# Multiple choice practice test

 1. (22 p.) Let $$ABCD$$ be a trapezoid with $$AB\|CD$$, $$AB=11$$, $$BC=5$$, $$CD=19$$, and $$DA=7$$. The bisectors of $$\angle A$$ and $$\angle D$$ meet at $$P$$, and bisectors of $$\angle B$$ and $$\angle C$$ meet at $$Q$$. Find the area of the hexagon $$ABQCDP$$.    A    $$28\sqrt 3$$    B    $$30\sqrt3$$    C    $$32\sqrt 3$$    D    $$35\sqrt 3$$    E    $$36\sqrt 3$$    N

 2. (20 p.) Suppose that the sum of base-10 logarithms of the divisors of $$10^n$$ is 792. Determine $$n$$.    A    11    B    12    C    13    D    14    E    15    N

 3. (20 p.) A parking lot has 16 spaces in a row. Each of the twelve cars took one parking space, and their drivers chose spaces at random from among the available spaces. After that a big van arrived and it requires 2 adjacent spaces to park. What is the probability that the van will be able to park?    A    $$\frac{11}{20}$$    B    $$\frac{4}{7}$$    C    $$\frac{81}{140}$$    D    $$\frac35$$    E    $$\frac{17}{28}$$    N

 4. (17 p.) A function $$f$$ is defined by $$f(z)=(4+i)z^2+\alpha z+ \gamma$$ for all complex numbers $$z$$, where $$\alpha$$ and $$\gamma$$ are complex numbers. Given that $$f(1)$$ and $$f(i)$$ are both real, find the smallest possible value for $$|\alpha|+|\gamma|$$.    A    $$1$$    B    $$\sqrt2$$    C    $$2$$    D    $$2\sqrt2$$    E    $$4$$    N

 5. (19 p.) Two circles of radius $$1$$ are chosen in the following way. The center of the circle $$k_0$$ is chosen uniformly at random from the line segment joining $$(0,0)$$ and $$(2,0)$$. Independently of this choice, the center of circle $$k_1$$ is chosen uniformly at random from the line segment joining $$(0,1)$$ to $$(2,1)$$. What is the probability that $$k_0$$ and $$k_1$$ intersect?    A    $$\frac{2+\sqrt2}4$$    B    $$\frac{3\sqrt3+2}8$$    C    $$\frac{2\sqrt2-1}2$$    D    $$\frac{2+\sqrt3}4$$    E    $$\frac{4\sqrt3-3}4$$    N

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