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 (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) | ||
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Followed by Problem 2 | |
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