The following problems are typical of problems which you might try to solve during the 2000 Summer Mathematics Institute at Oakland University. We will discuss strategies for solving a wide range of such problems during the camp, but we would like to know more about your experiences and thoughts about problem solving at this stage in your mathematical career. You probably have experienced occasions where you worked on a problem and found it very hard initially, but a day or two later you tried again and solved it easily. So allow yourself a lot of time to work on these problems.
You do not need to work on all of the problems. For each problem you do attempt, you should submit a sheet of paper (or more) containing your solution if you have one, and a summary of yourserious thoughsts and efforts on the problem. This summary should indicate partial results, conjectures you considered and why, how you tested your conjectures and the results, of these efforts, and anything else you wish to say. Even if you have no solution, a partial solution, or something you think is wrong (say so!), send in your work on that problem.
Exercise your problem-solving skills. Sometimes it helps to look at similar but simpler problems, or special cases. Sometimes a problem can be given graphical, geometric, or other visual representation which suggests an approach. Keep good, organized data. Be playful and try to help your mind see connections between the special cases, your representations, and the general ideas needed for the actual proble,. Look for patterns. These patterns can sometimes provide insights into how to prove the general case.
You may look at material in libraries or other sources if you wish. Getting specific help from another person is not allowed. All help on these problems from any source should be acknowledged in your comments. We hope you enjoy working on these problems.
1. Do there exist irrational numbers r and s such that rs is rational? Do there exist irrational numbers r and s such that rs is irrational?
2. Does there exist an infinite collection of sets such that the intersection of every two distinct sets in the collection is nonempty, but the intersection of every three distinct sets in the collection is empty?
3. Consider the number 29876543. Determine (1) the number of digits it has; (2) its last three (rightmost) decimal digits; and (3) its first three (leftmost) decimal digits.
4. Find the sum of the 4536 numbers from 1,000 to 10,000 which have all their digits distinct. This is really two problems: Find a solution by using brute force on the computer; find a solution by doing it analytically by hand.
5. Three positive integers a, b and c form a Pythagorean Triple if a2+b2=c2. It is primitive if they have no common factors. For example, 3,4 and 5 form a primitive Pythagorean Triple, 5,12 and 13 form another primitive Pythagorean Triple. Find two more primitive Pythagorean Triples.
6. Suppose your worst enemy gives you n red points and n blue points in the plane, no three of them collinear. (They are not nicely arranged.) Show that it is possible to draw n line segments, each segment joining a red point to a blue point (so that each red point is paired with a unique blue point, and vice versa) in such a way that none of the line segments intersect.
7. A round-robin tournament was held among 13 players. Each player played one game against every other player. No game ended in a tie. Show that there exists a player K such that for every other player L, either K beat L, or K beat someone who beat L.
8. A round-robin tournament was held among 13 players. each player played one game against every other player. Each player won six games. How many triples (K,L,M) of players are there such that K beat L, L beat M, and M beat K?
9. Suppose n children holding loaded water pistols are standing in an open field, no three of them in line, such that all the distances between pairs of them are distinct. At a given signal, each child shoots the child closest to him or her with water. Show that if n is an even number, then it is possible (but not necessarily the case) that every child gets wet. Show that if n is odd, then necessarily at least one child stays dry.
10. A certain object consists of lots of tiny beads with pieces of string each one inch long joining certain pairs of beads. The attachments are such that no closed loops are formed. (In mathematical terms, we aregiven a tree.P There are many such objects. Find a procedure to determine the length of this object (or any other tree); i.e. The largest distance it can span.