General Practice Test

 1. (31 p.) The right circular cone has height 4 and its base radius is 3. Its surface is painted black. The cone is cut into two parts by a plane parallel to the base, so that the volume of the top part (the small cone) divided by the volume of the bottom part equals $$k$$ and painted area of the top part divided by the painted are of the bottom part also equals $$k$$. If $$k$$ is of the form $$p/q$$ for two relatively prime numbers $$p$$ and $$q$$, calculate $$p+q$$.

 2. (31 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 $$p+q$$.

 3. (8 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 $$p+q$$.

 4. (14 p.) Assume that $$A$$ is a 40-element subset of $$\{1,2,3,\dots,50\}$$, and let $$n$$ be the sum of the elements of $$A$$. Find the number of possible values of $$n$$.

 5. (14 p.) Find the sum of all positive integers of the form $$n = 2^a3^b$$ $$(a, b \geq 0)$$ such that $$n^6$$ does not divide $$6^n$$.

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