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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.

Source: http://www.artofproblemsolving.com

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