# Geometry

 1. (14 p.) The angle $$\angle C$$ of the isosceles triangle $$ABC$$ ($$AC = BC$$) has measure of $$106^o$$. M is a point inside the triangle such that $$\angle MAC = 7^o$$ and $$\angle MCA = 23^o$$. The measure of the $$\angle CMB$$ in degrees can be written as a fraction $$p/q$$ for relatively prime integers $$p$$ and $$q$$. Calculate $$p+q$$.

 2. (46 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 $$m+n$$.

 3. (19 p.) If $$ABCD$$ is a convex quadrilateral with $$AB=200$$, $$BC=153$$, $$BD=300$$, $$\angle BAC=\angle BDC<90^{\circ}$$ and $$\angle ABD=\angle BCD$$, determine $$CD.$$

 4. (4 p.) Let $$\alpha$$ be the angle between vectors $$\vec a$$ and $$\vec b$$ with $$|\vec a|=2$$ and $$|\vec b|=3$$, given that the vectors $$\vec m=2\vec a-\vec b$$ and $$\vec n=\vec a+5\vec b$$ are orthogonal. If $$\cos\alpha=\frac pq$$ with $$q>0$$ and $$\gcd(p,q)=1$$, compute $$p+q$$.

 5. (14 p.) The area of the triangle $$ABC$$ is 70. The coordinates of $$B$$ and $$C$$ are $$(12,19)$$ and $$(23,20)$$, respectively, and the coordinates of $$A$$ are $$(p,q)$$. The line containing the median to side BC has slope -5. Find the largest possible value of p+q.

2005-2020 IMOmath.com | imomath"at"gmail.com | Math rendered by MathJax