Problem 2

Find the total number of different integer values taken by

f(x)=\lfloor x\rfloor+\lfloor2x\rfloor+\left\lfloor\frac{5x}{3}\right\rfloor+\lfloor3x\rfloor+\lfloor4x\rfloor

for real

x

with

0\le x\le100

Step 5 of 5: Count the final eight values

Previous step Next step

x\in[99,100]:\quad 99,\ 99+\frac14,\ 99+\frac13,\ 99+\frac12,\ 99+\frac35,\ 99+\frac23,\ 99+\frac34,\ 100

Detailed analysis

The remaining interval has the eight distinct breakpoints $99$ , $99+1/4$ , $99+1/3$ , $99+1/2$ , $99+3/5$ , $99+2/3$ , $99+3/4$ , and $100$ . Each gives the next value of the nondecreasing step function, so there are 8 additional values. The total is $33\cdot22+8=734$ .