Math Circles at FIU
Big Circle

Fall '07 Problems suggested for work at home

Oct. 20 - Telescopic Sums and Products - Try these problems

Sep. 29
- Mathematical Induction

1. Given n lines in the plane, show that the regions into which the plane is divided can be colored with two colors (say Blue and Red), so that neighboring regions (regions that share a line segment boundary) have different colors.

2. Bertrand's postulate (now known as a theorem) states that for every number N>1, there exists a prime number between N and 2N. Use this to show that every integer greater that 6 can be written as the sum of one or more distinct prime numbers.

For example: 7=7, 8=3+5, 9=2+7, 10=3+7 (or 10=2+3+5), 11=11, etc.

Sept. 15 -

1. The integers from 0 to 104 inclusive are written clockwise on a circle. (It's a 105 hours clock!). A frog starts at 0 and jumps clockwise around the circle by a fixed difference d, until he ends up back at 0. For example, if d=35, he would jump to 35, 70 and then back to 0. For how many positive integers d such that d<105 will the frog jump onto every number before returning to 0?

2. (Josephus Problem, 1st century AD) Ten thousands sailors are arranged around the edge of their ship and they are numbered 1, 2, 3, ..., 10000. Starting the count with number 1, every other sailor is pushed overboard (by the mad captain) until they are all gone. Where should you be standing to be the last survivor.


2007 Calendar of meetings


Date
Topic Suggested Problems
Comments
Sep. 15
Arithmetic & Geometric Progressions
see above

Sep. 29
Mathematical Induction


Oct. 13 - no meeting
for Big Circle


Little Circle meets this day
Oct. 20
Telescopic Sums and Products
Try these problems

Nov. 3
The Pigeon Hole Principle