Before MCC confuse us more with his forestry analogy (an interesting topic nonetheless), which is not that relevant to this problem, perhaps I'd better demonstrate it myself (as I think most have essentially "grabbed" the main concept here)...

Q1:

ronk has already given 2 separate examples to show that Shaquillina

could or

could not has her FT% touching the .800 mark. So the answer is (b), neither impossible nor a must.

For it to

happen, either M/A=4/5 or M/(A+1)=4/5, i.e. 5M=4A or 4A+4, which is possible only if

A or

A+1 is a

multiple of 5 and

M is a

multiple of 4.

For it to

not happen, M/A > 4/5 > M/(A+1), i.e. 4A < 5M < 4A+4, so there must be a

multiple of 5 among

4A+1, 4A+2, 4A+3. For example, A=7, then 4A+2=30 and M could be 6, or A=38, then 4A+3=155 and M could be 31. (Check: 6/7 > 4/5 > 6/8, 31/38 > 4/5 > 31/39...)

Q2:

The answer is (c), i.e. Shaquillina's FT% must at one stage be

exactly .800. Suppose the contrary is true, i.e. she "jumped" from below .800 to above .800 without stepping on the line

, then we must have:

M/A < 4/5 < (M+1)/(A+1)

Which leads to the following 2 inequalities:

5M < 4A so 4A-5M > 0

4(A+1) < 5(M+1) so 4A-5M < 5-4 so 4A-5M < 1

Since both A and M are integers, 4A-5M must be an integer. There is no integer existing between 0 and 1, so we reach a contradiction. Therefore Shaquillina's FT% could not have "jumped" from .800- to .800+ without hitting .800 at least once...

Note that there is 1 exception to the above reasoning... Suppose A=0 and M=0 (i.e. she hasn't shot a FT at all), from basketball convention her FT% is defined as 0% (although mathematically speaking 0/0 is not a valid fraction). If she makes her 1st ever FT, then her new FT% is (0+1)/(0+1)=1=100%, and she has "jumped" over the .800 line...

However, for this particular problem we know it is not the case (she was trying to "jump" from .650 to .850), so this reasoning stands and we can be sure of the answer...

Therefore both tarek and MCC has got the correct answers to both (though not with the elegant proof I had in mind)... I'm sure ronk and others could more or less guess it anyway...