Problem Solving
Set 20
22 July 2012
1. Denote by Sn the group of permutations of the sequence
(1, 2, . . . , n). Suppose that G is a subgroup of Sn such that
for every G \ cfw_e there exists a unique
Problem Solving
Set 13
16 July 2012
1. Find all continuous functions f : R R such that
f (x) f (y) is rational for all reals x and y such that x y
is rational.
2. Is it true or false that for each rea
Problem Solving
Set 11
14 July 2012
1. Call a polynomial P (x1 , . . . , xk ) good if there exist 2 2
real matrices A1 , . . . , Ak such that
k
xi Ai .
P (x1 , . . . , xk ) = det
i=1
Find all values o
Problem Solving
Set 9
12 July 2012
1. Let x, y, and z be integers such that n = x4 + y 4 + z 4 is
divisible by 29. Show that n is divisible by 294 .
2. Find all continuous odd functions f : R R such t
Problem Solving
Set 10
13 July 2012
1. Let f : R R be a continuous function. Suppose that for
any c > 0, the graph of f can be moved to the graph of cf
using only a translation or a rotation. Does thi
Problem Solving
Set 5
08 July 2012
1. Let f be a real-valued function with n + 1 derivatives at
each point of R. Show that for each pair of real numbers
a, b, a < b, such that
f (b) + f (b) + + f (n (
Problem Solving
Set 8
11 July 2012
1. Let C be a nonempty closed bounded subset of the real line
and f : C C be a non-decreasing continuous function.
Show that there exists a point p C such that f (p)
Problem Solving
Answers to Problem Set 7
10 July 2012
1. Let n 2 be an integer. What is the minimal and maximal possible rank of an n n matrix whose n2 entries are
precisely the numbers 1, 2, . . . ,
Problem Solving
Set 6
09 July 2012
1. Let f be a polynomial of degree 2 with integer coecients.
Suppose that f (k) is divisible by 5 for every integer k.
Prove that all coecients of f are divisible by
Problem Solving
Set 14
17 July 2012
1. Denote by V the real vector space of all real polynomials in
one variable, and let P : V R be a linear map. Suppose
that for all f, g V with P (f g) = 0 we have
Problem Solving
Set 15
18 July 2012
1. Let p be a polynomial with integer coecients and let
a1 < a2 < . < ak be integers.
(a) Prove that there exists a Z such that p(ai ) divides
p(a) for all i = 1, 2
Problem Solving
Set 21
24 July 2012
1. (a) A sequence x1 , x2 , . . . of real numbers satises
xn+1 = xn cos xn
for all n 1. Does it follow that this sequence converges for all initial values x1 ?
(b)
Problem Solving
Set A01
2 October 2012
1. For every positive integer n, let p(n) denote the number
of ways to express n as a sum of positive integers. For
instance, p(4) = 5 because 4 = 3 + 1 = 2 + 2
Problem Solving
Set 17
20 July 2012
1. Two dierent ellipses are given. One focus of the rst ellipse
coincides with one focus of the second ellipse. Prove that
the ellipses have at most two points in c
Problem Solving
Set 20
23 July 2012
1. A, B are square complex matrices and rank(AB BA) = 1.
Show that (AB BA)2 = 0.
2. Consider the equation
f (x) = f (x + 1).
Is there a solution with f (x) as x ?
Problem Solving
Set 19
21 July 2012
1. Let 0 < a < b. Prove that
b
2
2
2
(x2 + 1)ex dx ea eb .
a
2. If n N has the property that for all x Z there exists
y Z such that x2 + y 2 1 (mod n), show that n
Problem Solving
Set 18
21 July 2012
1. Let n be a positive integer. Prove that 2n1 divides
0k<n/2
n
2k + 1
5k .
2. Find all ordered triples of primes (p, q, r) such that
p | q r + 1, q | rp + 1, r | p
Problem Solving
Set 12
15 July 2012
1. Let n > 1 be an odd positive integer and A = (aij )i,j=1,.,n
be the n n matrix with
2 if i = j
aij = 1 if i j = 2 (mod n)
0 otherwise
Find det A.
2. Show that
xy
Problem Solving
Set 16
19 July 2012
1. Let n, k be positive integers and suppose that the polynomial x2k xk +1 divides x2n +xn +1. Prove that x2k +xk +1
divides x2n + xn + 1.
2. Find all solutions in
Problem Solving
Answers to Problem Set 6
09 July 2012
1. Let f be a polynomial of degree 2 with integer coecients.
Suppose that f (k) is divisible by 5 for every integer k.
Prove that all coecients of
Problem Solving
Set 2
05 July 2012
1. (a) Let A be a n n, n 2, symmetric, invertible matrix
with real positive elements. Show that zn n2 2n,
where zn is the number of zero elements in A1 .
(b) How man
Problem Solving
Set 4
07 July 2012
1. Let f C 1 [a, b], f (a) = 0 and suppose that R, > 0
is such that
|f (x)| |f (x)|
for all x [a, b]. Is it true that f (x) = 0 for all x [a, b]?
2. What is the grea
Problem Solving
Answers to Problem Set 1
04 July 2012
1. Let f C 1 (a, b), limxa+ f (x) = +, limxb f (x) =
and f (x) + f 2 (x) 1 for x (a, b). Prove that b a
and give an example where b a = .
Answer
131-01-f11
Quiz 6
Name:
The function f (x), along with its rst and second derivatives, is given below.
f (x) =
2x
21
x
f (x) =
2(x2 + 1)
(x2 1)2
f (x) =
4x(x2 + 3)
(x2 1)3
1. Identify the domains of
131-01-f11
Quiz 2
Name:
(10 points total)
1. A particle starts by moving to the right along a horizontal line; the graph of the position y as a function of
time x is given below.
(a) Graph the velocit