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Are there any strictly moral objections to incest besides those based on genetic undesirability ?

I think the danger is more than genetic "undesirability". Incest multiplies the likelihood of the two parents both having recessive alleles for a rare, debilitating disease like cystic fibrosis. In fact, it is well documented that several generations of the English royal family were plagued with hemophilia because they kept inbreeding with each other.

I would think that any power imbalance may have more drastic effects if the members of a couple are part of the same family. A powerful person outside the family would have less of an influence than a relative. An older, richer or more influential relative (parent, sibling, uncle, aunt) may force a younger person into a relationship (with him/herself or another favored relative). It may be even worse in sparsely populated places, or where the movements and social contacts of young people (and especially females) are severely restricted.

Okay, so let's suppose we're talking about brother-sister relationships rather than one with some large age difference since otherwise the general arguments against statutory rape come into play and that's not the phenomenon I want to isolate; similarly, the possibility of passing on two pernicious recessive alleles is exactly what I mean by genetic undesirability. Also note that it is possible to implement, in practice, a situation where the genetic argument would be disallowed -- suppose we are only talking about sex with birth control safeguards and so forth.

I'm not necessarily looking for a terribly detailed answer; it is simply that I have asked, well, quite a few of my friends now and we have not been able to think of anything, yet there still seems like there should be some moral objection, so what have we overlooked ?

Is it not controversial on having a discourse on a topic like this, on AOPS? If not controversial , I am really surprised!

I think there are better places to discuss this topic (I know it is the Round Table), not meant to offend anyone.

I'm not sure I understand the question.  I assume you're not asking why, in a scientific or historical sense, incest is considered wrong.  Natural selection has programmed us to abhor incest (for the genetic reasons discussed), which leads to cultural taboos against incest, which then reinforces our natural disposition against incest.  In the end, incest turns out not to be a big problem, because no one is interested in doing it, and everyone wants to prevent it.

Since incest is forbidden by just about every religion and is considered taboo by just about every human culture, and most people consider it to be disgusting.  Therefore, in some sort of tautological sense, incest is immoral.

But I'm guessing that you're trying to get at something like, "Is there any purely logical reason to consider incest immoral?"  That's still hard to answer, because you could ask the same thing about (unjustified) murder.  Why is murder immoral?  At some point, you need to start accepting some axioms, right?

So without actually stating what our secular "universal" morality axioms are or should be, the question is something like, "Is the taboo against incest logically independent of other axioms?"  I would say yes.  For example, as disgusting as it is, there's no purely logical reason why siblings, using birth control, shouldn't be permitted to have sex, purely as a matter of law.

Obviously, one might consider running the same analysis on homosexuality, and the obvious question is why natural selection "successfully" programs us to abhor incest but "unsuccessfully" programs (some of) us to abhor homosexuality.  I leave that explanation to a biologist, but it's not hard to come up with various hypotheses.

[quote="MysticTerminator"]Okay, so let's suppose we're talking about brother-sister relationships rather than one with some large age difference since otherwise the general arguments against statutory rape come into play and that's not the phenomenon I want to isolate; similarly, the possibility of passing on two pernicious recessive alleles is exactly what I mean by genetic undesirability. Also note that it is possible to implement, in practice, a situation where the genetic argument would be disallowed -- suppose we are only talking about sex with birth control safeguards and so forth.[/quote]If there is no imbalance of power and no risk of genetically "damaged" offspring, I do not see an objection. Religions and morals are based on perception of the past, when the possible arguments against incest could not be so easily dismissed. Maybe in a couple of centuries incest will no longer be taboo.
Disclaimer: My great-grandparents were cousins, but their offspring (grandma and four siblings) seemed perfectly normal to me.

[quote="buzzer11"][quote="beta"]Well it is definitely possible that opposite-sex siblings get separated at when they are young, and then they get married without knowing that they are siblings. (in fact this has happened before) What should they do then... it's not their fault?[/quote]

Where has this happened before besides on House?[/quote]

[url=http://www.cnn.com/2008/WORLD/europe/01/11/twins.married/index.html]http://www.cnn.com/2008/WORLD/europe/01/11/twins.married/index.html[/url]

Well, I guess its happened. But how [i]often[/i] does this happen, I should have asked.

There's nothing [i]technically[/i] wrong with incest. But its another one of those moral issues that can't be looked at in a right or wrong, black or white way. I think in case of an "accidental" incest, it should really be the couple's choice. Some will want to split. Others (maybe) would rather remain married. But is the concept of incest wrong? I think so. Why? Because I am stubborn. Frankly.

