Competition problem — Geometry — Maths Olympiad Prep

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Geometry Difficulty 5.4 AIME, harder Prove it Italy

Problem:

Let $ABC$ be an equilateral triangle, and denote by $D, E, F$ the midpoints of the sides. How many non-degenerate and pairwise non-congruent triangles can be obtained by choosing 3 of the points $A, B, C, D, E, F$ ?

How many ordered pairs $(x, y)$ of integers satisfy the equation $y^{4}-8 y^{2}+7=8 x^{2}-2 x^{2} y^{2}-x^{4}$ ?

Solution

Solution:

The answer is 4. Up to isometries it is possible to determine an obtuse triangle, a right triangle, and two different equilateral triangles.

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Source: MathNet, licensed CC-BY-4.0. Statement translated into English from it; metadata (topic, difficulty) added by this project.