Competition problem — Geometry — Maths Olympiad Prep

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Geometry Difficulty 6.5 National olympiad Prove it Greece

Let $AB\Gamma$ an equilateral triangle of side $\alpha$ . Let $\Delta$ , $E$ and $Z$ the midpoints of the sides $AB$ , $B\Gamma$ and $\Gamma A$ , respectively. Let $H$ the symmetric point of $\Delta$ with respect to the line $B\Gamma$ . We color the points $A$ , $B$ , $\Gamma$ , $\Delta$ , $E$ , $Z$ , $H$ using one of the two colors $\kappa$ = red and $\mu$ = blue.

a. Find how many equilateral triangles are defined with vertices from the seven points $A$ , $B$ , $\Gamma$ , $\Delta$ , $E$ , $Z$ , $H$ .

b. Prove that, if the points $B$ and $E$ will be colored with the same color, then for every coloring of the remaining points there exists an equilateral triangle with vertices from the points $A$ , $B$ , $\Gamma$ , $\Delta$ , $E$ , $Z$ , $H$ whose all vertices have the same color.

c. Can we have the same conclusion, if the points $B$ and $E$ will be colored by different colors?

Solution

a.
There are defined totally seven equilateral triangles from the given points. Since $\Delta E = EZ = Z\Delta = \frac{\alpha}{2}$ the seven equilateral triangles are

Figure 7

Figure 8
$AB\Gamma$ , $A\Delta Z$ , $B\Delta E$ , $E\Gamma Z$ , $\Delta EZ$ , $BHE$ (symmetric of $\Delta BE$ with respect to the line $B\Gamma$ ), $\Delta H\Gamma$ (it has $\Gamma H = \Gamma \Delta = \frac{\alpha\sqrt{3}}{2}$ = altitude of equilateral triangle, $\Gamma \hat{\Delta} H = 60^\circ$ ).

b.
Let $B$ and $E$ are colored red. If the points $\Delta$ or $H$ are also red, then we have the wanted triangle. Suppose that $\Delta$ and $H$ are colored blue. If the point $\Gamma$ is blue then we are done. Let the point $\Gamma$ is colored red. If the point $Z$ is colored red, then the triangle $E\Gamma Z$ has its vertices red. Let point $Z$ is colored blue. In that case for any coloring of $A$ one of the triangles $AB\Gamma$ , $A\Delta Z$ will have its vertices of the same color.

Figure 9

c.
In that case we have not the same conclusion. In figure 9 we give a coloring with all equilateral triangles having their vertices with different colors. The points $B$ , $\Gamma$ , $Z$ have been colored red and the remaining points blue.

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