Comments to follow refer to problem #2: From the set {1,2,3,...,n}, k distinct integers are selected at random and arranged in numerical order (lowest to highest). Let P(i,r,k,n) denote the probability that integer i is in position r. For example, observe that P(1,2,k,n)=0. (a) Compute P(2,1,6,10). (b) Find a general formula for P(i,r,k,n).

``General formula??? There IS no general formula!''

``Reason: P(2,1,6,10) denotes the probability that integer 2 is in position 1. The reason for that is because P(i,r,k,n) denotes the probability that integer i is in position r. Substitute 2 for i and 1 for r and it reads THE FIRST SENTENCE THAT I WROTE DOWN. So the probability that 2 is in position 1 is 100. I HAVE NO CLUE WHAT I AM TALKING ABOUT.''

``I forgot about sets three years ago, but I only forgot about probabilities two months ago.''

``Please note: I have difficulty understanding the problem, so I have restated the problem.''

``Passion comes from the heat, insanity comes from mathematics.''

``Even though I write many silly comments, I still DO think, so please don't skip #5.''

``The probability that, in a set of integers in numerical order, the number 1 comes after the number 2 is 0, for by any definition of numerical order 1 comes before 2.''

``This problem is worded such that I can't even make a non-trivial restatement of this problem.''

I also like the comments pertaining to problem #5: Let XYZ be a triangle which is not a right triangle. Prove that there exists circles C1, C2, and C3 such that C2 is tangent to C3 at X, C3 is tangent to C1 at Y, and C1 is tangent to C2 at Z.

``The circles are tangent at that spot. You did not specify that they had to be 100% tangent.''

``Given any 3 circles, the closest they can be placed together without overlapping or changing shape, is when each circle is tangent to the other.''

``I don't think they make circles that can do that!''

``If you look at the diagram, you will notice absolutely nothing... argh...this is a trashy problem....''

``I know it's true by intuition.''

4. Given any five distinct real numbers, one can compute the sums of any two, any three, any four, and all five numbers and then count the number N of DISTINCT values among these sums.
(a) Give an example of five numbers yielding the smallest possible value of N. What is this value?
(b) Give an example of five numbers yielding the largest possible value of N. What is this value?
(c) Prove that the values of N you obtained in (a) and (b) are the smallest and largest possible ones.

``Since the 26 sums of set (b) are all different...(I'd make a chart, but I don't have room/time/willpower/desire to) Fine, I will'' [and then they proceed to show all 26 sums]

``I don't know what is meant by 'distinct' so I'll change 'distinct' to 'prime.'''

``Once upon a time in the dead of winter some fool convinced me to allow my bran (sic) to organize 100 minutes over five poorly explained problems that really have no bearing upon life in the 20th century. Why does it matter how many sums one can find by adding 2, 3, 4, and 5 numbers together?''

``This problem is a joke! They all are. I'm totally faking it. I don't even know how I got this far in the first place. Oh well, I might as well have fun while I'm here.''

``Can't prove it because it's just a wild guess.''