SAT1000 - P900
A group of members from brilliant.org decide to plant trees on the playground defined by coordinate plane, and the schedule is as follows:
The k t h tree is planted at P k ( x k , y k ) , where x 1 = 1 , y 1 = 1 , and when k ≥ 2 :
⎩ ⎪ ⎪ ⎨ ⎪ ⎪ ⎧ x k y k = x k − 1 + 1 − 5 ( ⌊ 5 k − 1 ⌋ − ⌊ 5 k − 2 ⌋ ) = y k − 1 + ⌊ 5 k − 1 ⌋ − ⌊ 5 k − 2 ⌋
What's the coordinate of the 2 0 0 8 t h tree?
Let the coordinate be x 0 , y 0 . Submit 2 y 0 − x 0 .
Have a look at my problem set: SAT 1000 problems
The answer is 801.
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⌊ 5 k − 1 ⌋ − ⌊ 5 k − 2 ⌋ = { 1 0 if k ≡ 1 ( m o d 5 ) otherwise
From this we see that x k − 1 is the remainder when k − 1 is divided by 5 and y k − 1 is the corresponding quotient. Hence the coordinates of the 2 0 0 8 th tree are 3 , 4 0 2 and the answer is 8 0 1 .