# General Practice Test

 1. (8 p.) Real numbers $$x,y,z$$ are real numbers greater than 1 and $$w$$ is a positive real number. If $$\log_xw=24$$, $$\log_yw=40$$ and $$\log_{xyz}w=12$$, find $$\log_zw$$.

 2. (51 p.) Let $$\triangle ABC$$ have $$AB=6$$, $$BC=7$$, and $$CA=8$$, and denote by $$\omega$$ its circumcircle. Let $$N$$ be a point on $$\omega$$ such that $$AN$$ is a diameter of $$\omega$$. Furthermore, let the tangent to $$\omega$$ at $$A$$ intersect $$BC$$ at $$T$$, and let the second intersection point of $$NT$$ with $$\omega$$ be $$X$$. The length of $$\overline{AX}$$ can be written in the form $$\tfrac m{\sqrt n}$$ for positive integers $$m$$ and $$n$$, where $$n$$ is not divisible by the square of any prime. Find $$m+n$$.

 3. (21 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 $$p+q$$.

 4. (10 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.

 5. (8 p.) Let $$a$$, $$b$$, and $$c$$ be non-real roots of the polynimal $$x^3+x-1$$. Find $\frac{1+a}{1-a}+ \frac{1+b}{1-b}+ \frac{1+c}{1-c}.$

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