Difference between revisions of "1996 IMO Problems/Problem 1"
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First we define the rectangular board in the cartesian plane with centers of the unit squares as integer coordinates and the following coordinates for the squares at the corners of <math>A</math>, <math>B</math>, <math>C</math>, <math>D</math>, as follows: <math>A=(1,1)</math>, <math>B=(20,1)</math>, <math>C=(20,12)</math>, <math>D=(1,12)</math> | First we define the rectangular board in the cartesian plane with centers of the unit squares as integer coordinates and the following coordinates for the squares at the corners of <math>A</math>, <math>B</math>, <math>C</math>, <math>D</math>, as follows: <math>A=(1,1)</math>, <math>B=(20,1)</math>, <math>C=(20,12)</math>, <math>D=(1,12)</math> | ||
| − | Let <math>(x_i,y_i)</math> be the coordinates | + | Let <math>(x_i,y_i)</math> be the coordinates of the piece after move <math>i</math> with <math>(x_0,y_0)=A=(1,1)</math> the initial position of the piece. |
| + | |||
| + | Let <math>\Delta x_i = x_i-x_{i-1}</math>, <math>\Delta y_i = y_i-y_{i-1}</math> | ||
Revision as of 15:52, 20 November 2023
Problem
We are given a positive integer 
and a rectangular board 
with dimensions 
, 
. The rectangle is divided into a grid of 
unit squares. The following moves are permitted on the board: one can move from one square to another only if the distance between the centers of the two squares is 
. The task is to find a sequence of moves leading from the square with 
as a vertex to the square with 
as a vertex.
(a) Show that the task cannot be done if is divisible by 
or 
.
(b) Prove that the task is possible when 
.
(c) Can the task be done when 
?
Solution
First we define the rectangular board in the cartesian plane with centers of the unit squares as integer coordinates and the following coordinates for the squares at the corners of , , 
, 
, as follows: 
, 
, 
, 
Let 
be the coordinates of the piece after move 
with 
the initial position of the piece.
Let 
, 
Alternate solutions are always welcome. If you have a different, elegant solution to this problem, please add it to this page.
See Also
| 1996 IMO (Problems) • Resources | ||
| Preceded by First Problem |
1 • 2 • 3 • 4 • 5 • 6 | Followed by Problem 2 |
| All IMO Problems and Solutions | ||
