function
algebra
polynomial
geometry
trigonometry
complex analysis
real analysis

[b](a)[/b] Suppose there exist a constant $ M > 0$ and a positive integer $ n$ such that an entire function $ f(z)$ satisfies
\[ |f(z)| \leq M \cdot |z|^n
\]
for all $ z \in \mathbb{C}$ with sufficiently large modulus. Must $ f$ necessarily be a polynomial whose degree does not exceed $ n$? Prove or disprove.

[b](b)[/b] Which entire functions are uniformly continuous on the complex plane?

[b](c)[/b] Let $ f(z)$ be an entire function such that
\[ \left|f\left(\frac1{\log(n \plus{} 2)}\right)\right| < \frac1n
\]
for all positive integers $ n$. Must $ f$ necessarily be identically zero?

[b](d)[/b] Must an entire function whose real part is positive troughout the complex plane be constant?

[b](e)[/b] Must the range of a nonconstant entire function be dense in the complex plane?

[b](f)[/b] Can an entire function which is not a polynomial have no roots? Exactly $ 2008$ roots? Infinitely many roots? Uncountably many roots?

To solve these questions, please use elementary arguments (otherwise some are completely trivial).

This means: you can use Cauchy-Goursat and Cauchy, Morera, the Cauchy estimates, the maximum modulus principle, ... but nothing more advanced.

$ \textbf{(a)}$
Let $ |f(z)|\leq M|z|^k$.
If $ f(z)$ is entire, then it can be written as a Taylor series with center at the origin:
\[ f(z)=\sum_{k=0}^{\infty}a_n z^n\]
with each coefficient given by Cauchy's Formula:
\[ a_n=\frac{1}{2\pi i}\mathop\oint\limits_{C^+}\frac{f(z)}{z^{n+1}}dz;\quad |C|=R.\]
Then:
\begin{align*}
|a_n| &\leq \frac{1}{2\pi}\frac{M R^k}{R^{n+1}}2\pi R \\
&\leq \vspace{10pt}\frac{M R^k}{R^n}.
\end{align*}
Note for very large $ C^+$, the expression $ \frac{R^k}{R^n}$ tends to zero when $ n>k$.  This means that $ f(z)$ must be a polynomial of degree not greater than k.

(b)
Let's assume $ f(0) \equal{} 0$, by translating f(0) back to origin if necessary.
Then $ \frac{f(z)}{z}$ is bounded near 0.
The only other place where $ \frac{f(z)}{z}$ may blow up is the behaviour towards infinity.
Suppose $ |z\minus{}w| < \delta$ implies $ |f(z) \minus{} f(w)| < 1$.
For an arbitrary $ z$, consider $ z_1 \equal{} 0, z_2,...,z_n\equal{}z$, where $ z_i \equal{} \frac{iz}{n}$ are all on the line segment from 0 to z, and $ \frac{\delta}{2} < \frac{|z|}{n} < \delta$. Then
$ |f(z)| \leq \sum |f(z_{i\plus{}1}) \minus{} f(z_i)| \leq  n < \frac{2|z|}{\delta}$
So $ \frac{f(z)}{z}$ is entire and bounded, thus it is constant.

That means $ f$ must be linear.

(c)
Yes. Let us consider the local power series at 0, say $ f(z) \equal{} z^k g(z)$, with $ g(0) \neq 0$. Plugging int $ \frac {1}{log(n \plus{} 2)}$, it would mean that $ |g(\frac {1}{log(n \plus{} 2)})| \leq \frac {log(n \plus{} 2)^k}{n}$
Since L.H.S. converges to 0 as $ n \Rightarrow \infty$, by continuity of $ g$, g(0) = 0, contradiction.

(d)
Consider $ g(z) \equal{} e^{ \minus{} f(z)}$. By the given condition, it is claer that $ |g(z)| < 1$. Thus by Liouville's theorem, $ e^{ \minus{} f(z)} \equal{} C$ a constant. In particular, the real part of $ f$ is constant. Then by Cauchy Riemann equations, the imaginary part is also constant. Thus f is constant.

(f)
$ e^z$ no root.
$ e^z\minus{}1$ countably many roots.

Regarding $ \textbf{(d)}$, why is there no in-between?  That is, either an entire function is constant or it's real and imaginary components oscillate wildly with increasing amplitude and without bounds.  That is, any (constant) bounds upper or lower on it's real or imaginary part implies it's constant.  What property implicitly granted by Laplace's equation imparts this geometry to harmonic functions?  I don't know.

An entire function is either a polynomial or has en essential singularity at infinity. I mean, we have to look at neighborhoods of infinity in order to be able to get the denseness, so certainly this argument brings some understanding of the issue. Essential singularities are complicated. On the other hand, the polynomials do not behave like you said.

[quote="Kalle"] the polynomials do not behave like you said.[/quote]

ok, thanks . . .

Hi Arne.  I could think to use the proof of Picard's First Theorem to show non-constant entire functions are dense in $ \mathbb{C}$ but that's not on the list above.  I wouldn't know how to do it otherwise.   I'll review the theorems you suggested and try and figure it out though.  I wouldn't mind a hint either.

Picard is overkill :)

Let $ f$ be entire. Suppose that $ c \in \mathbb{C}$ is not in the closure of $ f(\mathbb{C})$. That means that there exists $ \varepsilon > 0$ such that $ |f(z) \minus{} c| \geq \varepsilon$ for all $ z \in \mathbb{C}$...

[quote="Arne"][b](d)[/b] Must an entire function whose real part is positive troughout the complex plane be constant?

[b](e)[/b] Must the range of a nonconstant entire function be dense in the complex plane?

[b](f)[/b] Can an entire function which is not a polynomial have no roots? Exactly $ 2008$ roots? Infinitely many roots? Uncountably many roots?

To solve these questions, please use elementary arguments (otherwise some are completely trivial).

This means: you can use Cauchy-Goursat and Cauchy, Morera, the Cauchy estimates, the maximum modulus principle, ... but nothing more advanced.[/quote]

d)  it suffices to look at polynomials (otherwise there is an essential singularity at infinity), but if $ \text{deg}p \geq 1$, then $ p(z) \equal{} \alpha$ has a solution for every $ z\in\mathbb{C}$.

e)  yes, for the same reason

f)  $ e^z$, $ z^{2008}e^z$, $ \sin z$, no - uncountably many roots implies an accumulation point of roots which implies (as a general property of analytic functions - that is, functions having a power series expansion) that the function is $ 0$ everywhere.


[quote="now a ranger"]Suppose a circle is centered at the origini with radius larger than one, and the arc length of the arc of the circle above the line y=1, is 4pi/3. Find the radius of the circle.[/quote]
[hide]So if you actually draw it out, you can kinda I guess say that the angle is 120 because its 1/3. So the radius will be 2. I don't know the real way though. :( [/hide]


algebra
polynomial
calculus
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number theory unsolved
number theory

Consider the polynomials
\[f(p) = p^{12} - p^{11} + 3p^{10} + 11p^3 - p^2 + 23p + 30;\]
\[g(p) = p^3 + 2p + m.\]
Find all integral values of $m$ for which $f$ is divisible by $g$.

$f(-1) = 0$, hence $f(p)$ is divisible by $p + 1$.
First we want $p + 1$ to be a factor of $g(p)$. So $m = 3$ and $g(p) = (p + 1)(p^2 - p + 3)$. We see that $p^2 - p + 3 \mid \frac{f(p)}{p + 1}$.
Now we have tp prove that $p^9 - p^8 + p^7 - p^6 + p^5 - p^4 + p^3 - p^2 + p + 10$ is irreducible over the integers.

Assume that $g(p)|f(p)$ (1). Setting p=0 in (1), the result is $m|30$. Hence $m$ is one of the 12 divisors of 30. Futhermore f(1)=66 and g(1)=m+3, thus $m+3|66$ (2). The only divisors of 30 satisfying (2) are m = -6, -5, -2, -1, 3, 30.  Now p=-2 in (1) gives $m-12|9108$, which reduces the possible values of m to -6, 3 and 30. But m = -6 and m = 30 is impossible since $g(3)|f(3)$, i.e. $m+33|531828$. So $m = 3$ is the only integer that may satisfies (1). The signature "Andreas" has already verified that m = 3 indeed make g(p) a divisor of f(p).


function
number theory unsolved
number theory

Find  all sequence $(a_n)$ such that :
      $a_1<a_2<...<a_n<...$ and

a)$a_n$ is square if and only if $n$ is square.
or
b)$a_n$ is prime if and only if $n$ is prime. :ninja:

I hope some one will post solution :lol:

if noone then  late three day i will post solution :blush:

noone? :(   :( .I don't think it be hard

It had very many solutions, and i think it is uncorrect problem.
For examle Let $g(n): N\to N$ any increase function, then $a_n=f(n)=g(n)^2n$ is solution a). If $f(n)=p_{g(n)}$ when n is prime and $f(n)=2p_{g(n)}$ if n- is not prime, and $g(n+1)/g(n)>2$ give one of solutions b).


This one's from 2 other inequality which I see it's really hard.

Find the best constant $C(k,l)$ for which this inequality is always true :

\[ a_1^k+a_2^k+...+a_n^k-n \ge c(k,l)(a_1^l+a_2^l+...+a_n^l-n) \]

For all $a_1,a_2,...,a_n$ are non-negative real and $a_1+a_2+...+a_n=n$.

I only some it in some particual value of $k,l$. Find the form of $c(k,l)$ is a super hard work.

I found only the result
$c(2,l)=\frac{1}{1+n+...+n^{l-2}}$.

I find we should only to solve:Find the minimum of:
\[ \frac{(n-1)x^k+\left(n-(n-1)x\right)^k-n}{(n-1)x^l+\left(n-(n-1)x\right)^l-n},1\le x \le \frac n{n-1},k,l>1 \]

Can you please make clear you hint, Zhaobin?

sorry,I have something wrong.but I think the thoerem I post [url=http://www.mathlinks.ro/Forum/viewtopic.php?t=64933]here[/url]can help us to reduce the inequality to 3 variables inequality(one is $k$) ,am I right :?:

Zhaobin, I think $x \geq 1$ for $k\geq l$, and $x\leq 1$ for $k \leq l$.


Circle B has radius 6. Circle A ,centered at point C, has radius 1 and is contained in B.Let L be the locus of centers C such that there exists a point D on the boundary of B with the following property: if the tangents from D and circle A intersect circle B again at X and Y, then XY is also tangent to A. Find the area contained by the boundary of L.

[hide="PS"] Very short solution.[/hide]

Why nobody :wink:

[hide="HINT"] Euler formula[/hide]


algebra
polynomial
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inequalities


calculus
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real analysis solved


Let $ a_1;a_2;a_3;a_4;a_5$ be non-negative real numbers whose sum equal 1.
Prove that $ a_2a_3a_4a_5\plus{}a_1a_3a_4a_5\plus{}a_1a_2a_3a_5\plus{}a_1a_2a_3a_4 \leq \frac{1}{256}\plus{}\frac{3275}{256}a_1a_2a_3a_4a_5$

[quote="robbenmaths"]Let $ a_1;a_2;a_3;a_4;a_5$ be non-negative real numbers whose sum equal 1.
Prove that $ a_2a_3a_4a_5 \plus{} a_1a_3a_4a_5 \plus{} a_1a_2a_3a_5 \plus{} a_1a_2a_3a_4 \leq \frac {1}{256} \plus{} \frac {3275}{256}a_1a_2a_3a_4a_5$[/quote]
Problem from M&Y magazine.
Please don't post solution!


