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 4 of 5: Count the complete periods

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100=33\cdot3+1\Longrightarrow 33\cdot22\text{ values on }[0,99)

Detailed analysis

The interval $[0,99)$ consists of $33$ complete periods of length $3$ . Because each period's values are shifted by $35$ , these sets contribute $33\cdot22$ distinct integer values.