# Information

The Bay Area Mathematical Olympiad (BAMO) is an annual competition for hundreds of Bay Area middle and high school students, consisting of 4 or 5 proof-type math problems with a time limit of 4 hours. It is typically held on the last Tuesday of every February. This year's test will be on**February 24, 2015**. The exams are proctored at schools and at several open sites around the Bay Area. They are graded the following weekend by a group of Bay Area mathematicians, teachers, and Math Circle enthusiasts.

The BAMO awards ceremony, which takes place the weekend after the grading, has become an annual focal point for the Bay Area middle and high school math activities; about 200 students, teachers and parents gather for an exciting day of Mathematics, including:

- a math talk by a distinguished mathematician
- the various awards distributed to three age groups
- lunch for everybody.

The BAMO organizing committee members are Ian Brown (Proof School), Zvezdelina Stankova (Mills College and UC Berkeley), Paul Zeitz (University of San Francisco), and Joshua Zucker.

## Sample BAMO problems

**Problem 2**, 2007 The points of the plane are colored in black and white so that whenever three vertices of a parallelogram are the same color, the fourth vertex is that color, too. Prove that all the points of the plane are the same color.

**Problem 4**, 2005 There are 1000 cities in the country of Euleria, and some pairs of cities are linked by dirt roads. It is possible to get from any city to any other city by traveling along these roads. Prove that the government of Euleria may pave some of the roads so that every city will have an odd number of paved roads leading out of it.

Please visit our BAMO Archives for solutions to these problems and the complete collection of problems and solutions from past Olympiads.

## Sample problems for BAMO for Teachers

### Sample pedagogical problem

A student stubbornly insists that (x+1)^{2} = x^{2} + 1^{2}. Demonstrate that you understand what the student might be thinking, and that you have several different methods at your disposal to convince the student that this formula is not correct and lead them to the correct formula. (Explain at least three ways you could approach this situation.)

### Sample math problem

(From Devlin’s Angle, Dec 2009):A new assistant accidentally left open the cages at the pet shop, and over 100 birds escaped. There were exactly 300 birds to begin with. The next morning, the local newspaper carried a report that gave the following figures:

Of the birds that remained, a third were finches, a quarter were budgies, a fifth were canaries, a seventh were mynah birds, and a ninth were parrots.

However, the reporter got one of the fractions wrong. How many parrots were left?

### Sample utility problem

A triangle with sides of length 13, 14, and 15 also has at least one integer altitude. You want a similar problem to use on a quiz. Explain how to produce more triangles like this, with integer sides and at least one integer altitude. (Similar triangles don’t count; your example triangles should have sides with greatest common factor 1.)