GEOMETRY PROBLEM OF OLYMPIAD LEVEL

Interesting problem. But did you read the posting guidelines?
 
Is there no more information?
I would have thought it is the 'range of possible values' for x, rather than 'the value' of x.
 
lex said:
Is there no more information?
I would have thought it is the 'range of possible values' for x, rather than 'the value' of x.
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No there is nothing much given..Were you able to find the range.if yes, then the can you provide the solution
 
It seems to me you can put a circle of 'any size' in there - tangential to the vertical line and the semi-circle (limited by the size of the semi-circle) - just attach the circle to the vertical line and slide it up until it is just tangential to the semi-circle. x will vary accordingly, from 0, to the height of the vertical line. The height of the vertical line can easily be calculated using Pythagoras' theorem - joining the centre of the semi-circle to the top of the vertical line (and you know the radius of the semi-circle).
It does seem suspiciously easy. That, and the language of the question ('value of x'), led me to question was there any more information given. Perhaps it is worth posting an image of the full question.
 
For some reason, I have taken x to be the height. Perhaps it is meant to be the first part of the blue diagonal line. In that case it would range from 2, to the distance to the top of the vertical line (easily calculated using Pythagoras’ thm).
 
lex said:
For some reason, I have taken x to be the height. Perhaps it is meant to be the first part of the blue diagonal line. In that case it would range from 2, to the distance to the top of the vertical line (easily calculated using Pythagoras’ thm).
Click to expand...
Based on the colors of other segments, I would say x is the length of the entire blue segment.
 
Yes, I think you're probably right. I thought that too as yet another possibility when I posted the last time. It's a pity to have to keep guessing!
 
lex said:
It seems to me you can put a circle of 'any size' in there - tangential to the vertical line and the semi-circle (limited by the size of the semi-circle) - just attach the circle to the vertical line and slide it up until it is just tangential to the semi-circle. x will vary accordingly, from 0, to the height of the vertical line. The height of the vertical line can easily be calculated using Pythagoras' theorem - joining the centre of the semi-circle to the top of the vertical line (and you know the radius of the semi-circle).
It does seem suspiciously easy. That, and the language of the question ('value of x'), led me to question was there any more information given. Perhaps it is worth posting an image of the full question.
Click to expand...
This is the full question.I know it's confusing, but I have posted the complete question which is Nothin but the diagram.
 
I calculated the length of the blue line for the smallest possible 'inner small circle' (0 radius!) and for the largest 'inner small circle' which just touches the diameter line, and got the same answer for both [MATH]3\sqrt{2}[/MATH]. Perhaps there is a single value after all?
 
AdkAdi said:
Thank you so much.I was stuck on this problem for more than three days. I am extremely grateful to you.
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Oh dear. That is perseverance! It was a difficult problem - good experience.
You're welcome.
 
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