2011 U of I Mock Putnam Problems

1. Let x=0.1.234567891011121314151617181912021...
(a) Find the 2011th digit after the decimal point of x.
(b) Show that x is irrational

2. There are 92 airports in Illinois. Suppose a flight takes off at each airport and lands in the nearest neighbouring airport. Assuming that mutual distances between the airports are all distinct, prove that there is no airport at which more than five planes land.

3. Find a simple formula (with  proof) of the sum S_n = \sum_1^n k/(k+1)!

Math Challenge Problems

1. Let P(x) = x^2011 + x^1783 -3x^1707 +  2x^341+ 3x^2-3  be a polynomial. Find the remainder obtained when P(x) is divided by x^3 - x.

2. How many ways can you make change for 50 cents using pennies, nickels, domes and quarters?  For example, 50 pennies is one way, 1 quarter and 5 nickels is another way.


If you have a solution one or both of the above problems, then please submit it in the math challenge problem drop box in STV 313.  

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.