Question

Please Help !!!

Serena took two numbers which may or may not be integers, rounded
each of them up to the nearest integer, multiplied the results, and
got 100. When she took the original numbers, rounded each of them
down to the nearest integer and multiplied the results, she got X.
Find the largest possible value of X.
( The Answer is 202, please give a step by step solution)
Thank You

Answer 1

Recall (or "define", if you're not familiar) the so-called ceiling and floor functions, or least/greatest integer functions, respectively; for any integer n,

• ceiling function:

`\lceil x \rceil = n \text{ if } n-1 < x \le n`

• floor function:

`\lfloor x \rfloor = n \text{ if } n \le x < n+1`

Let x and y be the two numbers Serena picks. She finds that

`\lceil x \rceil \lceil y \rceil = 100`

`\lfloor x \rfloor \lfloor y \rfloor = X`

Assume x and y are not integers. Let m and n be two integers such that m - 1 < x < m and n - 1 < y < n. Then

`\lceil x \rceil \lceil y \rceil = mn = 100`

so m and n are divisors of 100. There's a finite number of distinct divisor pairs (m, n) to consider,

(± 1, ± 100), (± 2, ± 50), (± 4, ± 25), (± 5, ± 5)

whose sums can be one of

± 101, ± 99, ± 52, ± 48, ± 29, ± 21, ± 10, 0

Similarly,

`\lfloor x \rfloor \lfloor y \rfloor = (m-1)(n-1) = 101 - (m + n) = X`

which is maximized if the sum m + n is negative; the largest of these is m + n = -101, so the maximum value of X is 101 + 101 = 202.

The reasoning for the case of either x or y, or both, being integers is nearly identical.