Practice Exam, Math 194
Thursday, Nov. 7, 2013
1. Suppose S is a set with 10 elements. How many subsets of S have an odd number of
elements?
2. For every positive integer k , let f1 (k ) denote the sum of the squares of the base 10
digits of k . For n 2 l
Math 194, solutions for problem set #4
For discussion Thursday October 24
(1) What is the remainder when the polynomial f (x) is divided by (x a)2 ? by x2 a?
Solution: Let n be the degree of f (x). The Taylor expansion of f (x) at x = a
1
1
is f (x) = f (
Math 194, solutions for problem set #6
For discussion November 14
(1) The horizontal line y = c intersects
the curve y = 2x 3x3 in the rst
quadrant as in the gure. Find c
so that the areas of the two shaded
regions are equal. (Putnam, 1993)
Solution: Let
Putnam problems and solutions A1
(Many solutions are taken directly from http:/www.unl.edu/amc/a-activities/a7-problems/
putnamindex.shtml where the authors are properly attributed. Others are from The William
Lowell Putnam Mathematical Competition, 1985-
Math 194, solutions for problem set #7
For discussion November 21
(1) Let 0 < xi < , i = 1, ., n and set x = (x1 + + xn )/n. Prove that
n
i=1
sin xi
xi
Solution: Let f (y ) = ln
sin x
x
sin(y )
y
n
.
(Putnam, 1978)
= ln(sin(y ) ln(y ), and compute f (y )
Math 194, solutions for problem set #3
For discussion Thursday October 17
(1) Prove that 3636 + 4141 is divisible by 77.
Solution: Notice that 3636 + 4141 (1)36 + (1)41 0 (mod 7). Also,
since (3)5 243 1 (mod 11), 3636 + 4141 (3)36 + (3)41 (3)36 +
(3)36+5
Math 194, solutions for problem set #2
For discussion Thursday October 10
1. Show that if the fraction a/b is expressed as a decimal number (where a, b are positive
integers), it either terminates, or begins repeating after most b 1 decimal places.
(Hint:
Practice Exam, Math 194
Thursday, Nov. 7, 2013
1. Suppose S is a set with 10 elements. How many subsets of S have an odd number of
elements?
Fix one element s of S , and let S be the set obtained by removing s from S . There
is a one-to-one correspondence
Pre-Putnam Exam Solutions
1. Find all polynomials p(x) with real coecients satisfying the dierential equation
d
7 [xp(x)] = 3p(x) + 4p(x + 1),
< x < .
dx
Solution:
Suppose we have a solution of degree n, so that p(x) = an xn + an1 xn1 + + a1 x + a0 . By
Math 194, solutions for problem set #1
For discussion Thursday October 3
1. Show that some multiple of 1232123432123454321 contains all 10 digits (at least once)
when written in base 10.
Solution 1: Let n = 1232123432123454321. Consider the n consecutive
Pre-Putnam Exam
This exam was designed to be taken in 3 hours without notes, books, calculators, collaboration, or
interruption. Good luck.
1. Find all polynomials p(x) with real coecients satisfying the dierential equation
7
d
[xp(x)] = 3p(x) + 4p(x + 1)
Putnam problems A1
2006. Find the volume of the region of points (x, y, z ) such that
(x2 + y 2 + z 2 + 8)2 36(x2 + y 2 ).
2005. Show that every positive integer is a sum of one or more numbers of the form
2r 3s , where r and s are nonnegative integers an