When I move my cursor over the menu on the left side, it becomes highlighted, while anywhere else, it doesn't get highlighted, it just turns blue...


I have a few questions on the 2003 USNCO National Exam Multiple Choice.(http://acswebcontent.acs.org/education/olympiad/2003/natl_03_part1.pdf for an online copy).  Any help would be greatly appreciated.

Problem 7:
Which would produce the largest change in the H2O level
when added to water in a 25 mL graduated cylinder?
(A) 10.0 g of Hg (d = 13.6 g·mL-1)
(B) 7.42 g of Al (d = 2.70 g·mL-1)
(C) 5.09 g of iron pyrite (d = 4.9 g·mL-1)
(D) 2.68 g of oak (d = 0.72 g·mL-1)

My thoughts:  [hide]This looks like a simple density problem, find the sample with the greatest volume.  I calculate that the aluminum has a volume of 2.75 mL and the oak has a volume of 3.72 mL.  However, the correct answer is B.  My only thought is that the oak might absorb water, but really how are you supposed to know how much it will absorb?[/hide]

Problem 53:
Which diatomic species has the greatest bond strength?
(A) NO (B) NO+ (C) O2 (D) O2-

My thoughts:  [hide]You're just looking for the one with the highest bond order.  NO+ has a triple bond.  NO and O2 only have double bonds.  But I have no idea what kind of bond O2- has.  Any ideas on what the Lewis dot structure for this looks like?[/hide]

Once again, thanks for any ideas you may have.[/hide]

[hide]The bond order of O2- is 1. You can try drawing out the molecular orbital to see it[/hide]

First question:
[hide]Wood floats, so it only displaces its weight (equivalently, mass) in water, by Archimedes' principle, so it displaces 2.68g of water, which is 2.68 mL of water.[/hide]

Saturn:
[hide]Are you sure?  It looks to me like it has bond order 1.5: $: \overline{\text{O}}: :\overline{\text{O}}: \cdot$, with the last dot being an antibonding electron.[/hide]

[hide]You're right, I drew the molecular orbital for O2 (2-) when I did it by accidnet =([/hide]

Oh right, thanks to both of you.  Any idea about the best way to learn molecular orbital theory :)?


Let a, b, c be positive real numbers such that $ abc \equal{} 1$. Prove that:
$ a(b \minus{} \sqrt {b}) \plus{} b(c \minus{} \sqrt {c}) \plus{} c(a \minus{} \sqrt {a}) \ge 0$

I'm sorry. I've edited my post.

Do you mean $ a(b\minus{}\sqrt{b})\plus{}b(c\minus{}\sqrt{c})\plus{}c(a\minus{}\sqrt{a}) \ge 0$?

Brut3Forc3//  What about a=1/4, b=1/100, c=400??

NguyenDungTN// Maybe problem is wrong...

First 'b' should be 'a' and First 'c' should be 'b'

It is just my opinion...

I think it should be
\[ a(b^2\minus{}\sqrt{b})\plus{}b(c^2\minus{}\sqrt{c})\plus{}c(a^2\minus{}\sqrt{a}) \geq 0\]  :wink:


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I have heard it said that analysis has become a mostly closed field with few open problems, aside from some very specialized things like in C* algebras. Other than that "It's only good for PDEs" as some folks claim. Could someone verify this or please convince me otherwise? I love the subject and I hate to think that should I decide to pursue it in greater depth it'll become something of a dead-end.

As long as there are open problems how can it be a dead end :?:

Here is a simple rule: if the statement about the field is made by a professional in that field, you can believe it with reservations; if it is done by somebody who understands almost nothing about the field, ignore it. I've heard the statement that analysis is dead or at least not what it used to be in the 60s quite many times (mostly from topologists and algebraic geometers, who do not seem to have the slightest idea about what analysis is but never miss an opportunity to comment on how what they are doing is superior to what other people are doing).

The statement about absence of open problems is a straightforward lie. What is true is that there are almost no "big and famous" problems like Riemann Hypothesis (erm..., isn't Riemann Hypothesis a statement from complex analysis by the way?) that, if solved, would revolutionise the field and result in rewriting all textbooks. The main way analysis develops is not the way of big shots coming up with very bold (though often poorly supported) conjectures, which the rest of the mathematical community then tries to work on and which are advertised as "big programs destined to change all our current knowledge" or something like that. Usually, in analysis you just ask innocently looking questions and try to answer them to the best of your abilities hoping that the methods you discover in the process will be useful in other problems as well. There are great breakthroughs occurring now and then, of course, (the latest two were the proof of subadditivity of analytic capacity by Tolsa and "phase-plane decomposition techniques" by Lacey and Thiele; some recent work of Tao can probably be mentioned here as well) but the main development comes from slow and hard to notice work on questions that seem of no particularly great importance and the methods developed are often more important than the statements obtained.

Another thing that is correct is that the borders of analysis are very vague: I fancy myself to be an analyst, but I frequently work with probability, convex geometry, and PDE, so I honestly cannot tell you where the analysis ends end other fields begin (and I couldn't care less about this question). It is true that, if you try to remove everything that can be also considered to be parts of other fields from analysis, the remainder will not look impressive, indeed. On the other hand, if you claim everything that uses analytic techniques "analysis and nothing but analysis", the remainders of most other fields after such a surgery will look even less impressive.

I'm sure Kent and mlok can comment more on how dead analysis is today. I'll just finish with the general advice to be wary of people who claim that something is a dead end. More often than not, it just means that they are totally blind :).

Again plugging my own university, what about the resolution of nirenberg-treves recently? That was pretty big  :wink:

Yeap, you're right, of course :) I'm just trying to stick to things I know reasonably well and those that cannot be claimed "a result in PDE/number theory/probability/whatever rather than in analysis" (otherwise I would also include Perelman's proof of Poincare, Tao&Green's proof of the existence of arbitrarily long arihtmetic progressions in the set of primes, recent works of Rudelson and Vershinin on random matrices, and plenty of other things; all of which can be justifiably claimed analytic in nature but I anticipate the objection that they "really belong to other fields", which may start a really long argument I would like to avoid).

I don't want to start a new topic, I'll just add it here:

Is it realistic to do research both into analysis and algebra. Are there some subfields that have strong synergies in algebra and analysis. Or will both never go hand in hand.

I like both... but the further I go the more work I need to do... As a student I already feel that I must start to specialize in one area... because perfecting both areas is too much work :)

Theory of analytic functions of several complex variables may be a reasonably good example of something that requires both algebraic and analytic thinking.

As to "the further I go, the more work I have to do", that will always be true. Choosing one area will bring a short relief but a year later or so you'll discover that you need to make a choice between several subareas again, and then again, and so on :) On the other hand Gauss and Euler have never made choices like that, they just were thinking about almost everything that came their way and the outcome was not that bad :P.

In practical terms, it means that it is probably time for you to start combining your regular studies with some research experience (yeap, that'll be even more work). Try both an algebra and an analysis project (not necessarily simultaneously, of course) and see how each of them goes. If nothing else, you'll get a better feeling of what  both fields are really about.

Good to hear another opinion fedja, thanks for responding.

Jacobson: my general impression about operator algebras is that it combines a lot of ideas from analysis and algebra, and it also has some really neat applications to quantum mechanics.

What are some of the exciting areas of research in algebra?

What part of algebra? Group theory, ring theory (commutative and non), algebras, homological, etc.

Right now a big thing in group theory is to say if a group has order $ n$ (of a certain form), and there exists a fixed point free automorphism, can we say the group is solvable? The answer is yes for lots of cases like $ n\equal{}p$, or $ n\equal{}pq$ and even $ n\equal{}pqr$ where $ p, q, r$ are arbitrary primes. I think the case $ p^2$ is also done.

[quote="HilbertThm90"]
Right now a big thing in group theory is to say if a group has order $ n$ (of a certain form), and there exists a fixed point free automorphism, can we say the group is solvable? The answer is yes for lots of cases like $ n \equal{} p$, or $ n \equal{} pq$ and even $ n \equal{} pqr$ where $ p, q, r$ are arbitrary primes. I think the case $ p^2$ is also done.[/quote]hmm are you sure? Groups of those orders are always soluble (squarefree order or $ p$-groups).

Woah! I typed that wrong. Those should be the order of the automorphism not the group.

I agree with fedja. This depends very much on what you consider to be "analysis". For example, I think the ideas of Gowers in the 1990s about notions of randomness and his "uniformity norms" was very important, and those ideas are still being explored today (in particular there are some conjectures about inverse theorems for his norms which give decompositions of functions into "random", "highly structured", and "small" parts). On the face, these appear to be purely analytic statements. You are estimating arbitrary functions by some other functions which have certain nice properties related to averaging and Fourier coefficients. One could argue, however, that these actually belong to some other field, because the main application is in these arithmetic results about structure of sets of integers.

But anyway, yes, there's still stuff going on in analysis. Speaking of that: fedja, mlok, what ever happened with that problem of constructing the doubling measure on $ \mathbb{R}^2$ that was supported on a set of dimension less than 2?

[quote="Xevarion"]Speaking of that: fedja, mlok, what ever happened with that problem of constructing the doubling measure on $ \mathbb{R}^2$ that was supported on a set of dimension less than 2?[/quote]
I think I outlined all the ideas needed for such a construction in my post, didn't I :? I don't know if mlok found it useful for anything though...

[quote="fedja"]I think I outlined all the ideas needed for such a construction in my post, didn't I :? I don't know if mlok found it useful for anything though...[/quote]
I talked to him about it one time. I don't think we ever figured out how to prove the 2nd step... (at least, I still have no idea about it)

Why didn't you ask questions then? If you both still want to know it, the three of us can either discuss it in PM's, or revive that thread. I'll gladly answer any particular questions you might have :).


calculus
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calculus computations

Thx a lot,Aashay THE ROKR.
It's a nice solution!!


function
inequalities unsolved
inequalities

find all fuction  f:R->R satisfy
$(x-y)f(x+y)-(x+y)f(x-y)=4xy(x^2-y^2) \forall x\in R$

[quote="nttu"]I don't know why you post functional problems here !!!
  This problem becomes easy with subtitution : $u =x-y$ , $v=x+y$[/quote]

I think we only need to observe $\frac{f(x)}{x} - x^2 = C$?


$ (x;y;z;t) \in R$ such as  $ x\geq\minus{}1$ ;  $ y\geq\minus{}1$  ;  $ z\geq\minus{}1$  ;  $ t\geq\minus{}1$   and   $ x\plus{}y\plus{}z\plus{}t\equal{}2$

prove that:      $ x^3\plus{}y^3\plus{}z^3\plus{}t^3\geq \frac{1}{2}$

Il est facile!

$ 4x^3 \minus{} 3x \plus{} 1 \equal{} (x \plus{} 1)(2x \minus{} 1)^2\geq 0$ e.t.c.

