Reach for the Summit - M-S6-A1

Given that $A(1,-1), B(-4,5)$ , $C$ is on line $AB$ , and $|AC|=3|AB|$ , then find all possible coordinates of point $C$ .

How to submit:

• First, find the number of all possible solutions $(x,y)$ . Let $N$ denote the number of solutions.
• Then sort the solutions by $x$ from smallest to largest, if $x$ is the same, then sort by $y$ from smallest to largest.
• Let the sorted solutions be: $(x_1,y_1), (x_2,y_2), (x_3,y_3), \cdots ,(x_N,y_N)$ , then $M=\displaystyle \sum_{k=1}^N k(x_k+y_k)$ .

For instance, if the solution is $(-1,2), (-1,1), (1,3), (0,4)$ , the sorted solution will be: $(-1,1), (-1,2), (0,4), (1,3)$ , then $N=4$ and $\\ M=\displaystyle \sum_{k=1}^4 k(x_k+y_k)= 1 \times (-1+1) + 2 \times (-1+2) + 3 \times (0+4) + 4 \times (1+3) =30$ .

For this problem, submit $\lfloor M+N \rfloor$ .

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Reach for the Summit problem set - Mathematics