I don't like this justification that as long as contraception is used it should be fine. Contraception is not perfect. So I think as long as some "perfect" form of contraception is used, then it is not immoral (like hysterectomy).

I don't think it is hard to come up with a reason why natural selection has made incest in some sense a naturally mostly repulsive activity. In order for natural selection to work at its best, there needs to be genetic variety. Thus we are attracted to people that are genetically different from us. In the case of homosexuality, it is much easier to find someone of the same sex that is genetically distinct from you than an immediate family member.

This is all just speculation based on the basics of natural selection that I know. I have no sources, so this could all already have been scientifically shown false without my knowledge. But at least it makes sense intuitively.

No, there's nothing wrong with it. Why should two people be barred from having sex just because they share genetic material?

But I'm not sure it will ever be accepted by society (even though I'm pretty sure that society is persistently improving its ability to recognize good and bad), at least when the couple grows up as brother and sister. The idea that siblings shouldn't be having sex feels too deep-set to be overcome by morality arguments. Maybe it's like gay marriage -- the inherent weirdness goes away as you familiarize yourself with the concept -- but I don't really think it is.

First off, we're talking morality not barring something. And yes, there is a moral problem (and reason to bar it) with people that share genetic material having sex. Look at the third post. If pregnancy occurs, there is a severely high risk of debilitating disease.

I do not know of any official moral objections to incest, but I have my own moral objections:

If two people are not of the same generation, and they are at least 10 years apart in age, they should not marry. For obvious reasons. :)

If two people of the same age wish to marry, then they should be able to, provided that they are old enough to carry the burdens that marriage provides.

Afterward, if the child that they rear carries a rare disease, it's the parent's fault. They shouldn't have done the bad thing they did. It is their responsibility to rear that child, and they should love it as they would love any other child of theirs.

@HilbertThm90: It is really dangerous to start discussing that it should be banned because of the poor genetic material of the offspring. This opens a whole can of worms about eugenics type thinking. Why is not ok (anymore) to discuss eugenics in other contexts? Is it ok to fornicate with a person from a culture with, say, low intelligence? (However controversial it may be, there are areas where the average IQ is like 70).  Should we allow children with down's syndrome? Thinking about some examples, clearly there are more variables to consider than just poor genetic material (and the offspring may be fine in many cases), so your argument is simply not good.

Secondly, the reasons for banning incest has originally nothing to do with science. I believe there is often something wrong when you start justifying the same moral law for different reasons depending on the time period. Compare with illegal narcotics. At least care needs to be taken to think about why the law is really in place.

I don't hold a view on the issue. Probably I think it's wrong, but I don't know why. As yenlee says, on a grand scale this is really a non-problem because people don't want to marry their sibling (but there is perhaps a real problem with parent/child relationships. I haven't thought about this). Out of some liberal harm principle I think it's probably ok for siblings to have a relationship, unless you think they are harming their potential children. I'm sure christian nuts who love nothing more than to stigmatize and bully "different" people thinks this.

@1=2: Your view is ridiculously naive. I detest that you think people who are 10 years apart in age shouldn't marry. Nonsense. Also, parents are responsible for their child, and playing some kind of blame game accomplishes nothing but to show your personal repulsion. There is a bigger risk that the child will carry a rare disease, but this doesn't imply that it's the parents' fault. What exactly would be the % risk for the assessment that it's the parents' fault? There are many other genetic issues where maybe we should blame the parents?

[quote="HilbertThm90"]First off, we're talking morality not barring something.[/quote]

Then you're talking about nothing. "Is incest moral?" has no right answer, or even any answer that makes sense outside of somebody's personal moral framework. "Morality" in the universal sense is not well-defined. It makes a lot more sense to discuss the reasons we can think of for banning incest -- which encompasses morality -- than to discuss morality directly.

[quote]And yes, there is a moral problem (and reason to bar it) with people that share genetic material having sex. Look at the third post. If pregnancy occurs, there is a severely high risk of debilitating disease.[/quote]

It's not clear to me why bringing somebody into the world with a debilitating disease is necessarily a bad thing. They might not enjoy life much, but I think it might still be nicer for them than never existing.

Regardless: effective methods of contraception and abortion exist.

I find that "something is better than never existing" argument rather useless. You may also conclude that humans should produce as many children as possible to create more life  :wink:
One simply cannot evaluate nonexistance.

@SuperThread: "Then you're talking about nothing. "Is incest moral?" has no right answer, or even any answer that makes sense outside of somebody's personal moral framework."

