2020 CIME I Problems/Problem 1
Problem 1
A knight begins on the point 
in the coordinate plane. From any point 
the knight moves to either 
or 
. Find the number of ways the knight can reach the point 
.
Solution
Let 
denote a move of 
units north and 
unit east, and let 
denote a move of unit north and units east. To get to the point using only these moves, say 
moves in direction and 
moves in direction , we must have 
because both the 
- and 
-coordinates have increased by 
since the knight started. Solving this system of equations gives us 
. This means we need the knight to make 
moves, 
of which are headed in direction , and the remaining are headed in direction . Any combination of these moves work, so the answer is 
Solution 2
We can draw lines using and . Calculating the lines, we see that they are from 
and 
respectively. The point is on the line 
, which is in the "middle" between both, since by multiplying or dividing the slope by we can get the other two lines. This means to get to , for every we do, we do one to balance it. Call this system of moves , and by performing once, we get to 
.
If we repeat five more times, we get to . Thus this is now a word arrangement problem where we have to arrange five s and s which represent each move (letter name is arbitrary). We get 
~ neeyakkid23
Video Solution
https://www.youtube.com/watch?v=SFVt0JYLkHY ~Shreyas S
See also
| 2020 CIME I (Problems • Answer Key • Resources) | ||
| Preceded by First Question |
Followed by Problem 2 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
| All CIME Problems and Solutions | ||
The problems on this page are copyrighted by the MAC's Christmas Mathematics Competitions. 
