$ as end points of the arc. Inscribe a circle $ \Gamma$ touching arcs $AC$ and $BD$ both externally and also touching the side $CD$. Find the radius of $ \Gamma$.
Problem
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Tags: geometry
subham1729
02.12.2012 13:37
Letting radius as $r$ we get $4(1-r)^2+1=4(1+r)^2\implies r=\frac {1}{16}$
cmtappu96
03.12.2012 06:30
The level of RMO has really dipped down. This is an IIT JEE type 3-minute problem, to be solved in 30 minutes
Pascal96
04.12.2012 16:17
Agree with tappu. Especially the geometry problems don't have that Olympiad feel. From the papers I have seen all the geometry problems are either trivial or can be bashed in 5 minutes with coordinates. Someone who has never done any "pure" geometry beyond tenth grade can still solve these problems. The geometry problems in 2009 and 2008 were much better.
Raj2000
07.10.2017 11:04
Isnt the answer 3/8
math_and_me
20.09.2018 15:34
Yeah , the answer is $\frac{3}{8}$
Vasu090
05.09.2019 16:16
What is the correct answer? Anybody with sure correct solution? Sorry for the long bump.
jayme
13.02.2020 13:38
Dear Mathlinkers, http://jl.ayme.pagesperso-orange.fr/Docs/Miniatures%20Geometriques%20addendum%20VIII.pdf p. 60... Sincerely Jean-Louis
Jishnu4414l
16.09.2024 16:53
math_and_me wrote: Yeah , the answer is $\frac{3}{8}$ I'm $6$ years late, but that's a another similar problem. Let $r$ be the radius of $\Gamma$. Pythagoras gives $(r-1)^2+\frac{1}{4}=(r+1)^2\implies r=\frac{1}{16}$
Agrawal001
21.09.2024 10:45
I got r=1/16