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Omni-MATH / Larry and Rob are two robots travelling in one car from Argovia to Zillis. Both…

Problem

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problem

Larry and Rob are two robots travelling in one car from Argovia to Zillis. Both robots have control over the steering and steer according to the following algorithm: Larry makes a 90∘90^{\circ} left turn after every ℓ\ell kilometer driving from start; Rob makes a 90∘90^{\circ} right turn after every rr kilometer driving from start, where ℓ\ell and rr are relatively prime positive integers. In the event of both turns occurring simultaneously, the car will keep going without changing direction. Assume that the ground is flat and the car can move in any direction. Let the car start from Argovia facing towards Zillis. For which choices of the pair (ℓ,r)(\ell, r) is the car guaranteed to reach Zillis, regardless of how far it is from Argovia?
Plain-text mathematical notation (without MathML)
Larry and Rob are two robots travelling in one car from Argovia to Zillis. Both robots have control over the steering and steer according to the following algorithm: Larry makes a 90^(∘) left turn after every ℓ kilometer driving from start; Rob makes a 90^(∘) right turn after every r kilometer driving from start, where ℓ and r are relatively prime positive integers. In the event of both turns occurring simultaneously, the car will keep going without changing direction. Assume that the ground is flat and the car can move in any direction. Let the car start from Argovia facing towards Zillis. For which choices of the pair (ℓ,r) is the car guaranteed to reach Zillis, regardless of how far it is from Argovia?
Original LaTeX notation
Larry and Rob are two robots travelling in one car from Argovia to Zillis. Both robots have control over the steering and steer according to the following algorithm: Larry makes a $90^{\circ}$ left turn after every $\ell$ kilometer driving from start; Rob makes a $90^{\circ}$ right turn after every $r$ kilometer driving from start, where $\ell$ and $r$ are relatively prime positive integers. In the event of both turns occurring simultaneously, the car will keep going without changing direction. Assume that the ground is flat and the car can move in any direction. Let the car start from Argovia facing towards Zillis. For which choices of the pair $(\ell, r)$ is the car guaranteed to reach Zillis, regardless of how far it is from Argovia?

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Source and history

task_id
c1d14016-a320-5c64-9815-a056eff9eb9a
task_key
test--c1d14016-a320-5c64-9815-a056eff9eb9a
task_revision_id
1
config / split
/ test

Official source

initial import