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The smallest right-angled triangle where the lengths of all the sides are integers, has sides of length $3$, $4$ and $5$ units. This triangle has an area of $6$ square units and a perimeter of $12$ units:

  • How many right-angled triangles are there with sides that are all integers less than $100$ units?
  • In how many of these triangles is the area one half of the perimeter?