In a potato race, a bucket is placed at the starting point, which is $5$m from the first potato, and the other potatoes are placed $3$m apart in a straight line. There are $20$ potatoes in the line. Each competitor starts from the bucket, picks up the nearest potato, runs back with it, drops it in the bucket, runs back to pick up the next potato, runs to the bucket to drop it, and continues in the same way until all the potatoes are in the bucket. What is the total distance the competitor has to run?

The bucket is at the starting point, the first potato is $5$m away, and the remaining $19$ potatoes are placed $3$m apart in a straight line.

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The distance from the bucket to each potato:

Distance to Potato $1 = 5$m

Distance to Potato $2 = 5 + 3 = 8$m

Distance to Potato $3 = 5 + 3 + 3 = 11$m

Distance to Potato $n = 5 + 3(n-1) = 2 + 3n$m

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For each potato, the competitor must run to it and back to the bucket. The distance for each trip is twice the distance to that potato.

The total distance is:

$\text{Total} = 2 \times \sum_{n=1}^{20} (2 + 3n)$

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Expanding the sum:

$\sum_{n=1}^{20} (2 + 3n) = \sum_{n=1}^{20} 2 + \sum_{n=1}^{20} 3n$

$= 2 \times 20 + 3 \sum_{n=1}^{20} n$

Using the formula $\sum_{n=1}^{20} n = \dfrac{20 \times 21}{2} = 210$:

$= 40 + 3 \times 210$

$= 40 + 630$

$= 670$

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The total distance:

$\text{Total distance} = 2 \times 670 = 1340$ meters