Berkeley Math Circle

Monthly Contests
2007-2008

Each month you will be given a set of five problems, and four weeks later we will collect your written solutions. (Please consult the Calendar for precise deadlines. Late solutions will be excluded from competition.) When the winners of the contests are announced, please click the little icon to the right of each week to see their names, scores, and prizes.

For contest rules, please click here.

Mira Bernstein: "A Few Words About Proofs" (PDF, PS)
	Problems	Solutions	Winners
MC1	(PDF,PS)	(PDF,PS)
MC2	(PDF,PS)	(PDF,PS)
MC3	(PDF,PS)	(PDF,PS)
MC4	(PDF,PS)	(PDF,PS)
MC5	(PDF,PS)	(PDF,PS)
MC6	(PDF,PS)	(PDF,PS)
MC7	(PDF,PS)	(PDF,PS)
MC8	(PDF,PS)

Your solutions will be evaluated by BMC staff. We ask you to write clearly your name, school and city, and to identify the contest number and problem you are solving on top of each of your solution sheets. If you present solutions to more than one problem, it is important that you write the solutions on separate sheets, since your solutions may be graded by different people.

It is even more important that your solutions are correct, complete, and clearly written. You should not expect much credit if your proof refers to a false statement, or even if all your statements are true but you forgot to tell us "why?" It is one of the purposes of the Circle to help you improve your "essay-proof" writing style as well as your logical skills.

Please remember that the Contest is individual. Although we strongly encourage cooperation and help among the participants of the Circle, the Monthly Contest will be one exception to this rule: you may consult your notes, books and other published literature, but you may not ask other people to help you. Other than that, the Contest should offer you the joy of problem solving in an atmosphere close to individual scientific research where the pressure of time limitations is reduced to the minimum. Our intention is to select problems whose solutions will eventually lead you to important pieces of mathematical theory. The more effort you invest into the solutions of your favorite problems, the more you will be able to appreciate the theory behind them. This remains true regardless of whether you manage to complete your own solutions or not. So, you should not give up too easily!

Each month several winners will be awarded prizes at the Math Circle. At the end of the year, a few grand prizes will be given for highest overall success, and probably for brightest solutions.

The Monthly Contest will be coordinated by Ivan Matic (left). Please direct any questions to him. Evan O'Dorney (right), a BMC student, will be helping him this year.

All materials here are copyrighted by Berkeley Math Circle; this includes both problems and solutions.

We allow other math circles to use these materials in their own sessions, however, these circles must observe the schedule for the Monthly Contest given below and never discuss these problems before the student solutions will be due in the Berkeley Math Circle. Selling or publishing the materials in this or any other section of the website without the explicit permission of Berkeley Math Circle is prohibited.

Contest Rules

LOGISTICS

You normally have 3 to 4 weeks to solve the five assigned problems; any schedule changes will be announced. Late solutions will not be accepted; if you know in advance that you cannot come to the Circle on the due date, please arrange to submit solutions earlier.
You may consult any aids - books, notes, etc. - but you may not receive help from other people. Of course, you are on an honor system in this regard.
Please submit your solutions to different problems on separate pieces of paper, with your name, school, and city, the contest and problem number on each sheet.

WINNER'S HANDICAP

Depending on funds, there will be serveral awards given for each Monthly Contest to the best papers in each of three age groups.
In order to encourage everybody to participate and in order to make our competition more fun and challenging for stronger participants, we have enacted a "Winner's handicap" rule.

Anyone who has been awarded a prize in a particular contest will receive a 7-point handicap (a deduction of seven points by default) in every subsequent competition as long as he or she is receiving a prize. If someone manages to win a prize in a contest despite the handicap of 7 points, next time he or she will have a handicap of 14 points, etc. If someone doesn't receive a prize in a contest next time, he or she will compete without handicaps.

GRADING

The solutions will be graded on a scale of 0 to 7 points, based on completeness, accuracy, and quality of presentation. All five problems are equally weighted.
Extra credit may be given for a substatial generalization, but not frequently
If a submitted solution is incorrect, the grader will indicate why (if the reason is not presumed to be obvious.
Because the objective of the contest is to encourage mathematical ingenuity, solutions based on excessively lengthy calculations or solutions which are trivialized by the problem using a computer program may not receive full credit.

You may refer to well-known results (or less-known results, if you provide a reference), so long as they do not trivialize the problem. For example, if a problem asks for a proof of Fermat's Little Theorem, a solution that says "This is Fermat's Little Theorem; a proof is given in [book, page]" will receive no credit. On the other hand, in a much more difficult problem, it could be appropriate simply to refer to Fermat's Little Theorem when it must be used. When in doubt, include as many proofs as possible.
The graders' decisions are final and cannot be appealed

These rules are subject to change; any modifications will be thoroughly announced.