# General Practice Test

 1. (29 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 $$p+q$$.

 2. (12 p.) Find the least positive integer $$n$$ such that when its leftmost digit is deleted, the resulting integer is equal to $$n/29$$.

 3. (26 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 $$p+q+r$$.

 4. (24 p.) Find the largest possible integer $$n$$ such that $$\sqrt n + \sqrt{n+60} = \sqrt m$$ for some non-square integer $$m$$.

 5. (7 p.) Let $$a$$, $$b$$, $$c$$ be positive integers forming an increasing geometric sequence such that $$b-a$$ is a square. If $$\log_6a + \log_6b + \log_6c = 6$$, find $$a + b + c$$.

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