John

The points obviously can be connected to form a triangle. What is the probability that they form an obtuse triangle? (An obtuse triangle has one angle that exceeds 90 degrees.)

This question is not homework, merely one that I found to be of interest. I hope others will enjoy it too.

Consider the family of all straight lines which intersect the 4th degree polynomial

P(x) = x^4 - 11*x^3 + 41*x^2 - 61*x + 30

in FOUR DISTINCT points. If we were to compute those intersection points, assume they fall at the locations (x_1,y_1), (x_2,y_2), (x_3,y_3), (x_4,y_4).

It is interesting to note that the sum of the x_i, thus

x_1 + x_2 + x_3 + x_4

is fixed and easily computable over the entire family of such lines. (Feel free to prove this for extra credit.)

Of course, the product of the x_i will not be held constant over all such intersecting lines, but my question is how to find the line that maximizes the product of the x_i? What is the maximum possible product

x_1*x_2*x_3*x_4

Feel free to write it as a floating point number, as it will be a bit messy.

This is NOT a homework problem, merely one that I hope others might enjoy solving.

Given an arbitrary set of distinct positive integers, {X_1, X_2, ..., X_n}, such that no member of the set has a prime factor that exceeds 7. Choose ANY two distinct elements of the set, and form the product of those two elements. How large must the set be to ensure that there must always be some pair of distinct elements with a product that is a perfect square? Worded differently, what is the minimum number of elements of any such set such that there will always be some pair of elements that generates a perfect square?

For example, given the set {2, 3, 14, 15, 21}, there is no product of a pair of distinct elements that generates a perfect square. However, the set {3, 8, 15, 60} is a set that does have a pair of elements, since 15x60=900=30^2.

I'm not looking for a specific example of a set that has the desired property, as I've already shown you one such example. I want to know, for a general, arbitrary set, what size must it be to ensure that some such pair of elements must always exist.

For a slightly more challenging but similar problem, suppose I ask you to take three distinct elements from the set and again form the product? How large must the set be to ensure that there will always be some triad of distinct elements that yields a perfect cube? Again, assume that no element of the set has a prime divisor exceeding 7.

... will always be a composite number.

{n, 7n + 4, 11n - 2}

A composite number is a positive number that has at least one positive integer divisor other than 1 and itself. Thus 6 is a composite number, as it has divisors of 2 and 3. 17 is not composite, since it is prime.

For example, when n = 7, both 7 and 7x7+4=53 are prime, but 11x7-2 = 75 is composite.

http://en.wikipedia.org/wiki/Composite_number

This is not a homework question, merely one that I found of moderate interest, and I hope a challenge to some. Enjoy.

Ok, its nearly time for the olympics to start, but I'm not looking for the gold here, just the GCD. GCD stands for greatest common divisor of a set of numbers, so the largest number that divides each member of the set with no remainder. So consider the set of numbers (n^17 - n), where n is an integer. There are infinitely many members of this set, but is there some greatest common integer divisor that divides all of them? Can you prove that must be true?

(This is not a homework problem, just something I've chosen to post for interest to those who are mathematically inclined. Enjoy.)

I've chosen to pose an occasional mathematical problem for fun. This is not homework, but designed to be of moderate interest. It does not require sophisticated methods or even a powerful computer to solve.

I'll give you a pile of 1000 pieces of paper, labeled in turn with the first 1000 odd numbers {1, 3, 5, 7, ..., 1999}, one number per slip of paper. Shuffle the pile into random order, then I'll ask you to remove any two of those slips of paper from the pile, labeled with say the numbers x and y. Replace those two slips you have removed with a single new slip that has written on it the amount (xy+x+y). After 999 iterations of this process there will be only a single piece of paper that remains. What is the number written on that paper?

I'm not sure what extra credit can be worth here, but for extra credit, how many digits are in that final number?

We use our regular microwave oven to dry sweet potatoes. It works well enough, producing a batch in a reasonable time, but we need to buy a new oven now and my wife has no place she wants to put a regular fruit dryer. So if we get a microwave-convection combo, will it help dry them faster, i.e., remove more moisture from the slices of sweet potato?