$ \therefore x^3 \plus{} y^3 \plus{} z^3 \plus{} t^3\geq \frac {3}{4}(x \plus{} y \plus{} z \plus{} t) \minus{} \frac {1}{4}\cdot 4 \equal{} \frac {1}{2}$.

[quote="peine"]$ (x;y;z;t) \in R$ such as  $ x\geq \minus{} 1$ ;  $ y\geq \minus{} 1$  ;  $ z\geq \minus{} 1$  ;  $ t\geq \minus{} 1$   and   $ x \plus{} y \plus{} z \plus{} t \equal{} 2$

prove that:      $ x^3 \plus{} y^3 \plus{} z^3 \plus{} t^3\geq \frac {1}{2}$[/quote]

Here is my solution:

let $ x \equal{} a \minus{} 1,y \equal{} b \minus{} 1,z \equal{} c \minus{} 1,t \equal{} d \minus{} 1 ;$

wa have to prove that:

$ (a \minus{} 1)^3 \plus{} (b \minus{} 1)^3 \plus{} (c \minus{} 1)^3 \plus{} (d \minus{} 1)^3 \geq \frac {1}{2}$ with $ a,b,c,d \geq 0$ and $ a \plus{} b \plus{} c \plus{} d \equal{} 6,$

$ \Leftrightarrow a^3 \plus{} b^3 \plus{} c^3 \plus{} d^3 \minus{} 3(a^2 \plus{} b^2 \plus{} c^2 \plus{} d^2) \plus{} \frac {27}{2} \geq 0,$

from C.S we have:

$ a^3 \plus{} b^3 \plus{} c^3 \plus{} d^3 \geq \frac {(a^2 \plus{} b^2 \plus{} c^2 \plus{} d^2)^2}{a \plus{} b \plus{} c \plus{} d} \equal{} \frac {(a^2 \plus{} b^2 \plus{} c^2 \plus{} d^2)^2}{6},$

It suffices now to prove that:

$ \frac {(a^2 \plus{} b^2 \plus{} c^2 \plus{} d^2)^2}{6} \minus{} 3(a^2 \plus{} b^2 \plus{} c^2 \plus{} d^2) \plus{} \frac {27}{2} \geq 0,$

$ \Leftrightarrow (a^2 \plus{} b^2 \plus{} c^2 \plus{} d^2 \minus{} 9)^2 \geq 0,$

which is true.

[quote="peine"]yes, but question is not to answer with indications, but it is necessary to give of oû idea comes.[/quote]

One wants to find a condition in the form

$ x^3\ge ax\plus{}b$, because then we could add those up and we'd be done.

So we know equality occurs for $ x\equal{}\frac{1}{2}$, so

$ \frac{1}{8}\equal{}\frac{a}{2}\plus{}b$

Then we need to have a double root at $ x\equal{}\frac{1}{2}$, otherwise the function would switch signs at $ x\equal{}\frac{1}{2}$.

So $ 3x^2\minus{}a\equal{}0$ for $ x\equal{}\frac{1}{2}$, solving, we get $ (a,b)$. Then we check that it factors as kunny did.


Find the area inside the graph $ |x\minus{}2|\plus{}|y\minus{}4|\equal{}2$, but outside the graph $ (x\minus{}2)^2\plus{}(y\minus{}4)^2\equal{}2$.

The graphs are a square (kinda in a diamond shape) with a circle of radius $ \sqrt2$ inscribed in it. Simply subtract the circle's area from the square's area to get the desired area. The answer is $ 8\minus{}2\pi$.


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I wonder whether I should post or not and if so how much to say.

The discussions about God here and on other threads on AOPS have raised a number of very good questions.  Many of which have been answered over the years.  Allow me to share what little I have learned on these matters.

I am a Christian who believes that the Bible is the inspired word of God.  Without a reference that is True, any human discussions are difficult at best and impossible otherwise.  Sorry to be so emphatic on this point, but for most of the centuries of the Church's existence the Bible was not only not available to most of the people, it was not even available to most of the priests.  Many died to bring the Bible into the common languages - in some areas they are still dying to make reading the Bible possible.

You may ask, can I prove the Bible is True.  The answer is no.  I believe it is true and I can point to the many miracles that were done by people who said that their miracles were God's witness that what they said was from God.  Note that God's standard for judging those that say they speak for God is harsh by our standards: if anything the person predicts doesn't happen they don't speak for God and should be killed.

All that I say will be my best efforts to reflect the God of the Bible.  Let me apologize initially for any mistakes or errors I will make in representing my understanding of this God of the Bible.

After judgement, people with live on the new earth, where God will live with people, so the label of Heaven is not quite right (See Revelations for details), but most people view Heaven as the place where the righteous go to spend eternity with God.  I'll use that convention here.  Also, currently people's souls go to Hades (Greek, a holding place having nothing to do with purgatory), not Heaven/the new earth or Hell but again, to simplify the discussion and make the points more understandable to most readers, I'll use the popular Heaven/ Hell convention, as the discussion is focusing on how people are judged and not the details of what happens when someone dies or is judged.

First, God is a loving God.  God loves people, but he loves his name even more.  He will not tolerate anything in heaven that is not perfect.  (It doesn't matter if he could have made a different decision, he didn't.) That presents a problem for all of us flawed, sinful people.  There are none righteous on their own merits - no one ever lived that was righteous on their own, not even one.  The angels and other beings are in Heaven with God right now are perfect.

Not only are we flawed, sinful people, but we had a representative, Adam, who rejected God.  He wasn't mislead by Satan as was Eve, but willingly decided to disobey God.  (The beginning of Genesis for those that want to find the scripture on this.)  This sin was attributed to us since Adam was our representative before God.  This attributed sin separates us from God and answers the question of how sin of those that die young is determined.  (Those that have a hard time believing this, look to Paul's discussion of Jacob and his twin brother Esau (Romans 9:10 and following) where before they were born or had done anything good or bad, God said that he "loved" Jacob and "hated" Esau.)

So we all deserve to be cast into Hell where, after judgment (and after death but before judgment) we would suffer constant, conscious torment - all in the absence of God's presence.  (I know God is everywhere, but being in his presence only occurs where his presence is felt by those who are in his presence.  Perhaps too much detail for some, but it addresses concerns others have raised in previous discussions.)

This presents a problem.  How can any of us sinful people ever get to Heaven?  If God decided not to punish our sin he decreases his glory or name - something he won't do.  The way God decided to have some people get to Heaven is by having a substitute be punished for our sins.  God decided (before creation or Adam's sin) to have his son Jesus - a third of the triune God - come to earth, be born a man, live a perfectly obedient life and die a perfectly obedient death.   This sin (crime against God) is paid by the more precious death of God himself (the Jesus third) - far better payment than the mere suffering of people.  (For those having trouble with this concept, think of a person who has a fine assessed against them for a crime and someone else comes forward to pay the fine.)  God's killing of his son represents sufficient payment for the sins of all people who have ever lived.  However, that is not enough - it merely brings us back to zero.  God requires perfection.  It is Jesus' perfect, sinless life that is also attributed to people that makes them acceptable to God and able to enter Heaven.

Now when Adam sinned, not only did all man (including all women as well) get Adam's sin attributed to them, all men were cursed by God as well.  This caused the loss of all men's will to seek God.  (Those that want more on this see the first chapters of the book of Romans or Martin Luther's Bondage of the Will - This is the 13th Century Monk/Doctor of Theology that started the Protestant movement and is the inspiration for naming Martin Luther King, Jr.)  The book is still in print.

The only people who can seek God or understand other telling them about God are the people whose hearts God has softened.  (He also wrote every one of his elect's names in his Book of Life, before the world was created.)  God has decided that some people from every language and tribe will be saved.  A remnant of mankind if you will, but most people will go to Hell.  This seems harsh to many and is the source of many refusing to believe, but the real puzzling thing is why God saved any of us sinful, wicked people.  (Those wanting more on this see the Romans citation given above.)  Every Christian is commanded to present the Gospel message, but God alone determines if anyone believes.  (See the last paragraph of Matthew for this command.)

So exactly how does one get saved?  After God softens your heart you must believe that Jesus was the son of God, that he died for your sins, accept and confess to God that you : 1) are a sinful person, 2) regret your sins and 3) intend to turn away from sin.

Note that this was the same way people were saved before Jesus was born and for those that never hear his name.  Those not believing this can read what Hebrews says about this.

Note that sins will continue - Christians are not perfect, sinless people - but they are trying to be less sinful.

Doing religious things, doing kind things, not doing antisocial things are not the path to salvation.  If you will good things are done by Christians because they are saved not in seeking salvation.  Scripture says that the salvation is by faith alone, not by man's good acts.

I know I've gone on too long for some.  I'm sorry for that, but I don't know how to fully express this message with fewer words in a manner I believe is accurate and addresses some of the questions I've seen in this area.

It is very hard for me to take anyone who claims the Bible to be the inspired word of God seriously. How do you choose what it true and what isn't true in the Bible? It seems rather arbitrary.

You most certainly don't believe every word to be true. See [url=http://www.youtube.com/watch?v=B5MkpzMAOZM]My year of living biblically[/url].

I'm sorry for your limitation that you say that you can't take someone who believes the Bible is true seriously.  In fact, I don't believe what you say.  If you were told that Sir Isaac Newton believed the Bible, would you suddenly not take Newton physics seriously?  I didn't think so.  If you look up his history, we wanted to be a minister and was unable to do so, his career was what else he did with his life.  There are many, many others whom you would write off based on their beliefs.  I believe what you are saying is that even though I believe the Bible is true, you do not.  That is reasonable and fair.

One example you might choose to ask about is the age of the earth.  There are many ways to estimate the age of the earth.  All involve assumptions based on things that we have not directly observed.  Let me suggest one for you to examine.  Salt contained in the steams flowing into the ocean doesn't leave when the water evaporates.  The US Great Salt Lake and the Israel Dead Sea are results of lakes without outlets.  The world's oceans are, by definition water bodies without outlets.  If one were to take the amount of salt flowing from the world's streams and assume that the flow was constant, the current salt content of the world oceans would lead one to conclude that the rivers were flowing into the oceans for how long?  I won't solve this problem for you.  I've already done so.

If your result isn't the billion and billions of years that Carl Sagan and others quote, my question to you is what happened to all that salt?

I don't choose what I believe it true in the Bible.  I believe it all is true.  I believe that I have to logically have that belief.  Any other belief sets man above God.  Allow me to say that I'm not saying that I understand it all, just that I believe it all must be true or I can't rely on any of it as necessarily being true.  I believe the stories related by folks that I believe were there over the thousands of years that the Bible was written correctly reflect the miracles that God performed to demonstrate that the people writing the Bible were writing for God.

You can't take the Year of Living Biblically - a humor book by its own admission - seriously?  According to the few pages I read in the book store and what my sister said about what she read in that book, he discounts some of the passages as not meaning what they say and doesn't look at the Bible as a continuing revelation, where some passages are explained or conditioned by later passages, which is what the Bible tells us is how it should be read.

Anyway, enough from me for now.  If you are seriously interested, and you think it helpful to you, I'll be glad to continue to discuss my beliefs.  I'm glad to explain what I believe and answer questions.  I'm sure my beliefs are not complete and would be surprised if I didn't totally miss some of what God meant.  However, following by beliefs is the best I can do.  I change them if I later believe that I was wrong before.  I have done that many many time in the past and expect to continue to do so as long as I'm blessed with life on this earth.

