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 1 of 5: Reduce the problem to one period

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f(x+3)=f(x)+35

Detailed analysis

For every integer $m$ , $\lfloor t+m\rfloor=\lfloor t\rfloor+m$ . Applying this to the five terms gives $f(x+3)=f(x)+3+6+5+9+12=f(x)+35$ . Thus the behavior repeats up to an additive shift of $35$ every interval of length $3$ .