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?
3. The expression 
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?
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 
and 
are drawn in a plane. Points 
are chosen on , and points 
are chosen on . All segments 
are drawn, such that 
and 
. Let the number of total intersections between these segments (not including endpoints) be denoted by 
. Given that no three segments are concurrent, besides at endpoints, prove that is divisible by 3.
and are drawn in a plane. Points are chosen on , and points are chosen on . All segments are drawn, such that and . Let the number of total intersections between these segments (not including endpoints) be denoted by . Given that no three segments are concurrent, besides at endpoints, prove that is divisible by 3. 5. In convex hexagon 
, 
, 
, and 
. Prove that the perpendicular bisectors of 
, 
, and 
pass through a common point.
, , , and . Prove that the perpendicular bisectors of , , and pass through a common point.
8. A convex 2012-gon 
has the property that for every integer 
, 
partitions the polygon into two congruent regions. Show that for every pair of integers 
, quadrilateral 
is a parallelogram.
has the property that for every integer , partitions the polygon into two congruent regions. Show that for every pair of integers , quadrilateral is a parallelogram.
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