M is any positive real number less than 2022.
Determine the total number of values of M for which the graphs of
y=abs(abs(abs(x-M)-x)-x) and y=x
intersect only at points with integer coordinates when graphed on the X-Y plane.
|b|=a implies b=a or b=-a. From the first equation y cannot be negative, so from the second, neither can x.
abs(abs(abs(x-M)-x)-x) =x
implies
abs(abs(x-M)-x)-x=x or abs(abs(x-M)-x)-x=-x
abs(abs(x-M)-x)=2x or abs(abs(x-M)-x)=0
abs(x-M)-x=2x or abs(x-M)-x=-2x or abs(x-M)-x=0
abs(x-M)=3x or abs(x-M)=-x or abs(x-M)-x=0
(x-M)=3x or (x-M)=-3x or (x-M)=x or (x-M)=-x
M=-2x or M=4x or M=0 or M=-2x
So for any positive M, the solutions for x are x=0 and x=M/4.
To be integers, M is a multiple of 4.
There are 505 such numbers.
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Posted by Jer
on 2022-12-09 12:29:07 |
Solution
