# Geometry

 1. (12 p.) Given a rhombus $$ABCD$$, the circumradii of the triangles $$ABD$$ and $$ACD$$ are 12.5 and 25. Find the area of $$ABCD$$.

 2. (10 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. (54 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. (17 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.

 5. (5 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 $$p+q$$.

2005-2018 IMOmath.com | imomath"at"gmail.com | Math rendered by MathJax
Home | Olympiads | Book | Training | IMO Results | Forum | Links | About | Contact us