All I'm saying is that you either believe the Bible to be completely true, or you don't. If you believe it to be completely true, then you must live by utterly absurd rules such as not wearing mixed fabrics, not shaving the edge of your beard, stoning adulterers and much more. Moreover, it is [i]impossible[/i] to live by all the rules (the point of the year of living biblically). So someone that claims the Bible is truth has to pick and choose which parts of the Bible to say is truth, and which parts to "interpret" and not live by. How is this done?

I agree with [u]Crazy[/u] [color=green][Moderator edit: see [url=http://www.artofproblemsolving.com/Forum/viewtopic.php?p=1211078#1211078]here[/url]][/color] as far as the need for faith to believe the big bang theory is correct.  There is no answer for what occurred before or what happens after.  What are the odds that we happen to be here between the beginning, infinite time before and before the end of the universe, after which there is an infinite amount of time.

[u]Hilbert[/u] - We have a couple points of misunderstanding - I apologize for not being as clear as I needed to be on these points.  Just because one believes that God has given commands doesn't mean that we follow them.  Not to do so is to sin.  Remember I said that all humans are sinners.  I certainly am.

I believe that the word of God is true and that it is reflected in the inspired writing of the original scripture.

What is commanded in the old testament (the Law) some 613 commandments.  (I haven't counted them myself, but I've been told by folks that say they know.)  You say that no one can ever live in obedience to all of these commands.  I agree.  So does scripture.  See Acts Chapter 15 where the council of Jerusalem is reported.  The question was whether to require the Gentile (non-Jew) Christians to follow the Law.  The answer was no.  See Acts 15:10, where Peter the head of the Church (Inspiration for the office of the Pope) said in part, "Why test God by putting on their necks a yoke that neither we not our fathers have been able to bear?"  The result was that the early church decided not to apply the Law to Gentiles.

Further, Paul explains the purpose of the Law and the fact that it no longer applies to Jews or Gentiles.  (Those that remained in the Jewish faith don't believe this, but the Jews that converted to Christianity do.)  See Romans 7:1-6 for Paul's explanation as to how the Law no longer applies in that God is requiring compliance.  Many of God's requirements (Laws) have time periods when they are applicable, for example the requirement to not work for a year every seven years and after the 49th year also take the 50th year off.  Similarly, God required compliance with the Law from Moses for a time.  This was not stated at the time, but Scripture is a revelation where earlier thing said are later explained.  One would not think that every father of a son named Isaac would have to follow the command in Gen 22:2 "Take you son, you only son, Isaac, whom you love, and go to the region of Mariah.  Sacrifice him there as a burnt offering on one of the mountains I will tell you about."

It is amazing that God has Scripture such that one can study it for years and still find new things to marvel about.  This complexity is one of the rationales used in years gone by that the church said that the people could not read the Bible on their own, they might be mislead.

Anyway, I hope this helps.  I'm not saying that my behavior is compliant with what God commands, I'm just saying that I believe it is True that he commands what the Bible says.

Again, read the introduction to the Living by the Bible book - the purpose was to write a humor book, not to try to live up to Scripture.  The author admits this was his goal when he started.  He also says that he picks different aspects to "try" to make it funny and sell his book.  (He may also teach some about the Bible or cause some to look something up.)

While I don't pick what part of the Bible to believe, I'm trying not to pick what part of the Bible to follow and which parts not to live by.  Again, I'm not perfect, not even good (Mark 10:18), I'm just trying to be better as time goes on.

I probably shouldn't keep feeding this, but I'll give it one last attempt to be as explicit as possible.

Some laws of the old testament are treated as laws that without a doubt are true: the 10 commandments fit here. Some laws are silly: you cannot sit in a chair for which a woman has sat while menstruating. This rules out essentially every chair in existence. You wouldn't be able to sit down.

No one, not even the most devout, try to follow laws like this. No one points to laws like this and says, "That is silly. You don't need to follow it." How do you make the decision that this law is pointless to try to follow, and this other one must be followed?

Note that I have attempted to switch from the impossibility of following all the laws (which you agree with), to laws that aren't impossible but that are generally agreed that it isn't a sin anymore.

@wizard

1. Is god omnipotent?

2. Do you believe that humans are a process of chemical reactions and hence a deterministic process?

3. Who controls the initial data (initial conditions) that govern any given human's life? (i.e. when/where s/he's born, what social status, exact brain composition, so forth)

(main 3 questions)

(lesser questions follow)

4. Why does god love his name so much? What does he care if some puny insignificant creatures begin to besmirch it?

5. What do you mean by  "He also wrote every one of his elect's names in his Book of Life, before the world was created." ? Is this the notion that it is already predetermined who goes to heaven and who goes to hell?

6. "Every Christian is commanded to present the Gospel message, but God alone determines if anyone believes." This also seems to indicate that god decides right from the beginning who gets to believe in him in order to get to heaven and who doesn't. So is it really the case that god, right at the beginning of the world, sat down and said yup, these people are going to make, and whoops, those people aren't? If yes, everything's predetermined anyway, so why don't we go ahead and do whatever the hell we want? If no, the quotation still seems to indicate that god's controlling whether or not we get to believe, so if we do happen to fail, that's more his fault than ours -- it doesn't seem to agree with the loving god who wants us to get into heaven. Granted, you pointed out that he has a bigger priority, namely making sure that his name doesn't get taken in vain, but again, if he's controlling this, why doesn't he just set stuff up so that his name is never sullied?

7. You really believe that large swaths of well-accepted modern science are simply incorrect? How exactly do you think that happened? Are people deluded just by formulating the scientific method? Are they terrible at doing experiments so that we've consistently gotten incorrect results? Are we failing to see obvious logical mistakes in our theories? Where are the errors we're making?

@hilbert

About the year of living biblically, it's a good link (I liked the vid) but you could surely have come up with some much stronger evidence for the same point! I'm also surprised that you so easily conclude what's silly and what's not. What exactly is so silly about not sitting in a chair in which a menstruating woman has sat? You could very easily just construct a chair and restrict it to males only, something that the video specifically mentioned, so it's hardly infeasible. Furthermore, as again the video specifically mentioned, if some martian were to come down, it's not like he would easily be able to say "oh yeah those people who don't sit in chairs that have been menstruated on are CRAZY but those other dudes who blow out fires on top of cakes -- they're all right". I think your point works better by simply noting that some laws are no longer followed without simultaneously passing utility judgments.

@mystic

just wondering, what side are you leaning towards

Ah. Woops. My last post was not meant to make a utility argument. The main point was that no one follows it anymore, probably because of how silly it would be.

I might as well come out with it instead of prodding.

The point I'm obviously after is that anyone that claims the Bible to be the word of God and must be followed has got to, at the very least, admit that some laws are no longer considered sinful. A conscious decision was made at some point that we no longer need to follow some of the laws.

Now to bring the logic into this. I'm after how this decision is made. Any justification I can think of inevitably goes something along the lines of, "Well, that's just silly. Why would that be sinful?" If that argument is brought up, or anything similar, then it is nothing more than an arbitrary judgment on what seems silly and what doesn't seem silly.

How do we then decide on more borderline cases? I think it is silly to say that homosexuality is a sin. Is this good enough evidence to say that it isn't?

As a side note, I am on a quest to rid my life of as many meaningless rituals as possible (this includes but is not limited to blowing out candles on a birthday cake...and celebrating a birthday...or any thing that occurs yearly...)

A lot to respond to.

First, I assume that the Mystic "@weird" was a typo and not an childish attempt to call names.  It would be below the level of your previous posts.  (This shows at least some of my concern with "my name," even in this trivial posting world.)

Hilbert - Your statements show real interest in these issue, but a lack of full understanding of the Bible.  For example, the statement about the woman menstruating chair issue - straight out of the book you earlier referenced, assumes today's world.  In the time of the old testament, menstruating women were considered ceremonially unclean and to themselves undergo a purification process before they would be ceremonially clean again.  If you want the details of the requirements, including similar rules for men, see Lev15:16-30  What you miss is that, just like people, chairs, beds and most things can be ceremonially cleaned then used by everyone just as if they weren't ceremonially unclean.  If women understood this and weren't trying to make trouble for others, as was the author's wife, this isn't a real big issue.

I must not have been clear before.  I don't decide which laws are "pointless to try to follow," and which, "must be followed."  The section I quoted before from Romans and many other sections of the Bible show that the Law does not apply to Gentile Christians.  Other sections of the Bible show that the Law does not apply to Jewish Christians.  There are none of the Law's details that have to be decided to follow or not.  For example, Jesus was often criticized for not following the Sabbath Command.  There are many commands regarding food (Kosher).  Examine what Paul says in Romans 14:1 regarding Kosher food - (In Rome, the meat for sale may well have been involved in a sacrifice to idols and a person with weak faith - perhaps a new convert - would see eating meat sacrificed at the alter of a "God" he formally worshiped as difficult.  Vegetables would avoid this issue completely, as well as the other Kosher concerns.)  See Romans 14:5 & 6 where Paul deals with the Sabbath.  Also note all the Law is treated in Romans 14:14 - God no longer sets things as unclean, but if a person regards something as unclean, for them it is unclean.  See Romans 14:22 for additional instructions as how this change should be managed among the believers.

An obvious question is why did God set forth the 613 commands, when he knew that none could comply.  One reason given, that I've already provided references is to demonstrate how sinful we all are.  No one can be righteous according to the law.  I'm glad that you see this is true.  Many Jews didn't see that.

A second reason (my opinion only) is perhaps this help fulfill God's desire for his name to be known.  Israel was taken to a land that was a narrow land area for travel (bound on the West by the Sea and the East by the desert).  Someone traveling trough the area seeing the Jews doing so many unusual things might ask, "Why are you doing (or not doing) this."  There was no way to provide an explanation without talking about their God at some point.

As far as the "pointless" view, I'm reminded of what Martin Luther said when asked a similar question.  (Not Scripture, but he was the father of the reformation.)  One is not insane to give up what he can not keep to obtain what he can not lose.

There are many very devoted Jews that believe in all 613 commands and they are trying very hard to follow all of them.  (The old testament of us Christians is their Scripture.)  You are right about the decision not to continue to fall under the Law.  We Christians believe that this decision was made by the same one who decided to make the initial commands - God.

As far as being silly, I don't believe that challenging God's decision process is a good idea.  You can look at the book of Job to see how God responded to one who did so.

I am not saying that I understand God's logic for all that he commands, I may agree or disagree with some.  However, if I am following any Scripture as the Word of God, I logically can't discard any of it myself.  If so, I've made the Scripture based on my support of it and not God's support of his Word.  Many try to "explain," "minimize," or "adapt" Scripture, not with a goal of finding out what God meant, but to change his words to reflect their will.

I hope this response helped.

Splatyango - If one were looking at probability and doing a cost/benefit analysis, I think that the fact that if one believes and is saved, one will avoid spending all of eternity in constant, conscious torment (Hell).  Even a small possibility of that occurring would cause some doing this analysis pause to be really sure that those freaks like me, talking about Jesus, are really wrong.

I hope this response helped.

