# General Practice Test

 1. (22 p.) We are given an unfair coin. When the coin is tossed, the probability of heads is 0.4. The coin is tossed 10 times. Let $$a_n$$ be the number of heads in the first $$n$$ tosses. Let $$P$$ be the probability that $$a_n/n \leq 0.4$$ for $$n = 1, 2, \dots , 9$$ and $$a_{10}/10 = 0.4$$. Evaluate $$\frac{P\cdot 10^{10}}{24^4}$$.

 2. (18 p.) The area of the triangle $$ABC$$ is 70. The coordinates of $$B$$ and $$C$$ are $$(12,19)$$ and $$(23,20)$$, respectively, and the coordinates of $$A$$ are $$(p,q)$$. The line containing the median to side BC has slope -5. Find the largest possible value of p+q.

 3. (11 p.) A triangle $$ABC$$ has sides 13, 14, 15. The triangle $$ABC$$ is rotated about its centroid for an angle of $$180^0$$ to form a triangle $$A^{\prime}B^{\prime}C^{\prime}$$. Find the area of the union of the two triangles.

 4. (22 p.) The sequence of complex numbers $$z_0,z_1,z_2,\dots$$ is defined by $$z_0=1+i/211$$ and $$z_{n+1}=\frac{z_n+i}{z_n-i}$$. If $$z_{2111}=\frac ab+\frac cdi$$ for positive integers $$a,b,c,d$$ with $$\gcd(a,b)=\gcd(c,d)=1$$, find $$a+b+c+d$$.

 5. (24 p.) Consider the polynomial $P(x)=(1 + x + x^2 + \dots + x^{17})^2 - x^{17}.$ Assume that the roots of $$P$$ are $$x_k=r_k \cdot e^{i2\pi a_k}$$, for $$k = 1, 2, ... , 34$$, $$0 < a_1 \leq a_2 \leq \dots \leq a_{34} < 1$$, and some positive real numbers $$r_k$$. The sum $$a_1 + a_2 + a_3 + a_4 + a_5$$ is equal to $$p/q$$ for two coprime integers $$p$$ and $$q$$. Determine $$p+q$$.

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