General Practice Test

 1. (5 p.) Let $$S$$ be the set of vertices of a unit cube. Find the number of triangles whose vertices belong to $$S$$.

 2. (23 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. (18 p.) The number $\frac1{2\sqrt1+1\sqrt 2}+\frac1{3\sqrt2+2\sqrt3}+\frac1{4\sqrt3+3\sqrt4} + \dots + \frac1{100\sqrt{99}+99\sqrt{100}}$ is a rational number. If it is expressed as $$\frac pq$$ for two relatively prime integers $$p$$ and $$q$$ evaluate $$p+q$$.

 4. (28 p.) Let $$ABC$$ be a rectangular triangle such that $$\angle C=90^o$$ and $$AC = 7$$, $$BC = 24$$. Let $$M$$ be the midpoint of $$AB$$ and $$D$$ point on the same side of $$AB$$ as $$C$$ such that $$DA = DB = 15$$. The area of the triangle $$CDM$$ can be expressed as $$\frac{p\sqrt q}r$$ for positive integers $$p$$, $$q$$, $$r$$ such that $$q$$ is not divisible by a perfect square and $$(p,r)=1$$. Find area $$p+q+r$$.

 5. (23 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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