I'll respond to Mystic in my next post, hopefully shortly.

dang I thought your name was weird2378

I remember bein all like

man I totally do not get that name

why would he ornament it with so many numbers if he's just going to call himself weird

well all right then thanks for clearing that up I'll edit it

no offense meant

although I find it curious that a devout christian such as yourself would hold themself to be a wizard -- is not such power an approximation to god's and hence blasphemous?

by the way, I don't agree with this cost/benefit analysis (in the pascalian vein) -- certainly, there is an infinite "payoff" (if one can express it in such crude terms) to be a true christian and ascend to heaven if such does in fact exist and an equally infinite "cost" in terms of going to hell if one refuses to accept the savior in the case that christianity is valid. one can nonetheless posit an entirely reversed theory, however, saying that christians get tormented forever and nonbelievers get to chill out and have a great time, and equally well postulate that these reversed expectation outcomes are infinite and might occur with nonzero probability. so one can't make any sort of sensible statement from this point of view without at least acknowledging the obvious problem of renormalization. (of course, I don't support splatyango's analysis either, which was patently nonsensical.)

[quote]The point I'm obviously after is that anyone that claims the Bible to be the word of God and must be followed has got to, at the very least, admit that some laws are no longer considered sinful. A conscious decision was made at some point that we no longer need to follow some of the laws.

Now to bring the logic into this. I'm after how this decision is made. Any justification I can think of inevitably goes something along the lines of, "Well, that's just silly. Why would that be sinful?" If that argument is brought up, or anything similar, then it is nothing more than an arbitrary judgment on what seems silly and what doesn't seem silly. [/quote]

I pretty much agree with your first paragraph... but not really the second. From what I've seen, the laws that you are referring to are from the Old Testament. But you forget about Jesus and the New Testament. When Jesus came he changed many of the old laws. People were no longer required to offer sacrifices as a token for their sins and so on. When I think of the laws we should obey now.. I mostly think of the "laws" Jesus talked about in the New Testament. There are still some laws we should follow in the Old Testament. i.e. the Ten Commandents. But I believe that many of the laws in the Old Testament such as being clean and unclean, only eating clean food, sacrifices, we are no longer required to follow.
    Everyone please tell me of anything I didn't explain clearly, anywhere I was unreasonable, etc.
   And wizard2378, I totally agree with everything you have said. So far, you have done a good job explaining Christianity. Finally someone who agrees with me! :D

@wizard. That was infinitely helpful. It was also my fault. I should have read the earlier post more carefully and that would have come across.

I guess my question is, if you are a Gentile Christian and the Law is not required to be followed, should things such as homosexuality that most Christians think of as sinful actually be taken seriously when it is not mentioned in the New Testament (to my knowledge)?

Mystic -

Sorry for the delay in responding.  I had a response worked out but the site timed me out and I lost it.

It will be easier the second time.

I'll try to get at least a bit done tonight.

Thanks for the support Kookamunga.

Hilbert - Let me respond to your posting, as my response is short and will only leave Mystic's for later.

Many people are amazingly absorbed by homosexuality.  It doesn't describe the entire individual, any more than any other inclination defines an individual.  Sin is sin.  It doesn't matter if one person commits five sins and another commits only one sin.  We are all sinful.  An individual inclined toward heterosexual activities with someone married to another is the same in the Bible as is an individual inclined toward homosexual conduct.  It is the behavior, not the individual that is condemned.  Yes, the new testament defines homosexual conduct as a sin.  See 1 Cor 6:9-10 where Paul lists Homosexual offenders in the list of other "wicked" that will not inherit the kingdom of God.  Note that it is not the person with the inclination, but the one who performs the prohibited act that is condemned.  See also Romans 1:26 and 27 where the history of mankind includes the sin of homosexuality.  Note that there is an unforgivable sin, but homosexual activity is not that sin.

Interested in what that is?  I could see where knowing this could be of some value to some of you.  See Matthew 12:31-32

At the council of Jerusalem the requirements to the new Gentile Christians were: to abstain from food sacrificed to idols, from blood, from the meat of strangled animals and from sexual immorality.  Acts 15:28-29 This frees up many things, but doesn't mean that God doesn't have expectations and that sins can't occur.

Other sections talk about other requirements that may apply, but different Christians have different views.  This is summarized in Romans, chapter 14.

Mystic - the only difference between the risk of hell you propose in your last posting is that it isn't supported by Scripture - or anything else.

Mystic - (I'll respond in a couple of posts to avoid the timout/lost message from yesterday.)

My answers to your questions: (I won't repeat the questions, folks can read them if interested.)

1. Yes

2. No.  Man's bodies are chemical reactions and might be deterministic (absent intervention).  Prayers are answered by God/angels and demons/Satan have the ability to impact things that would have otherwise occurred the other way, regardless of how well known the initial conditions and systems are.  Man's souls are not chemical reactions.

Support for my position:  Job - see Satan asking God for permission to change the direction of Job's life, which then changes directly - Note that God changed the direction again later.  Job 1:12, 13-19 and Job 42:12-17.

Science supports this view.  With the advent of brain surgery, since the brain feels no pain, the patient is often awake during the surgery.  The surgeon uses electrical impulses to determine if they are in the correct part of the brain, for freezing or cutting.  When an electrical current moves a part, such as an arm, the patient says something like, "Why did you move my arm?"  When the verbalization center is isolated from the half of the brain that moves an arm, so that the patient can't verbalize why they moved their arm, or if the arm is moved in response to a post hypothetic suggestion, the patient says something like, "I don't know why I moved my arm."  This shows that the brain itself is not the entire mind.  Perhaps the soul is the difference.

3. God.  In the call of the prophet Jeremiah  (Jer 1:5) God says to Jeremiah, before I formed you in the womb I knew you, before you were born I set you apart."  Similar Psalms 22:9, 58:3, 78:6.  Particularly detailed is Psalm 139:13-15 "For you created my inmost being; you knit me together in my mother's womb. . . . My frame was not hidden from you when I was made in the secret place.  When I was woven together in the depths of the earth. . ." See also Mark 10:30 and Luke 12:7, where Jesus says that God has numbered even the hairs on your head.  That seems like pretty intense interest.

I want to bring up something to discuss. (This might be irrelevant to the discussion currently going on, but I think it belongs in this topic)

God said that it was "not good" (Genesis 2:18; it showed that man was alone as well) after making the first man, but after making the first woman (so that both man and woman were made) he said that it was "very good" (Genesis 1:31; Genesis 1:27 showed that both man and women were made).

[quote="wizard2378"]Blahblahblah - Why? (Other than perhaps you do?)[/quote]

for several reasons, actually.  a newspaper doesn't purport to tell you how to live your life, for one (or to tell other people how to live their lives).  you can also easily check a newspaper against other newspapers, it's possible that they are all false but this is pretty implausible.

i think that skepticism was a poor choice of words because i think that even if one is a christian they should still question what the bible says.  i believe that people should question all of their beliefs to see if they hold water.  also, the bible is a particularly poor text to take at face value even if you believe it is the inerrant word of god (applying passages literally from leviticus may lead to some awkward situations) - again, even if one is a christian, i think that it is necessary to question exactly what it means for a christian to act 'biblically' in modern society.

more to the point though, people should question what the bible says because it is a document which is many thousands of years old and very possibly [i]not[/i] the inspired, inerrant work of god, because it is often used to promote policies and laws which do not respect the separation of church and state, and because many religious leaders themselves are unwilling to question it and spread a message of intolerance and conservatism which is hardly the default interpretation of the bible.

tl;dr: why? because the stakes are much higher.

bla^3 - I think we have a difference of opinion between us regarding the Bible telling people how to live their lives.  Perhaps this is semantics, but at most the Bible tells people what God says.

For example, in Luke 14:24-35 Jesus talks to “large crowds” telling them that there is a high price to being his disciple.  He relates this to several examples in the secular world telling folks if you don’t have what it takes, don’t start to be one of my disciples.  Before Jesus, the Bible only specified requirements to Jews.  After Jesus there are some extensions from the old testament, but the law doesn’t apply to gentiles.  As Paul said talking about the law, everything is permissible, but not everything is beneficial or constructive Co 10:23 and similarly in Romans 13:8 – 15:4 Paul, talking about the Sabbath and food and other old testament “requirements,” says that the requirement of the law do not apply, the requirement Paul talks about is the requirement not to place an obstacle in another’s way by how one acts and, “whatever you believe about these things keep between yourself and God.”  Romans 14:22

You do know that newspapers are not in the business of trying to tell the truth don’t you.  They try to sell as many papers as they can (or subscriptions/ hits if electronic) while not violating the civil law and creating liability for the paper.  One US standard is a careless disregard for the truth.  Same for the TV news shows.

Questioning what people say the Bible teaches is taught as a positive attribute, though this aspect is not heavily taught by some.  See Acts 17:11, where the residents of Berea are said to be noble because they carefully examine the message against scripture to see if it is consistent.

You seem to imply that the Bible has been changed over the years to reflect then policies and laws.  Records have shown that there seems to be a very unchanging of the Bible books after they are written.  First preserved by the Jewish priests and later by the Christian Monks and Priests.

The Bible has been misused, as has most other things.  The Bible can’t support the US separation of Church and State, as the last part of it was written over 1500 years before our little war with Britain started.  That being said, the Bible does have Jesus say, “Give to Caesar what is Caesar’s, and to God what is God’s.”  (Matthew 22:20) This was in response to a question was it right to pay taxes to Caesar.  This respect for separation between Church and State is more strongly reflected by Paul in Romans 13:1 “Everyone must submit himself to the governing authorities, for there is no authority except that which God has established. . . . Therefore, it is necessary to submit to the authorities, not only because of the possible punishment but also because of conscience. .  . . If you owe taxes, pay taxes; if revenue, then revenue; if respect, then respect; if honor, then honor. ”

As you note, many of the harmful things done in the name of the Bible are not from or of the Bible but from or of the person doing the harmful things.  One last Bible citation, from James, the oldest half brother of Jesus in James 3:1 where he says that those teaching the Bible will be judged more strictly than non-teachers.

I agree that the stakes of one beliefs in this area are higher than those in other areas.  I believe this is the most important thing anyone will ever decide.

ProtestanT – You are correct with God’s statement that an exception to things being good was man being alone.  But remember that these were the first two people.  Nothing here says that a man can’t avoid this being alone by being with anyone else who is human.  The woman was necessary for sex and additional people.

I note that the same Holy Sprit that inspired Moses to write Genesis, inspired Paul to write 1 Co 7:8 and 9 and 32 – 40 where he talks about the benefits of one not being married so one needs not to be concerned with pleasing one’s spouse and can be concerned with pleasing God.

I hope this is helpful – let me know if you have more questions in this area or others.  (We can start a new thread if this is too far off or goes on too long.)

[quote="kookamunga"]3) I didn't know about either of these claims. I had actually heard that the one about Dr. Spock was true.
...
5) I didn't occur to me that it could be false.

6) The person who sent me this e-mail wouldn't have sent it to me if they knew it was false.[/quote]  I think these points deserve some follow-up.  (Also, I'm in a bad mood, so picking on the seemingly defenseless seems like fun.)  For number (3), where did you hear this?  Do you think you should trust this source less in the future?  Do you usually accept comments of that sort without any sort of verification on your part?  On (5) -- chain letters are one of the most notoriously faulty forms of communication ever devised.  Why would you assume a chain letter you received was true?  Do you think in the future you might verify, especially before subjecting others to the letter?  Similarly, for number (6) -- you trust someone who evidently also doesn't bother to verify if the things they're sending you are true?  Shouldn't this make them suspect more generally?  Perhaps you should suggest to this person that they do their own verification before they send things out that claim to be true.  Maybe you should be asking them some of the same questions I've asked you.

