Reach for the Summit - M-S6-A1

Geometry Level 3

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

How to submit:

  • First, find the number of all possible solutions (x,y)(x,y). Let NN denote the number of solutions.
  • Then sort the solutions by xx from smallest to largest, if xx is the same, then sort by yy from smallest to largest.
  • Let the sorted solutions be: (x1,y1),(x2,y2),(x3,y3),,(xN,yN)(x_1,y_1), (x_2,y_2), (x_3,y_3), \cdots ,(x_N,y_N), then M=k=1Nk(xk+yk)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)(-1,2), (-1,1), (1,3), (0,4), the sorted solution will be: (1,1),(1,2),(0,4),(1,3)(-1,1), (-1,2), (0,4), (1,3), then N=4N=4 and M=k=14k(xk+yk)=1×(1+1)+2×(1+2)+3×(0+4)+4×(1+3)=30\\ 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 M+N\lfloor M+N \rfloor.

Reach for the Summit problem set - Mathematics


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