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Geometry Difficulty 5.0 AIME Prove it Ukraine

Is it possible to draw a square with area of 5050 units on a graphing paper using a ruler that can connect two edges with a line?

Solution

Consider a 10×1010 \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 12⋅10⋅10=50\frac{1}{2} \cdot 10 \cdot 10 = 50.

Figure 1

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