I know that critical thinking is [url=http://en.wikipedia.org/wiki/Doubting_Thomas]actively discouraged by Christianity[/url],* but it's a useful skill if knowing the truth is important to you and if you don't want to annoy people by sending them dishonesties.

* Note: this comment is intended to be interpreted as humorous, but the sort of humor that has a grain of truth to it.

JBL - I know you said that you were in a bad mood, but attacking all of Christianity based on a description in Wikepedia is a bit weak.  You are correct that I agree that there is at least a bit of truth that at least some, perhaps many Christian leaders are not engaged or open to critical thinking.

I think in fairness you can also agree that many campuses are also similarly not engaged or open to critical thinking, if there is a possibility that the political correctness views won't be supported.

Based on your attacks on others, I guess you should be ready for someone else who is in a bad mood to attack you for not even looking up the scripture referenced.  They might point out that there are on-line Bibles, if you don't own one yourself.  I'll defend you on this point that the article seemed to correctly reflect that part of the Bible.  I believe that the behavior being criticized by Jesus wasn't the lack of critical thinking.  (Thomas was being criticized for not accepting the eyewitness testimony and requiring that if he personally didn't handle something it didn't exist.)

One could suggest that this particular limit would be difficult for either the study of astronomy or history.

I hope you are feeling better soon.

[quote="wizard2378"]
I think in fairness you can also agree that many campuses are also similarly not engaged or open to critical thinking, if there is a possibility that the political correctness views won't be supported.
[/quote]

eh? what do you mean by this? I have heard lots of people at my college express potentially politically incorrect views, and I'm sure this is true more generally.

Mystic -- you've again fallen into the trap of assuming that wizard says things that make sense ;)

[quote="wizard2378"]I guess you should be ready for someone else who is in a bad mood to attack you for not even looking up the scripture referenced.  ...  I'll defend you on this point that the article seemed to correctly reflect that part of the Bible.  I believe that the behavior being criticized by Jesus wasn't the lack of critical thinking.  (Thomas was being criticized for not accepting the eyewitness testimony and requiring that if he personally didn't handle something it didn't exist.)[/quote]  Or perhaps it should occur to you that I [i]have[/i] read this section of the bible, and chose a reference that provided some explanatory commentary, not just the biblical text.  (And the reference I chose contains a link to the text, so anyone following my reference could easily verify the interpretation given by the secondary source I cited.)  Since you agree that I chose a reference that correctly explains this, you should perhaps consider apologizing for insinutating incorrectly that this was good luck rather than intentional choice.  Finally, I think the interpretation you offer here is clearly at odds with the most widespread interpretations of this portion of the bible which hold that Thomas was chided for not accepting the claims about some miracles [i]on faith[/i] -- specifically, Jesus says, [quote]blessed are they that have not seen, and yet have believed[/quote] a line widely accepted as the official biblical endorsement* of faith above evidence.  I suppose you would have it as the official biblical endorsement of hearsay and rumor above evidence (and, in fact, I suspect again that you are being disingenuous in your own interpretation of this section, although I have no way of proving it).

* This is, again, a joke phrase, just as the digressive sentence wizard is getting worked up about was.

I am a bit confused here

isn't religion, by definition, supposed to require faith as opposed to the scientific endeavor of logical conclusion ?

so you are basically attacking christianity for being a religion instead of being a science? I mean, I guess you're free to do so, but it seems like kind of a weird thing to do

Well, to preemptively add on to that. Most religious people will say that science takes just as much faith due to the problem of induction (darn you Hume!). I think the empirical verification of science (and lets add on, say, the logical consistency as well for good measure) clearly blows this argument away.

[quote="MysticTerminator"]isn't religion, by definition, supposed to require faith as opposed to the scientific endeavor of logical conclusion ?

so you are basically attacking christianity for being a religion instead of being a science? I mean, I guess you're free to do so, but it seems like kind of a weird thing to do[/quote]  No: this particular section of the bible suggests that Thomas should have believed in something without verification, even when such verification was available.  It is only fundamentalist religion that claims that secular truths (including everything that we can deduce by empirical methods, i.e. science) must submit to religious truths.  Most religious people don't believe this -- a typical solution (e.g., that of mainstream Christianity in this country) is to claim that religion offers "spiritual" or "moral" truths, rather than empirical truths.  The holdouts here are, for example, young-earth creationists and others who insist that if science tells them something that disagrees with their religious understanding, it is science that must be wrong.  It's a wonderful PR trick that this completely insane position has been represented as the mainstream religious view in this country.  (It's also a position that is not widely shared by religions outside of Christianity -- I imagine, for example, that many followers of Shinto or Buddhism consider themselves religious but simply don't conceive of their religious views as substitutes for, say, astrophysics.  And most Jews don't think the Old Testament is a textbook in evolutionary biology.)

[quote="HilbertThm90"]Most religious people will say that science takes just as much faith due to the problem of induction[/quote]  I think that the "most religious people" here is wrong in the same way that Mystic's "religion" was wrong -- this view is more-or-less limited to one particular group of people and that most religious people, if questioned closely, will submit that "things I drop fall towards the earth due to gravity" requires less faith to believe than "God exists."

Mystic – What I mean is that many campuses are not open to any discussion that doesn’t follow the liberal, multiculturalism, anti-male view that many in leadership positions in college administrations hold.  These are views that promotes victim hood and often deny personal responsibility for one’s actions.  Many professors, in particular, are not allowed to engage in open discussion of their personal beliefs of issues with students.

You may be blessed with a campus that doesn’t have this same reaction.

For those of you that want specifics, consider the case of the lacrosse team that was accused of sexually assaulting the woman.  Their faculty Senate made extreme statements against the students, while the investigation was proceeding and after the acquittal, the disbarment of the prosecuting attorney for violating the accused rights so much that he violated the minimum ethical requirement for being an attorney in the state, refused to issue a retraction or apology, even when requested by the students they wrongfully attacked.

I’ve heard from many students and many universities that they are not given the same freedom to politically disagree with the political correctness that you express your college blesses you with.

There are other cases I’m aware of at local colleges where, book selections, speakers and subjects are reviewed and raised in performance evaluations, simply because of the political disagreement of the administration with the professor’s views.  I understand that one major university, I believe it was Harvard has acknowledged this bias in on campus dialogue by designating one chair for conservative thought.

JBL – I’ll again ignore your personal attack.

I could assume many things about you and your behavior, but I restrict my posting to responding to what you have posted, rather than assume conduct.

Sorry you are concerned with what I was insinuating.  I was merely trying to avoid an attack that wouldn’t be appropriate here.  Regardless of what someone might think I was insinuating, I will expressly say that many of the commentaries are accurate in most of there statements, so no luck was needed, but I think most Bible scholars wouldn’t take that reference as a serious commentary.

I do believe that if you are seeking to talk about what the Bible says, it is always a good idea to start with the Bible itself.

Logically, your quote (John 20:28) doesn’t support your position.  There is nothing that says Jesus is asking people to accept anything without any support.

Let me set forth my thoughts on this.  (You can still think I’m being disingenuous, but logically I can’t type anything to address this view.)

1)	Thomas was not present when others he trusted saw the resurrected Jesus. Without being asked, Jesus showed them his hands and side. (John 20:19-20 & 24)
2)	Thomas said that he would not believe them unless he saw Jesus himself and . . .  (John 20:24)
3)	Thomas saw the resurrected Jesus and confessed belief. (John 20:26 & 28)
4)	Thomas was chided with Jesus saying, “Because you have seen me, you have believed: blessed are those who have not seen and yet have believed.” (John 20:29)

I believe it was Thomas’ lack of faith in others statements, that Jesus was contrasting the blessing other would have if they didn’t emulate Thomas’ behavior.

All of Scripture belief is based on hearsay.  (I don’t know where you get rumors.)  Who said that hearsay isn’t evidence?  Most evidence in the world and most of what we all believe is based solely on hearsay.  Why wouldn’t this be acceptable?

Without hearsay, how do you know who wrote the Declaration of Independence?  Or even that it isn’t a recent creation?

I’m not trying to be difficult, I just don’t understand what you are trying to say.

I jump past the posts of Mystic and Hilbert, having nothing to add here.

JBL- Please explain what verification was available to Thomas?

Assuming you are correct for this analysis, what relevance would the assumed fact that most religious people hold a view have to what a religion teaches?

I don’t think that anyone said that one’s religious views are a substitute for astrophysics.  I don’t believe that anyone would say that the Old Testament is a textbook on evolutionary biology.  (Many might deny evolution occurred, but this is different.)  (Incidentally, what Christians call the Old Testament, is called Scripture, normally with differing names, by most Jews, as they deny Scriptural authority to what we Christians call the New Testament.)

JBL It isn’t really fair to compare one science statement with the existence of God.  I think Hilbert is right.  The comparison is really between two world views (even though most of the views encompass things outside the world):  one with God and one without God.  You have to compare all things with and without God to evaluate Hilbert’s statement about the amount of faith required.

[quote]I think these points deserve some follow-up. (Also, I'm in a bad mood, so picking on the seemingly defenseless seems like fun.) For number (3), where did you hear this? Do you think you should trust this source less in the future? Do you usually accept comments of that sort without any sort of verification on your part? On (5) -- chain letters are one of the most notoriously faulty forms of communication ever devised. Why would you assume a chain letter you received was true? Do you think in the future you might verify, especially before subjecting others to the letter? Similarly, for number (6) -- you trust someone who evidently also doesn't bother to verify if the things they're sending you are true? Shouldn't this make them suspect more generally? Perhaps you should suggest to this person that they do their own verification before they send things out that claim to be true. Maybe you should be asking them some of the same questions I've asked you.
[/quote]

All of your questions are quite demanding... all I'm going to say is yes, I'll be more careful about such things as chain mails in the future. If you want any more of an answer please be more polite and ask more reasonable questions. I would then be glad to answer.

Does it usually take you three lines of text to ignore something?  (Also, your non-quoting relevant sections of posts makes it largely unclear what you're responding to.)

[quote]I believe it was Thomas’ lack of faith in others statements, that Jesus was contrasting the blessing other would have if they didn’t emulate Thomas’ behavior.

All of Scripture belief is based on hearsay. (I don’t know where you get rumors.) [/quote]  I thought this portion of my comments were self-explanatory, but apparently not, so lets try again.  You are arguing that this is what happened:

A bunch of Thomas' buddies come to him and say, "Yo dude, the most amazing thing just happened -- Jesus like, totally rose from the grave and stuff!  It was awesome, you shoulda been there!"  Thomas, quite reasonably, calls bs on this outrageous claim, and demands proof.  Then Jesus shows up and [i]chides Thomas for not accepting the hearsay and rumors of his reincarnation on faith.[/i]

Now, I was giving a charitable (and more sensible, frankly) interpretation of Jesus' words here.  But, you appear to be intent on arguing that the moral of the story of Thomas is that one should accept hearsay and rumors on faith instead of seeking to verify said rumors.  I don't really want to waste more of my life discussing this point with you, so hopefully if you still don't understand, someone else can fill you in.

