A Wetpaint Site | TV Fandex 100

FIU Math Club

Weekly Problems

Problems for October 6th-October 10th, 2008:

1. Is there an infinite sequence

of nonzero real numbers such that $n=1,2,3,\ldots$ the polynomial
$p_n(x)=a_0+a_1x+a_2x^2+\cdots+a_nx^n$
has exactly

distinct real roots?

2. Let

be a finite group of order

generated by

and

. Prove or disprove there is a sequence

$g_1,g_2,g_3,\ldots,g_{2n}$

such

(1) every element of

occurs exactly twice, and
(2) $g_{i+1}$ equals

, for $i=1,2,\ldots,2n$ . (Interpret $g_{2n+1}$ as

3. To each positive integer with

decimal digits, we associate the determinants of the matrix obtained by writing the digits in order across rows. For example, for , to the integer 8617 we associate

. Find, as a function of n, the sum of all the determinants associated with

-digit integers. (Leading digits are assumed to be nonzero; for example, for n=2, there are 9000 determinants.)

4. Let

be differentiable (real-valued) functions of a single variable t which satisfy

for some constants

. Suppose that for all i,

. Are the functions

necessarily linearly dependent?

Problems for September 28th-October 3rd, 2008:

1. Evaluate

Weekly Problems - FIU Math Club

Express your answer in the form

where a,b,c,d are integers.

Hint:

2. Sum the series

3. Find the minimum value of

for x >0.

Problems for September 22-26th, 2008:

(-1) Let p be an odd prime and let

denote (the field of) integers module p. How many elements are in the set

0. Let f be an infinitely differentiable real-valued function defined on the real numbers. If

for

Compute the values of the derivatives

, for

(1) Let S be a finite set of integers, each greater than 1. Suppose that for each integer n there is some

such that

. Show that there exist

such that

is prime.

2. Prove that the expression

is an integer for all pairs of integers

(

) Let

denote the set of all permutations of the numbers

. For

, let

is an even permutation and

is an odd permutation. Also, let

denote the number of fixed points of n.

Show that

4. Let

be a polynomial of degree n, all of whose zeros have absolute value 1 in the complex plain. Put

. Show that all zeros of g'(z)=0 have absolute value 1.

5. Let n be an integer greater than 1. The positive divisors of n are

where

. Define

(a) Prove that

.
(b) Determine all n for which D is a divisor of

.

6. Show that for any positive integer n, there is an integer N such that the product

can be expressed identically in the form

where the

's are rational and each

is one of the numbers -1,0,1.

7. Let S be a nonempty set with an associative operation that is left and right cancellative ( xy=xz implies y=z, and yx=zx implies y=z). Assume that for every a in S the set

is finite. Must S be a group?

Problems for September 15-19th, 2008:

Prove the following inequalities:

1.

Problems for September 8-12th, 2008:

1. Let T be an acute triangle. Inscribe a pair R, S of rectangles in T as shown:

Let

denote the area of polygon X. Find the maximum value, or show that no maximum exists, of A(R)+A(S)/A(T)

, where T ranges over all triangles and R, S over all rectangles as above.

2. Define polynomials for

for $n\geq0$ by f_0(x)=1

, and

$\frac{d}{dx}(f_{n+1}(x))=(n+1)f_n(x+1)$ for

.

Find, with proof, the explicit factorization of

into powers of distinct primes.

Problems for July 21-25th, 2008

1. Find all functions

such that:

for all real

satisfying

2. (Proposed by Rafael Badui) If

and

are two invertible

matrices such
that

and

, find all possible values of

.

3. Consider the lattice of points in the plane having both coordinates integers.
Show that for any positive integer

, there exists a circle containing exactly

lattice points in its interior.

4. Consider a line in the plane, a point

on this line and

vectors of length

, with common origin

. Suppose that all the vectors are situated in one of the
semi-planes determined by the line. Prove that if

is an odd, then the sum of the
vectors is a vector of length at least

Problems for July14-18th, 2008

(Solutions by Rafael as a pdf)

(Comment from Prof. T. Draghici: The first two are logic/strategy problems given to me by Professor S. Hudson.)

1. There are five pirates, call them

, according to their seniority
(

is the oldest,

is the next, and so on). They have a capture of

golden coins
and they want to share it. The sharing procedure runs as follows: First, pirate

makes a sharing proposal. This goes to a vote among the pirates. If the proposal
passes, then each takes its share and that is it. If the proposal fails, then pirate

is killed, pirate

gets to make a proposal and the procedure continues as before.
It is known (to the pirates also) that each pirate has the following priorities, in
this order: (1) wants to be alive; (2) wants to be rich; (3) would like the other
pirates dead.

If you were pirate

, what would you propose?

(One more detail: In case of a tied vote, the vote of the most senior pirate breaks
the tie.)

2. Ten people are lined up in a row and hats are put on their heads. The hats
are either black or white. Each person can see the color of the hats of the people in
front of him, but cannot see the the hat on his head, nor those behind him. Then
each person is allowed to say ”Black” or ”White” and nothing else. The group gets
a point for each correct guess (i.e. an answer matches the color of the hat of the
person who gave the answer). If the group is allowed to discuss a strategy prior to
the experiment, what is the maximum number of points they can score and how do
they do it?

3. Prove that the equation

has infinitely many integer solutions.

4. Prove that there exists infinitely many positive integers a, so that

is
not prime for any positive integer

a314sces	Latest page update: made by a314sces , Oct 9 2008, 1:17 PM EDT (about this update About This Update new problems - a314sces 56 words added 1 word deleted 18 images added view changes - complete history)
	Keyword tags: problems suggested problems More Info: links to this page