What? Are you saying that the study of ethics is nothing? I know some people who make their living off being ethicists (well, philosophy professors that specialize in ethics). Of course it has no right answer. That doesn't mean there aren't good arguments on both sides that are worth discussing. You are a relativist in the most extreme sense. I tend toward a more absolutist view of ethics. Slavery is wrong. This does not depend on someone's moral framework.

In addition, arguments about policy are often greatly different than theoretical arguments.

@Kalle: OK. So scratch banning. I said the line above, and it fits perfectly in this category (which is why I brought it up). In traditional ethical language: One ought not engage in sexual relations with a close relative due to the genetic argument.

In religion, the potential genetic defects and the usual natural repulsion are, I think, manifested [i]evidence [/i]for the immorality of incest, as opposed to [i]reasons [/i]for the immorality of incest.

Actually, I have developed particular beliefs about the ideal characteristics and intrinsically present duties of certain relationships. I don't think a familial relationship ([i]especially[/i] a sibling relationship) and a romantic relationship can simultaneously exist between two people. Family members have certain unique duties towards each-other which are jeopardized by a serious romantic relationship. It also leads to a very awkward role for the parents, and this can lead to heightened family tension and possible unwanted conflicts of interest.

What about casual sex? On one hand, I do not object to it on the same moral terms as I would a serious relationship. On the other hand, one's choice of a romantic partner is much more particular than one's choice of a ƒ- friend, so I see no reason why someone can't find a ƒ- friend who's not in their family. In other words, while it's not as problematic, it is also much less justifiable before moral/family considerations are made.

[b]Summary:[/b] Incest? Don't do it.

[quote="TZF"]

Actually, I have developed particular beliefs about the ideal characteristics and intrinsically present duties of certain relationships. I don't think a familial relationship ([i]especially[/i] a sibling relationship) and a romantic relationship can simultaneously exist between two people. Family members have certain unique duties towards each-other which are jeopardized by a serious romantic relationship. It also leads to a very awkward role for the parents, and this can lead to heightened family tension and possible unwanted conflicts of interest.
[/quote]

Wait why can't a familial relationship and a romantic relationship  simultaneoulsy exist?  Which unique duties are you talking about, and why are they being jeopardized by a romantic relationship? The parents have nothing to do with a romantic relationship (their opinions are completely irrelevant -- I'm sure interracial marriages can lead to heightened family tension and possible unwanted conflicts of interest as well... is that an argument against interracial marriages?).

For me, I think it's totally okay for step-siblings or adopted-siblings ( not blood related)  to marry.

[quote="1=2"]
Afterward, if the child that they rear carries a rare disease, it's the parent's fault. They shouldn't have done the bad thing they did. It is their responsibility to rear that child, and they should love it as they would love any other child of theirs.[/quote]

That raises another interesting, and different question.  If a parent carries a gene for a disease and there is a high chance of it being passed off to children...should they not have children?  In particular, I'm thinking of Huntington's, which doesn't manifest itself until the individual is older.

@Sunny: Is it better to have lived for a short while than never to have lived at all?

@beta: Incest involves sex. Sex involves emotional romantic attachment. Familial relationship involves fighting at some point. Fighting with an emotional romanticly attached lover involves brokenheartedness. Brokenheartedness = bad, unlikely to trust again, esp. since one normally can count on one's family to be good.

Interestingly, this topic was brought up in the Jan 2009 issue of First Things. From a Christian perspective, they believe that incest is spoken against in the Scripture, and even non-Christians do not innately believe incest is all right. Therefore, it is possibly good to set incest=bad as an axiom. Not all things are worth finding a logical ground for; some things are innately bad, we all know it, but we don't really know why. Just because we can't prove the non-existence of something doesn't mean it exists. Just because we can't prove the badness of something doesn't mean it is good.

I think setting the incest=bad as an axiom is certainly harsh. There are arguments on both sides (in fact, this thread seems to indicate only a single argument exists for the axiom you claim and not even all people are agreeing with it). It seems much more likely that incest is amoral. I think that before setting an axiom like that, it needs to be irrefutably self-evident, and it certainly isn't.

I also disagree with "sex involves emotional romantic attachment."

It's not harsh at all. Marking this prohibition as an independent axiom means you can exclude it without changing any of your other moral beliefs. Removing another principle, such as deference to authority, from your morality has far more significant consequences.

[quote="TZF"]I don't think a familial relationship (especially a sibling relationship) and a romantic relationship can simultaneously exist between two people.[/quote]
This is a consequence of how our brains work- we're not attracted to the people we grew up living with.
The mythological template here (family member sent into exile very young, later comes back and marries close relative, everybody finds out and tragedy ensues) basically gets it right.

