[QUOTE=jyb;212629]Whether or not you are wasting your time depends on what result you would consider worthwhile. If blob100 is learning but not indicating it here (or indeed acknowledging your help), would that be a waste of time? If he's learning nothing, but others on this forum are benefiting, would that be a waste of time?

My sense is that even if blob100 never comes back, there are enough people on this forum who would profit from your problems/examples that there is value in continuing in this vein. Lots of people need a brush-up or simply enjoy the puzzle aspect.

As for blob100, remember that he's young and needs time to absorb new concepts. Be patient with him; what can it hurt?[/QUOTE]

I'm trying to teach him. Part of teaching requires that the student
attempt problems for himself/herself, perhaps guided by hints from the
teacher. After attempts are made, the teacher can show the student
where the gaps are, where the errors are, and what needs to be studied
further. But a teacher can not possibly know where the difficulties are
until the student has made an attempt.

Learning math is hard. If blob100 (Tomer) really wants to learn, then he
needs to put in the effort. Otherwise, he is just another wannabee. I
(and others) have nearly laid out the entire solution for my problem (1),
but we have not seen any work from him. He *claimed* to know how to
do induction. But until we see his attempt, we can't possibly know where
he lacks understanding.

I am willing to commit my time to teach. He needs to show that either
he is willing to put in a strong effort, or tell us that he doesn't want to
bother.

[QUOTE=R.D. Silverman;212637]I'm trying to teach him. Part of teaching requires that the student
attempt problems for himself/herself, perhaps guided by hints from the
teacher. After attempts are made, the teacher can show the student
where the gaps are, where the errors are, and what needs to be studied
further. But a teacher can not possibly know where the difficulties are
until the student has made an attempt.

Learning math is hard. If blob100 (Tomer) really wants to learn, then he
needs to put in the effort. Otherwise, he is just another wannabee. I
(and others) have nearly laid out the entire solution for my problem (1),
but we have not seen any work from him. He *claimed* to know how to
do induction. But until we see his attempt, we can't possibly know where
he lacks understanding.

I am willing to commit my time to teach. He needs to show that either
he is willing to put in a strong effort, or tell us that he doesn't want to
bother.[/QUOTE]

Did we drive him (Tomer) away?

Does anyone else want to take a crack at the problems that I posed?

To go along with those problems, here is a set of exercizes in polynomial
algebra:

Throughout these exercizes let f(x) = a_n x^n + a_{n-1} x^(n-1) + .... a_0.

Suppose that its roots are x_0, x_1, .... x_n.
a_i are integers.

(1) Find the coefficients of a polynomial whose roots are 1/x_0, 1/x_1, ... 1/x_n.

Is this polynomial unique?

(2) Assume that we have the same polynomial and roots as in problem 1.
Express the coefficient of x^k as a function of the roots.

(3) Prove the remainder theorem:

The remainder left when f(x) is divided by (x-r) equals f(r).

(4) Prove the rational roots theorem:

If alpha = a/b is a rational root of f(x) then b divides a_n
and a divides a_0.

(5) Prove that any polynomial of odd degree must have a real root.

(6) Prove that if a_n = 1, (known as a monic polynomial), then
any rational root must be an integer.

(7) Solve x^4 + x^3 + x^2 + x + 1 = 0. Hint: use the result from problem
(1).

Note that this problem can be solved without knowing how to solve quartics
in general.

(8) Suppose all of the coefficients of f(x) equal 1. Prove that the absolute
value of all of the roots must equal 1.

[QUOTE=R.D. Silverman;212942]
(5) Prove that any polynomial of odd degree must have a real root.

[/QUOTE]

Try it [b]without[/b] using any results from Calculus. i.e. do not use
the intermediate value theorem.

Here are some questions about the circle/lines:

Is this correct?

[tex]regions = 1 + \sum_{n=0}^\infty n[/tex]

Verbally, can we make the claim that if each new line has two distinct intersections with the circle, and subsequent lines each have one intersection occurring x degrees CW from the previous and the other intersection occurring 2x degrees CW from the other intersection, each new line will intersect with all previous lines, creating the maximum new regions possible? From this, can we further claim that as x approaches 0, lines and therefore regions can approach infinity?

(BTW, I am very much an amateur "hobbyist" with no aspirations, merely a high interest, with an ancient background. Thank you for your review problems.)

[QUOTE=EdH;212952]Here are some questions about the circle/lines:

Is this correct?

