Problem Seminar. Fall 2013.
Problem Set 6. Algebra.
Classical results.
1. Vandermonde. Let
V =
Then det(V ) =
1i<j n (xj
1
x1
x2
1
.
.
.
1
x2
x2
2
.
.
.
.
.
n
x1 1 xn1
2
1
xn
x2
n
.
.
.
n
xn1
.
xi ) .
2. CayleyHamilton. Given an n n matrix A the charac
Math 345  Algebra I
September 14, 2016
1
Basic Group Theory
Abstract algebra is all about taking algebra and making it abstract. For example, we will be replacing
numbers by symbols; we will be replacing basic laws of arithmetic by axioms. Then we will f
Name
Problem Solving seminar  Team selection contest
Friday September 28th, 2012, 3 hours
Write down only your nal solutions in this booklet. Justify your answers.
Problem Your score
1
2
3
4
5
6
Total
1. Three pasture elds have areas of 10/3, 10 and 24 a
Problem Seminar. Fall 2013.
Problem Set 3. Inequalities.
Classical results.
1. Jensen. For any convex function f , real x1 , x2 , . . . , xn and nonnegative real 1 , 2 , . . . , n
such that n=1 i = 1
i
f (1 x1 + 2 x2 + . . . + n xn ) 1 f (x1 ) + 2 f (x2
Problem Seminar. Fall 2013.
Problem Set 2. Induction.
Classical results.
1. Finitely many lines divide the plane into regions. Show that these regions can be colored by two
colors in such a way that neighboring regions have different colors.
2. Ramseys th
Problem Seminar. Fall 2013.
Problem Set 1. Proofs by contradiction.
Classical results.
1. Prove that there are innitely many prime numbers. (Recall that an integer p > 1 is prime if its
only divisors are p and 1.)
2. Recall that
e=
n=0
1
1
1
1
= 1 + 1 + +
Problem Solving seminar

Team selection contest
1. A walker and a jogger travel along the same straight line in the same
direction. The walker walks at one meter per second, while the jogger runs at
two meters per second. The jogger starts one meter in f
Problem Seminar. Fall 2013.
Problem Set 4. Number theory.
Classical results.
1. Euler. For a positive integer n and any integer a relatively prime to n one has
a(n) 1 (mod n),
where (n) is the number of positive integers between 1 and n relatively prime t
Problem Seminar. Fall 2013.
Problem Set 5. Geometry.
Classical results.
1. Triangle area. Let ABC be a triangle with side lengths a = BC , b = CA, and c =
AB , and let r be its inradius and R be its circumradius. Let s = (a + b + c)/2 be its
semiperimeter
Problem Seminar. Fall 2013.
Problem Set 9. Miscellaneous.
Putnam 2007.
A1. Find all values of for which the curves y = x2 + x + 1 and x = y 2 + y + 1 are
tangent to each other.
A2. Find the least possible area of a convex set in the plane that intersects
Problem Seminar. Fall 2013.
Problem Set 8. Combinatorics.
Classical results.
1. The equation
x1 + x2 + . . . + xr = n
has exactly
n+r1
r1
nonnegative integer solutions.
2. Consider a convex polygon with n vertices so that no 3 diagonals go through the sa
Problem Seminar. Fall 2013.
Problem Set 7. Calculus.
Classical results.
1. Every continuous mapping of a circle into a line carries some pair of diametrically opposite points to the same point.
2. Mean value theorem. If f : [a, b] R is a differentiable fu
Math 345  Algebra I
December 1, 2016
1
Basic Group Theory
Abstract algebra is all about taking algebra and making it abstract. For example, we will be replacing
numbers by symbols; we will be replacing basic laws of arithmetic by axioms. Then we will for