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 k cfw_1, 2, . . . , n
for which (k) = k. (Here e is the
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 real number > 0 there exist
positive integers m and n such
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 of k for which all homogeneous polynomials
with k variab
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 that the
identity f (f (x) = x holds for all real x.
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 this imply that
f (x) = ax + b for some real numbers a and
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 (b)
ln
f (a) + f (a) + + f (n (a)
=ba
there is a number
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) = p.
(A set is closed if its complement is a union of
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, . . . , n2 ?
2. Show that every rational number p/q (0, 1) with
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 5.
2. (a) Given a triangle ABC show that there is a un
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 P (f ) = 0 or
P (g) = 0. Prove that there exist real nu
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, . . . , k.
(b) Does there exist an a Z such that the
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) A sequence y1 , y2 , . . . of real numbers satises
yn+1
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 = 2 + 1 + 1 =
1+1+1+1. Also dene p(0) = 1. Prove that p
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 common.
2. The set of pairs of positive real (x, y) such
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 divides
12.
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 | pq + 1.
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 + y x > 1
for all real x, y > 0
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 integers of
x3 + y 3 + z 3 = 3xyz.
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 f are divisible by 5.
Answer: This is really very easy
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 many zero elements are there in the inverse of the
n n mat
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 greatest sum that cannot be paid for in 2c and
5c coins?
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: I misread this when I rst read it (without my
glasses
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 each of f (x), f (x), and f (x):
(a) Domain of f (x):
(
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 velocity in the blank graph provided.
(b) For what x-values is