[quote="beta"]The parents have nothing to do with a romantic relationship (their opinions are completely irrelevant -- I'm sure interracial marriages can lead to heightened family tension and possible unwanted conflicts of interest as well... is that an argument against interracial marriages?).[/quote]
It [i]can[/i] be an argument against a [i]particular[/i] interracial marriage.

Look, unless you plan to run away and cut off contact with parents, they are a part of your life. I'm sure if your parents vehemently opposed the thought of you marrying someone, you would take that into consideration. It is worth vetoing only if you think you (and your potential) partner are willing to put in the effort to resolve the conflict and patch up the family, or if their opposition is greatly outweighed by the benefits of the relationship, or some combination. Keep in mind that it can be very strenuous on a partner if your parents are bitter about her. All of these judgments should be made first.

But there is also a big difference. There is also no inter-family conflict of interest in an interracial marriage. Husband's family will support husband, wife's family will support wife. What if the two families are the same? It would be like the plaintiff and the defendant having the same lawyer.

I'll respond to your other concern later -- it's packing time :)

[quote="solafidefarms"] From a Christian perspective, they believe that incest is spoken against in the Scripture, and even non-Christians do not innately believe incest is all right. Therefore, it is possibly good to set incest=bad as an axiom. Not all things are worth finding a logical ground for; some things are innately bad, we all know it, but we don't really know why. Just because we can't prove the non-existence of something doesn't mean it exists. Just because we can't prove the badness of something doesn't mean it is good.[/quote]

Hmm I'm not a Christian myself so it's very interesting to hear the Christian perspective. But it's clear that Adam and Eve's children were incestuous (either that or "God" created new people to avoid incest from happening...I don't know enough about the Bible).

I do consider incest between blood-related sibling or close cousins to be strange, wierd, and unnatural(Even though according to Freud, we did have incestuous thoughts when we were young- we wanted to have sex with our mothers) but I wouldn't go as far as saying it's immoral as long as they don't have kids.

Some Christians believe the creation account was a metaphor, and evolution actually happened. Others believe creation happened, but Adam and Eve were perfectly heterozygous and we've only been losing genetic information since Eve ate the apple. I believe Moses delivered God's warning to Israel against incest, as after >1000 years and two genetic bottlenecks down to small populations [Adam and Eve and Noah, his three sons, and their respective wives] there was the potential for ill.

I will recuse myself from the morality debate since I have hardly any experience with philosophy or religion so the term "strictly moral objections" doesn't really mean much to me. I basically have no idea what “morality” is. Perhaps I will figure it out one day. The word “ethical” means a lot more to me than the word “moral” since “ethical” refers more to a social and societal rule which is to me more tangible than whatever “moral” refers to. So, instead let me try to answer a policy question that I can hopefully better respond to:

[b] Modified Question: [/b] Should close relatives be legally allowed to marry each other (in the United States)? Should close relatives be legally allowed to have children (in the United States)?

I am limiting this to the United States because I don't feel like I know enough about foreign cultures and the general public opinion on this issue in other countries. And I do think public opinion and culture is important here. In fact, its hard to even gauge the feeling in the entire United States, since there is such a large and diverse and distributed population here.

There are several criteria that should be considered and weighed against each other in responding to the modified question:

1) The benefit to the close relatives in allowing them these rights.

Marriage and child-birth and child-rearing bestows a tremendous amount of pleasure and happiness and comfort on the parents. It is an experience that binds all humans past and present together. Many people consider these milestones in life as precious and inalienable and feel that removing them would be to infringe on a basic human right.

So, telling a couple that they do not share these privileges with everyone else may make them feel ostracized or second-class or less-worthy than everyone else. And I think that having couples with these feelings is immeasurably not good.  So, those privileges should only be removed when very serious risks are at stake. In issues 2) and 3) we will try to examine how serious the risks are.

2) The “tangible” cost to society of allowing them these rights.

Obvious, the major risk of incestual marriage is the risk of having a child with genetic abnormalities due to combining two copies of a rare genetic trait. I think historical examples, modern epidemiological studies, and modern genetics show that this risk is very real.

Whenever parents have a disabled or otherwise genetically abnormal child, it is not only those parents that suffer. The child might require enormous amounts of medical attention and expensive operations which put strains on the hospital system. The public school system might need to hire additional personnel and build additional facilities and provide additional services to accommodate the child. When the child becomes an adult, he may have difficulty finding employment and supporting himself, putting strains on his employers who are required to abide by the "Americans with Disabilities Act", and government programs such as Welfare whose aid he may require.

