General Practice Test

 1. (5 p.) Find the product of the real roots of the equation $$x^2+18x+30=2\sqrt{x^2+18x+45}$$ (the answer is an integer).

 2. (57 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. (3 p.) Let $$S$$ be the set of vertices of a unit cube. Find the number of triangles whose vertices belong to $$S$$.

 4. (24 p.) A bug moves around a triangle wire. At each vertex it has 1/2 chance of moving towards each of the other two vertices. The probability that after crawling along 10 edges it reaches its starting point can be expressed as $$p/q$$ for positive relatively prime integers $$p$$ and $$q$$. Determine $$p+q$$.

 5. (9 p.) Given a regular 12-gon D, determine the number of squares that have two or more vertices among the vertices of D.

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