[tex]regions = 1 + \sum_{n=0}^\infty n[/tex]
[/QUOTE]

Essentially, yes.

[QUOTE]

Verbally, can we make the claim that if each new line has two distinct intersections with the circle, and subsequent lines each have one intersection occurring x degrees CW from the previous and the other intersection occurring 2x degrees CW from the other intersection, each new line will intersect with all previous lines, creating the maximum new regions possible? From this, can we further claim that as x approaches 0, lines and therefore regions can approach infinity?
[/QUOTE]

This, on the other hand makes no sense to me. The last part
"lines and therefore regions can approach infinity" is total gibberish.

Thanks!

I'm growing fond of the term "gibberish" and am not in disagreeance. I just no longer have the terminology to express mental imagery in meaningful ways.

In an attempt to portray my "picture," let me try to describe my thought process:

The circle contains a region.

The first line, as long as it intersects and is not tangent, creates two regions.

The first line intersects at point1 and point2 on the circle.

No matter where we place a second line, as long as it intersects the first within the circle, we will create two more regions.

However, for maximum benefit later, let's choose a point of intersection one-third of the distance from point1 and intersecting at a small angle a. This will result in a line with two new intersections with the circle, point3 and point4. The distance between point3 and point1 will be ~one-half the distance from point4 to point2.

If we were to continue in the same direction around the circle with odd points ~one-half the distance as even points, each new line would intersect all others, until we reached the starting line.

However, the smaller we make angle a, the more lines that can be added before we reach the first line.

As a approaches 0, would not the number of lines be able to approach infinity?

Now, to try to figure out how to describe this in proper terms.....

[QUOTE=EdH;212964]
As a approaches 0, would not the number of lines be able to approach infinity?

.....[/QUOTE]

I already gave a construct that showed one can always place an unbounded
number of lines .

[quote=R.D. Silverman;212576]Am I wasting my time trying to lead him through these problems?
Am I wasting my time trying to point out things he needs to learn
[B]before[/B] he attempts learning Calculus?

While Number Theory is irrelevant to the latter, it does teach how to
put together simple proofs. blob100 is failing in this regard.

Is he just another wannabee who is unwilling to put in the effort needed
to actually learn mathematics? There are already too many of those in this
forum.

If he [B]does[/B] want to continue, then I will help. But I need to see
the work he has put in.

The next set of exercizes will involve polynomial algebra, since it is clear
that he lacks skills in this area.[/quote]

I can't understan one thing M.r Silverman.
I didn't enter the forum for a week becuase of reason I don't even need to tell about.
Yes, I solved the problem, and I try to solve the others.
I disappointed? If you think you waste your time, just don't try to help me. I'll learn by myself (of course, harder and I will get less...).
I had a problem translating the proof, but I'm close to finish.
You must understand that saying athing as you said, will just make people aviod you, this forum, and the most horrible, be avoided of mathematics.
Why will you give me exercizes on areas which I'm lack of skills on them? (last two senteces of your letter).
First I need to learn, then have some exescizes.
I must say this is horrible saying, mostly to a teenager with a passion:
"While Number Theory is irrelevant to the latter, it does teach how to
put together simple proofs. blob100 is failing in this regard."
Maybe I should leave the forum.

The solve of problem (1):
Let the formula be:
F(N)=F(N-1)+N.
For N=1, F(1)=F(0)+1=2, which is true.
From now, we try to prove that if the formula is true for N, it is true for N+1.
If we want a maximal areas for N+1 lines, the line number N+1, will be needed to cross the whole other lines and of course second times the circle.
*If we will add a new line, every area it crosses, will devided to two areas.
*If The new line is crossed by two consecutive places (places that between then there is no cross of a line with the line), we say, the new line made a new area (devided an area).
Easily we say there are N+1 consecutive pairs, and so, N+1 new areas.
We say F(N+1)=(F(N)=last areas)+(N+1=new areas).

[quote=R.D. Silverman;212576]
Is he just another wannabee who is unwilling to put in the effort needed
to actually learn mathematics?[/quote]
What do you mean by "wannabee"? I'm wannabee what?
I didn't say I'm good..
And I do make an effort learning mathematics, I even spend much time.
As I said, I was not able to be hear, the reasons are not your business...

I think my learning will need to get more organized:
1) exersizes with you.
2) books you recommend me to read (to spend my time on reading alone and increasing my knowledge).
Please call me Tomer, blob100 isn't my name.

Thanks, And please.
Tomer.