# Geometry

 1. (15 p.) Let $$ABC$$ be a rectangular triangle such that $$\angle C=90^o$$ and $$AC = 7$$, $$BC = 24$$. Let $$M$$ be the midpoint of $$AB$$ and $$D$$ point on the same side of $$AB$$ as $$C$$ such that $$DA = DB = 15$$. The area of the triangle $$CDM$$ can be expressed as $$\frac{p\sqrt q}r$$ for positive integers $$p$$, $$q$$, $$r$$ such that $$q$$ is not divisible by a perfect square and $$(p,r)=1$$. Find area $$p+q+r$$.

 2. (8 p.) Let $$X$$ be a square of side length 2. Denote by $$S$$ the set of all segments of length 2 with endpoints on adjacent sides of $$X$$. The midpoints of the segments in $$S$$ enclose a region with an area $$A$$. Find $$[100A]$$.

 3. (44 p.) Let $$\triangle ABC$$ have $$AB=6$$, $$BC=7$$, and $$CA=8$$, and denote by $$\omega$$ its circumcircle. Let $$N$$ be a point on $$\omega$$ such that $$AN$$ is a diameter of $$\omega$$. Furthermore, let the tangent to $$\omega$$ at $$A$$ intersect $$BC$$ at $$T$$, and let the second intersection point of $$NT$$ with $$\omega$$ be $$X$$. The length of $$\overline{AX}$$ can be written in the form $$\tfrac m{\sqrt n}$$ for positive integers $$m$$ and $$n$$, where $$n$$ is not divisible by the square of any prime. Find $$m+n$$.

 4. (8 p.) A triangle $$ABC$$ has sides 13, 14, 15. The triangle $$ABC$$ is rotated about its centroid for an angle of $$180^0$$ to form a triangle $$A^{\prime}B^{\prime}C^{\prime}$$. Find the area of the union of the two triangles.

 5. (21 p.) Let $$K$$ and $$L$$ be the points on the sides $$AB$$ and $$BC$$ of an equilateral triangle $$ABC$$ such that $$AK=5$$ and $$CL=2$$. If $$M$$ is the point on $$AC$$ such that $$\angle KML=60^o$$, and if the area of the triangle $$KML$$ is equal to $$14\sqrt3$$ then the side of the triangle $$ABC$$ can assume two values $$\frac{a\pm \sqrt b}c$$ for some natural numbers $$a$$, $$b$$, and $$c$$. If $$b$$ is not divisible by a perfect square other than 1, find the value of $$b$$.

2005-2021 IMOmath.com | imomath"at"gmail.com | Math rendered by MathJax