RMO 2015

Geometry Level 5

There are two circles with centres $X$ and $Y$ .The circle with centre $X$ passes through $Y$ . $A$ is any point on the circle with centre $Y$ and $M$ is the midpoint of $YA$ . $B$ is another point on the circle with centre $Y$ , such that $AB$ is parallel to $XM$ . AB intersects the circle with centre $X$ at $L$ . If $AL=\sqrt3 \text{ cm}$ , then find the value of $AB$ .

Give your answer to 2 decimal places.

The answer is 3.46.

1 solution

Alan Guo
Dec 21, 2015

Let the image of a dilation of factor 2 of $\Delta YMX$ from $Y$ be $\Delta YAK$ . Then $YK$ is a diameter of the $X$ -circle. As $MX\|AK$ , it follows that $A,L,B,K$ are collinear.

Therefore, $\angle KLY = 90^{\circ}$ , meaning $LY$ is the perpendicular bisector of $AB$ , and so $AB=2\sqrt{3}$ .

RMO 2015

The answer is 3.46.

1 solution

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