Hey ant, I just want a word for you: the word is stop!

Geometry Level 4

In the figure shown above A B C ABC is a right triangle with ∠ B = 9 0 ∘ \angle B = 90^\circ , A B = 10 − π AB=10-\pi and B C = 6 BC=6 . There is a semi-cylinder with radius 1 1 and its axis is perpendicular to A B AB . That little ant in point A A must scape to point C C soon! What is the length of shortest way for him to going to his nest in point C C ?

7 + π 7+\pi 10 10 136 \sqrt{136} 136 − π \sqrt{136}-\pi 11 11

Kazem Sepehrinia by Kazem Sepehrinia , 31, Iran

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3 solutions

Horisadi Afyama
Apr 18, 2015

2 Helpful 0 Interesting 0 Brilliant 0 Confused
Kushal Patankar
Apr 12, 2015

Is answer ecactly equal to 10 ????

I got 10.051 so i saw 10 as closest.

1 Helpful 0 Interesting 0 Brilliant 0 Confused

I made a silly mistake while calculating lengths. The intended way to solve this problem is to flatten the cylindrical part. Imagine the terrain to be made of paper, now just hold the ends of the paper and pull them apart to get a flat terrain. Now the new distance between A A and B B will be 8 8 (I went wrong here,took it as 10 10 ). The distance B C BC doesn't change. Hence, now we have our familiar 6 , 8 6,8 right triangle whose hypotenuse is the straight line(shortest distance) joining A A and C C , which is 10 10 units long.

Terrain-ception!

Raghav Vaidyanathan - 6 years, 2 months ago

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That is how it should be done!

Kazem Sepehrinia - 6 years, 2 months ago

Yes Kushal, Exact answer is 10.

Kazem Sepehrinia - 6 years, 2 months ago

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May be, cuz I did all calculations in decimals I got that answer. Anyway, problem was really beautiful.

Kushal Patankar - 6 years, 2 months ago

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can you please tell how did you solve it??

Yash Sharma - 6 years, 2 months ago

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@Yash Sharma – Calculated AC

Subtracted the part ( you know what (

And then added the curved length.

Kushal Patankar - 6 years, 2 months ago
Arian Tashakkor
May 4, 2015

I don't quite remember the year but it was for sure one of the best 1st-level questions in the history of Iranian MO...

0 Helpful 0 Interesting 0 Brilliant 0 Confused

Yeah, It is from 1383/2005 first level of 23rd Iranian mathematical olympiads.

Kazem Sepehrinia - 6 years, 1 month ago

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