1. In a 10 by 10 grid of dots, what is the maximum number of lines that can be drawn connecting two dots on the grid so that no two lines are parallel?
2. If r_1, r_2, and r_3 are the solutions to the equation x^3 – 5x^2 + 6x – 1 = 0, then what is the value of r_1^2 + r_2^2 + r_3^2?
3. The expression \circ \ 1\ \circ \ 2 \ \circ 3 \ \circ \dots \circ \ 2012 is written on a blackboard. Catherine places a + sign or a – sign into each blank. She then evaluates the expression, and finds the remainder when it is divided by 2012. How many possible values are there for this remainder?
4. Parallel lines \ell_1 and \ell_2 are drawn in a plane. Points A_1, A_2, \dots, A_n are chosen on \ell_1, and points B_1, B_2, \dots, B_{n+1} are chosen on \ell_2. All segments A_iB_j are drawn, such that 1 \le i \le n and 1 \le j \le n+1. Let the number of total intersections between these segments (not including endpoints) be denoted by Q. Given that no three segments are concurrent, besides at endpoints, prove that Q is divisible by 3.
5. In convex hexagon ABCDEF, \angle A \cong \angle B, \angle C \cong \angle D, and \angle E \cong \angle F. Prove that the perpendicular bisectors of \overline{AB}, \overline{CD}, and \overline{EF} pass through a common point.
6. The positive numbers a, b, c satisfy 4abc(a+b+c) = (a+b)^2(a+c)^2. Prove that a(a+b+c)=bc.
7. For how many positive integers n \le 500 is n! divisible by 2^{n-2}?
8. A convex 2012-gon A_1A_2A_3 \dots A_{2012} has the property that for every integer 1 \le i \le 1006, \overline{A_iA_{i+1006}} partitions the polygon into two congruent regions. Show that for every pair of integers 1 \le j < k \le 1006, quadrilateral A_jA_kA_{j+1006}A_{k+1006} is a parallelogram.
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