## Task $5 / 69$

Given is a parallelogram with the perimeter $u=30$ units, whose angles are all not less than $60^{\circ}$. In this parallelogram, an equilateral triangle is inscribed such that one side of the triangle coincides with the parallelogram side $A B=a$, and the opposite vertex of the triangle lies on the opposite side of the parallelogram.

How large are the parallelogram sides $a$ and $b$, if their measures are integers?
The height of the parallelogram is, according to the problem statement, equal to that of the equilateral triangle and thus \( h = \frac{a}{2} \sqrt{3} \).

The side \( b \) is longest when the parallelogram is a rhombus and shortest when it is a rectangle. In the first case, \( u = 4a \) and thus \( a = 7.5 \, \text{LE} \), in the second case, \( u = 2a + 2h = 2a + a\sqrt{3} \) and thus \( a = 8.038 \, \text{LE} \). From this, one can infer that \( 7.5 \, \text{LE} \leq a \leq 8.038 \, \text{LE} \).

Due to the required integer values for \( a \) and \( b \), only \( a = 8 \, \text{LE} \) can be inferred from this. Thus, \( b = \frac{u}{2} - a = 7 \, \text{LE} \).