Problem:
A triangle with integral sides has perimeter . The area of the triangle is
Answer Choices:
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Solution:
Let be the lengths of the three sides of the triangle. Since the longest side must be an integer that is less than half the perimeter and at least one third of the perimeter, . Since and , the only triangle with integral sides and perimeter has sides of lengths and . The altitude to the base of length of this isosceles triangle is , so its area is .
Query. If any integral perimeter larger than were used, the area would not be unique. Can you prove it?