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Difficulty: 1/102024 NMTC 2024 (QII-27)

All the ________ in plane are similar

Guide / Hint

Hint 1: Recall that similarity means having the same shape, regardless of scale.

Hint 2: Which geometric shape has only one degree of freedom (its scale/radius) and always retains the identical shape?

Hint 3: Conclude that all circles (or squares, or equilateral triangles) are similar.

Solution

Step 1 (Definition of Similarity): Two geometric figures are similar if they have the same shape, even if they have different sizes. This means one can be obtained from the other by scaling (and possibly rotation or translation).

Step 2 (Circles in a Plane): Every circle is completely determined by a single parameter: its radius. Changing the radius scales the circle but preserves its perfect circular shape. Therefore, any two circles in a plane can be scaled to match each other perfectly, meaning all circles are similar.

Step 3 (Conclusion): The blank is filled by circles (or squares). The most standard geometric term is circles.

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    2024 NMTC 2024 QII-27 - Olympiad Math Olympiad Question | Leminno