HMMT 二月 2012 · TEAM1 赛 · 第 6 题

英文解析

6. [ 30 ] It has recently been discovered that the right triangle with vertices (0,0), (0,2012), and (2012,0)
   is a giant pond that is home to many frogs. Frogs have the special ability that, if they are at a latticepoint ( x, y ), they can hop to any of the three lattice points ( x +1 , y +1), ( x − 2 , y +1), and ( x +1 , y − 2),
   assuming the given lattice point lies in or on the boundary of the triangle.
   Frog Jeff starts at the corner (0 , 0), while Frog Kenny starts at the corner (0 , 2012). Show that the setof points that Jeff can reach is equal in size to the set of points that Kenny can reach.
   Answer: see below
   Team A
   √
   3 y
   ( x, y ) 7 → ( x + , y )
   2 2
   We transform the triangle as follows: map each lattice point ( x, y ) to the point
   √ √
   x (1 , 0) + y (1 / 2 , 3 / 2) = ( x + y/ 2 , y 3 / 2).
   This transforms the right triangle into an equilateral triangle as shown above.
   Now, the three allowed movements
   ( x, y ) 7 → ( x + 1 , y + 1) ,
   ( x, y ) 7 → ( x − 2 , y + 1) ,
   ( x, y ) 7 → ( x + 1 , y − 2)
   become the movements
   √
   ( x, y ) 7 → ( x + 3 / 2 , y + 3 / 2) ,
   √
   ( x, y ) 7 → ( x − 3 / 2 , y + 3 / 2) ,
   √
   ( x, y ) 7 → ( x, y − 3)
   √
   °
   That is, each step is a movement of 3 in any of these three directions, which are separated by 120
   °
   angles. The pond is now completely symmetrical with respect to 120 rotations, so it does not matterwhich vertex you start at. The lower left vertex corresponds to the original point (0 , 0), and the topvertex corresponds to the original point (0 , 2012).