Harvard has recognized a bias, but I would be hesitant to say that expressing a conservative viewpoint is [i]not allowed[/i]; many of our most popular introductory government classes, for example, are taught (dominated!) by noted conservative Harvey Mansfield, I believe. Your citing the DA in the Duke lacrosse case is not particularly relevant; Nifong was obviously not affiliated with Duke in any way.

This, of course, isn't a rebuttal to your statements, as at this point I can only present at best anecdotal evidence, having only attended a few colleges.


geometry trigonometry percent calculus integration function algebra

REsults are out guys....

as for me--didnt do well...not willing to comment on my marks

dei Rohit it was me agian who saw the results first and told ranga to see.... :D

is the server stuck or can firefox not display the result page
after entering the roll no d page goes blank.. any clue??

dei i did worst again  :D Got more than what i expected but lesser percentile  :o  :o

Pathetic FIITJEE indeed.. I got 195(after expecting 228)..
94.7 percentile..
and yeah Chem Rock got 196..

How come the 90 percentile score increased?  :huh:  I felt that this test was tougher than FT3  :maybe:

that's what i said was shocking da  :o  This time 90 percentile score it seems is 172 and i thought this exam was tougher than FT3  :maybe:   :(

Depressed. 204 96.4 percentile.

WTH how did 90 percentile cutoff become so high???? I think our preparations are much lagging.

[hide]
Was kidding. I didn't write the test.  :rotfl:

[/hide]

I am almost afraid to look :( Didnt check yet... But I am sure its going to be pathetic... And will PM your NSEP marks to you guys if I get to talk to Shiv Ku.... He isnt picking his phone guys....

Yeah I got 196 obviously for the first time 10 marks higher than expected :)
But still I was shocked to see the 90 percentile score becoming so high :o
Anyway it is mainly due to physics I got a total of 60 in physics(by Pardesi's standards quite very ordinary) I got only 65 percentile which goes to show that physics kattans are stamping their authority on FIITJEE paper :maybe:

[quote="pardesi"]ok but i had seen the problem before...and i am  sure u wud have too :D
i don't see any point in having such a question[/quote]

99 Percent of such problems have trivial (I mean elementry) solutions. So people get the simple solutions involving the first few natural numbers and then gunsa.... which turns out to be right.

[quote="pardesi"]ok but i had seen the problem before...and i am  sure u wud have too :D
i don't see any point in having such a question[/quote]
Thats why i said iitjee is killing maths with the objective pattern  :D

Don't start it all over again :roll:  :roll:
I told u a million times u are looking at olympiad standards da :o

Actually they are doing a great job considering how difficult it is giving obj sums in maths especially when ppl gunsa 3-4th of the paper.

LOL! I usually substitute and find the answer for atleast 50% of the maths paper....It may not be the correct method but I run out of time because I spend most of it doing physics and I'm tuned into the new pattern.

The worst part in this is that they give full length unsolvalable co-geo sums in objective with riduculous options and 99% sure to get -1 in those :mad:  :mad:  :mad:
[quote="valeriummaximum"]Even me, idevote upto 1-1.5 hours for physics.....
30 min for chemistry...
1 hour for maths....
If i don get sums in maths, i get so dejected that i leave them and go to unsolved problems in chem and phy..[/quote]
This is the (n+1)^n^n times I am saying this start with maths da for atleast a change :mad:

[quote="madness"]I knew you would have said this. Ok da let iitjee be objective  :) . But they have no right to give such sums which can be easily gumsa'd, they could rather give sums which cannot be gumsad ok?  :D[/quote]

True. But it is not easy setting such problems however hard you try because ppl always find some way to gunsa, except in Co-Geo. Even if you think its impossible to gunsa, ppl (Like srini) manage to find out techniques which really surprise. And anything generic you give, we will substitute n=1.

As sathej, said, we aren't really learning much or improving our standards, we are just preparing for IITJEE.

P.S Lets wait Anit Sahu and the rest to post their marks, shall we? Wonder if its a paltry 250 this time.

anit didn't give the exam :D  :D  :D  :D  :D  :D  :D  :D

Why? He's online now  :D  :D
And as usual he's raving that he's gonna get single digit marks in incho  :rotfl:

[quote="chemrock"]Shrids did the height of comedy:
In one integral with limits from x1,x2 belonging to some set A what is the value?
Shridhar did x1=x2 so it is 0
How foolish I looked solving that :mad:  :mad:[/quote]

dei! :D
what i did was logically and gunsa-ically correct only! if x1, x2 belong to the domain, who is stoppin u from putting x1= x2??
NOBODY CAN STOP YOU!!  :rotfl:

and besides, i had another logical proof for that which convinced me that the ans was 0 :
Every integral is just an area under the graph of that particular function. All the 4 choices given were constant values.
The only possible function in which area under the graph is constant between ANY 2 points on the curve is a f(x) = 0 function..

Q.E.D  :P


algebra
polynomial
function
calculus
integration
ratio
absolute value

Let $ 0<a<b<1$ and $ I$ a non-trivial interval (which is not reduced to a point). Prove that there exists a polynomial with integer coefficients such that all the values $ p(x)$ when x is between a and b fall in the interval I.

you can prove the following.
$0<a<b<1$ and $f$ a continuous function. Then there exists a sequences of integer polynomial that converge uniformly towards $f$ on $[a;b]$.
It is a good pb.

I know that theorem. Indeed, it is a nice problem. What about a complete solution?  :D

This second one might look stronger, but they are equivalent :).

Since the real polynomials are dense in $C[0,1]$, it suffices to show that the required conclusion holds when $f$ is a polynomial. Next, we observe that it suffices to show it when $f$ is constant, because after we do that, by multiplicatios with powers of $x$ and additions (of the approximations of some constant functions), we can approximate any polynomial.

On the other hand, approximating a constant function $f:[a,b],\ f(x)=t,\ \forall x$ with a sequence converging uniformly of integral polynomials reduces to finding, for any $I=(t-\varepsilon,t+\varepsilon)$, an integral polynomial which takes values in $I$ for all $x\in[a,b]$, and this is Harazi's problem.

Edit: working on that problem... :)

Actually, I thought I had a rather nice approach, looking first at series with integral coefficients which converge absolutely for $x\in[a,b]$, and then truncating them, but I think it only works for the case when $I$ contains $0$ (i.e. I can make the values of the polynomial as small in absolute value as I want). I may be close, but I have to go, and I bet someone will solve it before I get back :).

I think of the following construction:
If we take sufficiently large prime $p$ and take $\frac rp$ in the interval, then the polynomial
$r(p^{p-1}(x-\frac 1p)(x-\frac 2p)\cdots (x-\frac {p-1}p))+\frac 1p)$ is good.

Here are my solutions for the initial problem, posted by Harazi
and the generalization posted by Alekk. Of course the first
trivially follows from the second but it has a much simpler and
nicer solution:

 a)We see that $f(x)=2x^2-2x+1$ satisfies $|f(x)-\dfrac 12|=2(x-\dfrac 12)^2$ hence if $x \in [a,b]$ the $f(x) \in [0;1]$ and we see
 that the number $f(f(\cdots(x)))$ ($n$ times) converge to $\dfrac 12$
 and  $2|f(f(\cdots(x))-\dfrac 12|=(2 |x-\dfrac 12|)^{2^n}$ and it tends
 to $0$ as $2|x-\dfrac 12|\leq max \{|2a-1|, |2b-1| \}<1$.

 Thus we may get $x$ as close to $\dfrac 12$. Now we can easily
 find a polynomial $Q$ with integer coefficients s.t. $P(\dfrac 12)\in I$ (just write some number of form $\dfrac a{2^k}$ from the interval in binary form and associate the
 polynomial).
 By the continuity of $Q$, taking sufficiently big $n$ will ensure
 us $Q(f(f(\cdots (x))) \in I$ ($f$ repeated $n$ times) as
 desired.

 b) For this problem, we need some additional results:

 (*)Weierstrass Theorem: Any continuous function $f$ on some finite interval may
 be approximated as close as we wish by a certain polynomial $P$
 in $\mathbb R[X]$
 Here's the proof:

 Lemma 1: For all
sufficiently big $n$ we have for any $0<\alpha<2$

$|1-(\dfrac 1{1+\alpha^n}+\dfrac 1{1+(2-\alpha)^n})|<t$ for some $t>0$.

Proof: By clearing denominators we must show that $b=1-\alpha^n (2-\alpha)^n <t(1+\alpha^n)(1+(2-\alpha)^n)$. WLOG let $\alpha>1$.
Now we see that $\alpha(2-\alpha)=\sqrt[n]{1-b}<1-\dfrac bn$ by
Bernoully. Thus $\alpha>1+\sqrt {\dfrac bn}$ and hence
$\alpha^n>1+\sqrt {bn}>1+bn$ and hence we deduce that
$t(1+\alpha^n)(1+(2-\alpha)^n)>t(1+bn)$ and for $n>\dfrac 1{t}$ we
have the claim.

Lemma 2: For any $M>0$ the function $\dfrac 1{1+x}$ may be
approximated as well as we wish by a polynomial $f$ on the
interval $[0;M]$.

Proof: We want to find such a polynomial $q$ s.t. $q(-1)=-1$ and
$|q(x)|<\epsilon$ for any $x \in [0;N]$, then we can take
$f(x)=\dfrac {q(x)+1}{x+1}$. Now note that the function
$h(x)=-e^{-(x+1)^N}$ where $N$ is sufficiently  big satisfies all requirements we seek for $Q$ and moreover may be approximated as close as we wish by a polynomial $Q$ on $[0;M]$ since the Taylor expansion of $e^x$ has radius of convergence infinity hence may be approximated by a polynomial.

We prove Weierstrass Theorem for the interval $(0;l)$ where $0<l<1$. Pick up some $\epsilon>0$ .Since $f$ is continuous it's uniformly continuous and hence we may find $n$ s.t. if $|a-b|<\dfrac 1n$ then $|f(a)-f(b)|<\epsilon$. We may suppose that $f(x)>1$ for any $x$ in our interval. Now divide the interval by $n$ points $l \dfrac 1n, l \dfrac 2n,\cdots, l \dfrac {n-1}n, l$. Now let $a_i=f(l \dfrac in)$. Let $g_i(x)$ be functions that satisfiy
$g_i(\dfrac in)=a_i$, $g_i(\dfrac jn)<\dfrac {\epsilon} {2n}$. We construct the functions in such a way: $g_i(x)=\dfrac {a_i}{1+(\dfrac {2n(x-x_i)}l)^{2N}}$ where $N$ is chosen large enough. Now let's prove that $g(x)=\sum_{i=1}^{n} g(l \dfrac in)$ satisfies $|g(x)-f(x)|<2\epsilon$. Indeed, let $l \dfrac in<x<l \dfrac {i+1}n$. We see that $g_j(x)<\dfrac {a_i}{1+2^N}$, not semnificative for sufficiently big $N$. Thus we have to look only at $g_i(x),g_{i+1}(x)$. Recall that $|a_i-a_{i+1}|<\epsilon$ by our choice of $n$. However if we put $\alpha =\dfrac {n (x-l \dfrac in)}l$ then $2-\alpha=\dfrac {n(l\dfrac {i+1}n-x)}l$ and now the lemma 1 does the job for $t<\dfrac {\epsilon} {||f(x)||}$, where
$||f(X)||$ is the supremum of $f$ on the interval. Finally, note that we can approximate $g$ by a polynomial as good as it's possible by Lemma 2.

