# General Practice Test

 1. (32 p.) If $$ABCD$$ is a convex quadrilateral with $$AB=200$$, $$BC=153$$, $$BD=300$$, $$\angle BAC=\angle BDC<90^{\circ}$$ and $$\angle ABD=\angle BCD$$, determine $$CD.$$

 2. (22 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 $$p+q$$.

 3. (12 p.) Find the least positive integer $$n$$ such that when its leftmost digit is deleted, the resulting integer is equal to $$n/29$$.

 4. (17 p.) Given a rhombus $$ABCD$$, the circumradii of the triangles $$ABD$$ and $$ACD$$ are 12.5 and 25. Find the area of $$ABCD$$.

 5. (15 p.) Let $$X$$ be a square of side length 2. Denote by $$S$$ the set of all segments of length 2 with endpoints on adjacent sides of $$X$$. The midpoints of the segments in $$S$$ enclose a region with an area $$A$$. Find $$[100A]$$.

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