Droney the Drone

Level pending

Droney the Drone is a good robot and always follows his Master's commands. One day, his Master gave him the following command:

Move 1m forward (starting from point A). Then turn 12 0 0 120^{0} to the left

After 2014m (at point B) , Droney's battery stopped working. The distance from point A A to point B B can be expressed as x \sqrt{x} (in meters) ,where x is a natural number.What is the value of 2014 x − 8 \frac{2014}{x}-8 ?


The answer is 999.

Bogdan Simeonov by Bogdan Simeonov , 21, Bulgaria

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1 solution

Bogdan Simeonov
Jan 9, 2014

Since the sum of angles in a n-gon is ( n − 2 ) . 180 (n-2).180 degrees and Droney is moving left by 120 degrees, we get that he is revolving in a regular hexagon. Since 2014 is congruent to 4 modulo 6, we get that he has gotten 335 times to point A and moved another 4m to point B. In a regular hexagon, the radius of the circle it is circumscribed in is equal to the side of the hexagon, thus equal to 1m. Let M be a point 1m backwards from B.Thus, Δ A M B \Delta AMB is right angled, AM=MB=1. By the Pythagorean theorem, A B = 2 AB=\sqrt{2} . So x=2 and 2014 2 − 8 = 999 \frac{2014}{2}-8=\boxed{999}

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