(**) Any polynomial in $\mathbb{R}[X]$ may be approximated as close as we wish by a polynomial in $\mathbb{Q} [X]$. This is really obvious.

(***) For prime $p$, $\dfrac 1p$ can be approximated as close as we wish by ratio by a polynomial from $\mathbb {Z}[X]$ and $(a;b)$ (Note: approximation by ratio means that the ratio can be made as close to $1$ as we wish).

Proof: We have $\dfrac {(1-x)^{p^m}}p$ tend to zero as $m$ tends to $\infty$ (this is because $1-x<1-b<1$. However all coefficients in
$(1-x)^{p^m}$ are divisible by $p$, except the free term and the leading one. However we can ignore the leading term as it tends to $0$ as $m$ tends to $\infty$. Thus the polynomial $\dfrac {1-(1-x)^{p^m}+(-1)^{p^m} x^{p^m}}p$ is our desired polynomial that approximated $\dfrac 1p$.

(****) Any polynomial in $\mathbb{Q}[X]$ can be approximated as close as we wish by a polynomial in $\mathbb{Z}[X]$. This directly follows from (***).

Finally, the problem directly follows from the results above: by (*) we build an approximating polynomial $P$  from
$\mathbb{R}[X]$, by (**) we turn it to a polynomial from $\mathbb{Q}[X]$ and by (****) we transform it into a polynomial from $\mathbb{Z}[X]$.

[Note: (***) or (****) provides an alternative solution of the original problem]


I just took a take home contest that was due on Friday, and I found them pretty dificult (but I'm kinda stupid), so I was wondering how would AoPSers do these problems. they are proof based:

1.  You are standing in the center of a square room. Mirrors cover all four walls. Prove that
you cannot see the back of your head. [Technical restatement: a light ray leaving the
center point cannot next return to the center point from the diametrically opposite
direction. Assume a light ray that strikes a corner reverses direction.]

2.  Are there any natural numbers that can be represented both as the product of two and
also four consecutive natural numbers?

3.  In a convex quadrilateral ABCD, sides AB and CD are each divided into five equal
segments, and corresponding points are joined as in the figure. Prove that the area of the
center section EFGH equals one-fifth of the area of ABCD. (i've attached a picture of it)

4.  Suppose there are 2^n balls in several baskets. An allowable operation consists of
choosing two baskets, say A with p balls and B with q balls, p>=q (greater than or equal to), and moving q balls
from A to B. Show that after a finite number of allowable operations all of the balls can
be brought to one basket.

So if you can solve these problems and post your solutions/methods that would be great! :D

hmm...maybe i posted this in the wrong section, can a moderator move this to the pre-olympiad section under high school?

for problem 4, first prove if they are devided into 3 equal parts, then the area in the middle.

[quote="stupidkid"]Come on this is the AoPS, there are many people who are IMO medalists, these problems can't be too hard. So can someone please help me out?[/quote]Nobody is required to solve your problems, especially when you are trying to cheat on a contest. Sorry but topic is locked!


ARML geometry calculus integration trigonometry

1.
In right triangle $ABC$, $\angle C = 90^\circ$, $c=3$, and $a+b=\sqrt{17}$.  Compute the area of the triangle.

2.
If $(a+b):(b+c):(c+a) = 6:7:8$, and $a+b+c=14$, compute the value of $c$.

3.
Let $z$ be a root of $z^5-1=0$, with $z \not= 1$.  Compute the value of

\[z^{15} + z^{16} + z^{17} + \cdots + z^{50}. \]

4.
A regular polygon of $2n$ sides is inscribed in a circle of radius 5.  The sum of the squares of the distances from any point on the circle to the vertices of the polygon is 600.  Compute the value of the integer $n$.

5.
The circles, each of radius 5, lie in perpendicular planes.  They intersect in two points, $X$ and $Y$, thus determining a common chord $\overline{XY}$.  If $XY = 8$, compute the distance between the centers of the circles.

6.
How many positive integers less than 54 are equal to the product of their proper divisors?

7.
Find all positive integers $n$, less than 17, for which $n! + (n+1)! + (n+2)!$ is an integral multiple of 49.

8.
In triangle $ABC$, $a \ge b \ge c$.  If $\displaystyle \frac{a^3+b^3+c^3}{\sin^3{A} + \sin^3{B} + \sin^3{C}} = 7$, compute the maximum possible value of $a$.


If $ f(x,y,z) \equal{} \sum\frac {x(2y \minus{} z)}{1 \plus{} x \plus{} 3y},$ where $ x,y,z\ge0$ and $ x \plus{} y \plus{} z \equal{} 1.$

Find the maximum and minimum value of $ f(x,y,z).$

hello, is this your sum $ f(x,y,z)\equal{}\frac{x(2y\minus{}z)}{1\plus{}x\plus{}3y}\plus{}\frac{y(2z\minus{}x)}{1\plus{}y\plus{}3z}\plus{}\frac{z(2x\minus{}y)}{1\plus{}z\plus{}3x}$?
Sonnhard.

[quote="Dr Sonnhard Graubner"]hello, is this your sum $ f(x,y,z) \equal{} \frac {x(2y \minus{} z)}{1 \plus{} x \plus{} 3y} \plus{} \frac {y(2z \minus{} x)}{1 \plus{} y \plus{} 3z} \plus{} \frac {z(2x \minus{} y)}{1 \plus{} z \plus{} 3x}$?
Sonnhard.[/quote]

yes  :D

hello, simplifying your $ f(x,y,z)$ i have got
$ f(x,y,z)\equal{}\frac{49xyz}{(1\plus{}x\plus{}3y)(1\plus{}3x\plus{}z)(1\plus{}y\plus{}3z)}$ so we have $ f(x,y,z)\geq0$ for all $ x,y,z$ with $ x\plus{}y\plus{}z\equal{}1$ and $ x\geq0,y\geq0,z\geq0$, further we get
$ f(x,y,z) \le \frac{1}{7}$
Sonnhard.


induction
integration
calculus
limit
Euler
derivative
calculus computations

It can be shown by mathematical induction that:
\[ \sum_{k\equal{}1}^{n}k^2 \equal{} \frac{n(n\plus{}1)(2n\plus{}1)}{6}\]

Use this fact to give a formal proof that: $ \int_{0}^{1}x^2 dx \equal{} 1/3$

Using the definition of the Riemann integral partition [0,1] in to n pieces each 1/n units long. Then

$ \int_{0}^{1}x^2 dx \equal{} \mathop{\lim}\limits_{n \to \infty}\sum_{k\equal{}1}^{n} \frac{1}{n} \left( \frac{k}{n} \right)^2\equal{}   \mathop{\lim}\limits_{n \to \infty} \frac{1}{n^3} \sum_{k\equal{}1}^{n}k^2 \equal{} \frac{1}{3}.$

Hi;
First make use of the fact that:

$ \sum_{k\equal{}1}^{n}k^{2}\equal{}\sum_{k\equal{}0}^{n}k^{2}$

and with
$ f(k) \equal{} k^2$
Using the first couple of terms of the Euler Mclaurin summation formula which links sum, integrals and derivatives together.

$ \\ \sum_{k\equal{}0}^{n}k^2\equal{}\int_0^n f(k) \, dk \plus{} \frac{f(n)\plus{}f(0)}{2}  \plus{}\frac{f'(n)\minus{}f'(0)}{12} \plus{}\frac{f'''(0)\minus{}f'''(n)}{720} \dots$

We don't need the last term because it is zero.

$ \\ \sum_{k\equal{}0}^{n}k^2\equal{}\int_0^n f(k) \, dk \plus{} \frac{f(n)\plus{}f(0)}{2}  \plus{}\frac{f'(n)\minus{}f'(0)}{12} \dots$

Rearranging:

$ \\ \sum_{k\equal{}0}^{n}k^2\minus{} \frac{f(n)\plus{}f(0)}{2} \minus{}\frac{f'(n)\minus{}f'(0)}{12} \equal{}\int_0^n f(k) \, dk$

$ \frac{(n*(n\plus{}1)(2n\plus{}1))}{6}\minus{}\frac{n^2}{2}\minus{}\frac{2n}{12}\equal{}\int_0^n f(k) \, dk$

simplifies to:
$ \frac{n^3}{3}\equal{}\int_0^n f(k) \, dk$

Now with n=1:

$ \int_0^1 f(k) \, dk \equal{} \frac{1}{3}$


Hi

I am trying to solve this set of equations

2*x1 - x2 - x3 - x4 = 1
-x1 - 2.5*x2 - x3 - x4 = 1
-x1 - x2 - 3*x3 - x4 = 1
-x1 -x2 - x3 - 3.5*x4 = 1

Now I know this can be solved if the variables x1, x2 etc are negative, but is there a way to find a solution so that they are positive? If it helps the value that the equations are equal to (i.e. 1) can be modified if this helps to find a solution.

Thanks in advance for any help you can give me.

Jon

As written, the unique solution to that system is $x_1=\frac{10}{67},x_2=-\frac{20}{67}, x_3=-\frac{15}{67},x_4=-\frac{12}{67}$
Are you sure you wrote the right coefficients here? Changing the 1 would merely scale the solution.


group theory
abstract algebra
superior algebra
superior algebra solved

let $H$ be a group, $a \in H$ ,and $\phi:Z\rightarrow H$ the exponential homomorphism $\phi(n)=a^{n}$ . What would the possibilities of Kern $\phi$ be?

Sorry when I tell something that you didn't want to read:

You can have $ker( \phi )$ as every subgroup of $\mathbb{Z}$ (easy with homomorphism theorem)
More directly:
To get $ker( \phi) = m \mathbb{Z}, m \in \mathbb{Z}$ you have simply to chose $H=\mathbb{Z}_m,a=1$
In the other direction this are all subgroups of $\mathbb{Z}$ and since the kern has to be a subgroup, this are all possibilities.


We easily have $ mn\ge1, mn\ge m, mn\ge m$
Adding up these inequality and we find the solution


$ \begin{gathered}
  {\text{Determine all the solutions of the system}} \\
   \\
  x \plus{} y \plus{} z \equal{} w \\
  \frac{1}
{x} \plus{} \frac{1}
{y} \plus{} \frac{1}
{z} \equal{} \frac{1}
{w} \\
\end{gathered}$

We have:
$ (x\plus{}y\plus{}z)(\frac{1}{x}\plus{}\frac{1}{y}\plus{}\frac{1}{z})\equal{}1\Leftrightarrow\frac{(x\plus{}y)(y\plus{}z)(z\plus{}x)}{xyz}\equal{}0$
$ \Leftrightarrow x\equal{}\minus{}y,y\equal{}\minus{}z,z\equal{}\minus{}x$

Though the above solution is correct, it is not complete. One still has to mention stuff like $ x, y, z \ne 0$.

isn't that obvious from the quetsion itself that $ x,y,z \neq 0$
otherwise the question fails. :D