Some Problems in Number Theory

1.  Show that if n is a composite number then n divides (n-1)!

2.  Find the last two digits of the number 5^5^10000.

3.  Is log_10 2 rational or irrational?

4. How many zeros are there at the end of 100!

5. Can the number 111111111....1  (a string of n ones, where n >1) be a perfect  square?

An interesting geometry problem

This is not a Putnam problem, but it could be one. I have been struggling to prove this. If you can try and
let me know if have any success, that would be great.

Consider an acute angled triangle ABC. (This is a traingle in which all 3 angles are less than 90 degrees. ) Let PQR be a light triangle inside ABC. That is, it is a triangle inscribed by a light ray that starts on a point P on AB and then hits a point Q on BC, gets reflected and hits a point R on CA, and finally returns to a starting position A after a second reflection. These rays satisfy the usual property (angle of incidence = angle of reflection) of light at each of the points P, Q, and R.  Show that PRQ is an altitude triangle. That is, P, Q and R are the feet of the perpendiculars drawn from C, B and A (respectively) to to their opposite sides. 

Some exercises

Here are some problems for you. Note that the problems on the Putnam exam will be more difficult than these.

1. Show that for any positive integer N (no matter however large) there exists N consecutive composite number. For exam, if N  = 3, then  14, 15, 16 is a string 3 composite numbers (not primes).

2. Show that for any non-zero real number x, the absolute value of  x + 1/x is at least 2. (show this directly without using know inequalities such as GM < = AM, etc.

3.  This is the problem of the week:   If T_n is the nth triangular number, that is T_n = n(n+1)/2, then show that the T_n can never leave a remainder of 2 or 4 when it is divided by 5.

Putnam Exam 2008

This section is not just for problems and solution for Putnam 2008, but all stuff related to our second week.
By the way, now you should feel free to see solutions of the Putnam 2009 problems, especially those which you
tried and were not successful.

MAT 268: Introduction to Undergraduate Research in Mathematics

Here is a very exciting opportunuity in Spring 2011 for students with a strong mathematical background. MAT 268: "Introduction to Undergraduate Research in Mathematics" will be offered in Spring 2011. Please read this  announcement   about it and let know if you are interested. I think it would be a fun
course.

Putnam Exam 2009

Use this place to post your ideas, solutions, comments, generalisations for Problems from the 2009 Putnam exam.  The first one (A1) is not difficult. (As said before, the first problem is often the easiest problem.)

ISU Putnam homepage

All information you need to know and more about Putnam Competition is available at the  ISU Putnam Homepage.

Welcome to the Putnam training sessions

Dear Enthusiastic Problem Solvers!

I just created this blog to make our Putnam preparation more fun!  We will meet only once a week to go over some problems and to learn new problem solving techniques.  The main purpose of this blog is to serve as a forum for online discussions between our weekly meetings. Here,  you can explain your solutions (complete or partial) to the problems I will assign you every week. You can also post problems and brain-teasing questions and puzzles. So start blogging now ...  I hope this will be helpful and you will enjoy learning by participating in discussions here.  Please let me know if you have any suggestions for me to make this whole preparation more fun! Our goal is to prepare for the Putnam competition, but the key  to that is to enjoy solving problems no matter how hard the problems are.