# General Practice Test

 1. (32 p.) Let $$BC$$ be a chord of length 6 of a circle with center $$O$$ and radius 5. Point $$A$$ is on the circle, closer to $$B$$ that to $$C$$, such that there is a unique chord $$AD$$ which is bisected by $$BC$$. If $$\sin\angle AOB=\frac pq$$ with $$q>0$$ and $$\gcd(p,q)=1$$, find $$p+q$$.

 2. (19 p.) Let $$a$$, $$b$$, $$c$$, $$d$$ be the roots of $$x^4 - x^3 - x^2 - 1 = 0$$. Find $$p(a) + p(b) + p(c) + p(d)$$, where $$p(x) = x^6 - x^5 - x^3 - x^2 - x$$.

 3. (4 p.) The set $$A$$ consists of $$m$$ consecutive integers with sum $$2m$$. The set $$B$$ consists of $$2m$$ consecutive integers with sum $$m$$. The difference between the largest elements of $$A$$ and $$B$$ is 99. Find $$m$$.

 4. (23 p.) In a tournament club $$C$$ plays 6 matches, and for each match the probabilities of a win, draw and loss are equal. If the probability that $$C$$ finishes with more wins than losses is $$\frac pq$$ with $$p$$ and $$q$$ coprime $$(q>0)$$, find $$p+q$$.

 5. (19 p.) Let $$ABC$$ be a triangle with sides 3, 4, 5 and $$DEFG$$ a $$6 \times 7$$ rectangle. A line divides $$\triangle ABC$$ into a triangle $$T_1$$ and a trapezoid $$R_1$$. Another line divides the rectangle $$DEFG$$ into a triangle $$T_2$$ and a trapezoid $$R_2$$, in such a way $$T_1\sim T_2$$ and $$R_1\sim R_2$$. The smallest possible value for the area of $$T_1$$ can be expressed as $$p/q$$ for two relatively prime positive integers $$p$$ and $$q$$. Evaluate $$p+q$$.

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