Evaluate the greatest integer less than or equal to 100000 times 32 divided by (4 ln 2 + 3).

Step-by-step solution

1. 1

   Interpret the bracket as the floor function. In this notation $[\,\cdot\,]$ means the greatest integer not exceeding the value, so the task is to compute a quotient and then truncate downward. The whole difficulty is deciding the integer part correctly, which requires knowing how close the quotient sits to an integer.

2. 2

   Collapse the numerator. $100000\times32=3{,}200{,}000$, so we need

   $\left\lfloor\frac{3{,}200{,}000}{4\ln 2+3}\right\rfloor$

3. 3

   Evaluate the denominator to enough precision. With $\ln 2=0.69314718055994531$,

   $4\ln 2=2.77258872223978124,\qquad 4\ln2+3=5.77258872223978124$

   Truncating $\ln2$ to three decimals would shift the quotient by about $\pm 200$, which is far too coarse for a floor — this is why ten or more digits are needed.

4. 4

   Divide.

   $\frac{3{,}200{,}000}{5.77258872223978124}=554{,}344.0134010431$

5. 5

   Take the floor and confirm the margin. The fractional part is $0.0134$, so the floor is $554{,}344$. The result is genuinely close to the boundary: multiplying back, $554{,}344\times5.77258872224=3{,}199{,}999.923$, which is just under $3{,}200{,}000$, while $554{,}345\times5.77258872224=3{,}200{,}005.70$ overshoots ✓. So $554{,}344$ is correct and $554{,}345$ is not.