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Ukrainian National Mathematical Olympiad

Geometry Difficulty 5.0 AIME Prove it Ukraine

Is it possible to draw a square with area of $50$ units on a graphing paper using a ruler that can connect two edges with a line?

Solution

Consider a $10 \times 10$ square and connect the midpoints of the sides as shown in Fig. 17. The square obtained clearly has area that is a half of the original square's area, that is $\frac{1}{2} \cdot 10 \cdot 10 = 50$ .

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