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.