PROBLEM CORNER

The Department of Mathematics and Computer Science is pleased to announce a project to be known as “PROBLEM COR: NER” and to be coordinated. by Dr. Tomasz Kaczynski. The aim of the project is to make available to students a set of interesting and challenging problems. Another aim is the encouragement of problem solving.

The basic rules of operation of the project are as follows:

— A set of problems shall be posted at the beginning of each month.

— Solutions to any of the problems will be welcomed until the end of the month and should be submitted in writing to Dr. Tomasz

Kaczynski, Memorial-104. (Please slide your submission under the _

door if Dr. Kaczynski is absent).

— Queries about the project and questions should be directed to Dr. Kaczynski.

— Prizes will be offered to the authors of correct and/or origi- nal solutions of any problem at a social gathering at the end of the semester.

SET 1-—January 1989

1. Show that the product of three consecutive integers is never a perfect square.

2. There are K people in a class (K > 2). Show that at least two of them must know the same number of people from among those present. (Assume that if A knows B then B knows A).

3. Solve the simultaneous system of equations

1+ 22 73 274 =2

Z2+21 2324 =2

4+ 2, 22 274 =2

tata: 22 23 =2

(Explain your work so to make it clear that you obtained all solutions).

Peri lous Advent. e ; KR

Th

4; If eo — my and for n 1, sa = on acd, find

limn—oo Zn. Prove your answer.

Solutions accepted until Wednesday, February 15 (date of post- ing solutions February 16) Only the first correct solution of a prob- lem qualifies for prize unless other solutions are based on different methods or ideas.

SET 2-FEBRUARY 1989

1. A hat contains 7 ballots labeled with the numbers 1, 2, 29. 23, 24, 25, 26, respectively. We randomly select ballots without re- placement until the sum of numbers printed on them exceeds 124. Which value of the obtained sum is most probable?

2. A French boat is fishing on Canadian waters without permis- sion. Each casting of their net into the sea brings a catch of equal value, but during each casting the probability of being caught by the coast guard is 1/k where k is a given integer. We assume that the event of being or not being caught during consecutive castings is independent of the number of previous castings. If the boat is caught by the coast guard, all the fish is confiscated and the fish- ing cannot continue.

The captain plans to return home after n castings. By taking account of the risk, the total gain during the cruise is a random variable. Find the number n which will make the expected value of that random variable greatest.

3. Decide whether the number z2=1+4+(1)(1+(1)1+9 \C.. 10 10? 10

is rational or irrational.

4. Prove that, for each positive integer n, the polynomial

P(z) = 1+ (2) + (22) + (#3) +... +(2%) . 1! 2! 3! n!

has no multiple roots (notation n! = 1x2x3x... x fp_-1 X Zn).

Solutions are accepted until the end of February.

The UPE! Business School

Accepts Donation

The UPEI Business School, j part of a four university grou including Memorial, Acadia, a the University of Moncton, 1 cently received the sum of 5 million dollars from the Atlant Canada Opportunities Agenc $1 to $1.25 million went direct to UPEI

Bob O’Raurke, Dean of UP] Business School, says the mond will be used to open the Busine Institute, a business which wi offer counselling, establish a dai base and provide development case studies for small business just starting off:

“The businesses will get cou selling in things such as marke ing plans, etc... and a case w be developed for each business.

The money was_ obtaing through a proposal submitted } the four universities in January | 1988. It was approved in Febr ary 1988.

- The Business Institute scheduled to open in April 198 and will be based downtown 4 Kent Street. It will prove bene cial to. UPEI business students, } some of them will be hired, alo with faculty members and othe to run the business.

By: Ellen Perry

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Elections

Student Union elections will be held, March 15 & 16, 1989. Advance polls will open on the 13th and 14th of March. Nominations for positions will open on Feb 17th at 9am. And close on Friday March 3rd at 4pm.

Nominations for the following posi- tions will be accepted beginning Feb 17th. They are as follows:

Four (4) Arts Representatives Two (2) Science Representatives Two (2) Senates — Mee One (1) MAPUS Representatives One (1) Student Ombudsman

Executive

Vice-President Communications Vice-President Finance Vice-President Operations Vice-President Academic President