# General Practice Test

 1. (17 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. (39 p.) Suppose $$m$$ and $$n$$ are positive integers with $$m> 1$$ such that the domain of the function $$f(x) = \text{arcsin}(\log_{m}(nx))$$ is a closed interval of length $$\frac{1}{2013}$$. Let $$S$$ be the smallest possible value of $$m+n$$. Find the remainder when $$S$$ is divided by $$1000$$.

 3. (7 p.) Find the least positive integer $$n$$ such that when its leftmost digit is deleted, the resulting integer is equal to $$n/29$$.

 4. (17 p.) A circle of radius 1 is randomly placed inside a $$15 \times 36$$ rectangle $$ABCD$$. The probability that it does not intersect the diagonal $$AC$$ can be expressed as $$p/q$$ where $$p$$ and $$q$$ are relatively prime integers. Find $$p+q$$.

 5. (18 p.) Let $$ABCD$$ be a convex quadrilateral with $$AB = CD = 180$$. Assume further that the perimeter of $$ABCD$$ is 640, $$AD \neq BC$$, and $$\angle A = \angle C$$. Then $$\cos \angle A$$ can be represented as $$p/q$$ for relatively prime positive integers $$p$$ and $$q$$. Calculate $$p+q$$.

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