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 3 of 5: Count the distinct breakpoints and values

Previous step Next step

3+3+6+6+4=22

Detailed analysis

After reducing fractions, the candidates contribute 3 integers, 3 fractions with denominator $2$ , 6 with denominator $3$ , 6 with denominator $4$ , and 4 with denominator $5$ . These sets are disjoint, giving $3+3+6+6+4=22$ distinct breakpoint positions in one period and hence 22 attained values on $[0,3)$ .