Therefore, it is not just a "personal decision" whether a pair of close relatives have children. It is a decision that could potentially affect many many people and the “tangible” cost could be measured in “dollars” or in “man hours” or in various other real physical services.

3) The social or moral cost to society of allowing them these rights.

And there is finally there is the issue of culture. What I mean by this is how other people would view the marriage. As we have seen a little bit in this thread, there are some people here who appear to have strong feelings towards incest. If a vast majority of Americans think incest is “morally” wrong, then allowing couples to engage in that practice would make these Americans think of their country as a less “moral” place which is probably not an opinion we want to be popular (I know I said I would try not to use morality here, but I am because I think it is a big issue for a lot of other Americans, even though not for myself).

An incestual couple might defend their desire to marry by saying: “our marriage doesn’t affect all of these people against incest…so who cares whether or not they support our marriage. We are not harming these people. We should be able to do what we want as long as we don’t inflict harm on anyone else.” Issues 2) and 3) both show why this argument is short-sighted and overly simple.

So, the question really becomes "how strong is the opposition to incest in the United States?". In my experience, the opposition is very little. Perhaps that is because I am from a liberal, progressive family and I have spent a lot of my life in academic communities tending to contain other people with similar viewpoints. I have definitely never heard anyone openly oppose incest and I had hardly heard any discussion of it before this thread. I would say the opposition to incest is much less than the opposition to gay marriage. Furthermore, I think the opposition has been and will continue to steadily decrease with time.

So, while I have admittedly very little experience with this topic and very little idea what the current policies are, I would probably propose to try to allow incestual couples to marry (and have sex) but try to prohibit the most closely related ones from having kids.

Of course there is then the issue of whether this is possible i.e whether it makes any sense to allow incestual marriages but try to prohibit incestual children. When a couple gets married, is it feasible to require them to use contraception of some sort and punish them if they have a child? What if the contraception does not work and they have a child but it is not there fault? Well, I really have no idea how well contraception works, but I think that the state of technology is advanced enough so that it is probably reasonable to hold an incestual couple responsible for having a child when not having a child was a condition of their marriage (especially with legalized abortion, but its probably not a good idea to bring another controversial idea into this discussion).

Perhaps another thing to consider is genetic testing. I think it will not be too far into the future before we can test whether a couple has genotypes that put them at risk for all of the major diseases. I think this is perhaps the most likely solution to the problem of incestual marriage and procreation. But then that raises the issue of whether all couples intending to get married or have children should be required to have genetic testing done.

Gosh, whenever you start talking about one ethical issue, it seems lots of others abound and the discussion gets exponentially more complicated. I feel like in the course of this post, I have only raised more questions than I have answered. This is very different from olympiad problem-solving!

FibonacciFan: Not many strict liberals would factor in the "moral cost" to society in determining if the harm principle can be applied. An individual should be able to do anything that doesn't hurt anyone else, and offending peoples' moral codex cannot be seen as hurting someone.

The way you talked about how it might be worth to deny other peoples' rights based on "society's moral well-being" is just so wrong! When something like that is said in a historic context people always identify it as something terrible, but it's not caught when it goes on today.

Government funded health care and other things has made the following a big argument: "I don't want people to be allowed to do X because I pay for X-doers health care". The X-doers also pay to the government. The government funded health care was not put in place so you could extend your power over other people by claiming you own part of them because you pay your taxes. Stop using this argument. It is communist nonsense.

Hm... I prefer FibbonacciFan's question. I also think "I have hardly any experience with philosophy or religion so the term 'strictly moral objections' doesn't really mean much to me."

First, what is moral? Are we trying to decide whether incest is wrong or right? Well, I don't think it makes sense. "some things are innately bad, we all know it" is false. [i]I[/i] don't know it. I can discuss the question only if it is well defined.

Now, if we are restricted to the USA, first we need somebody who understands the USA laws... Kalle, maybe individual freedoms should be maximized, so people should be allowed to make incest. But if you think "an individual should be able to do anything that doesn't hurt anyone else, and offending peoples' moral codex cannot be seen as hurting someone", why can't nude people walk on the streets? It is against US laws, and I suspect this is something related to moral.

[quote="Valentin Vornicu"][quote="buzzer11"]Where has this happened before besides on House?[/quote]It happened on Nip Tuck too. True, the siblings there only shared 1 parent (does that diminish the immorality of the situation?)[/quote]

Okay, I admit I definitely was making an understatement when I implied that incest occurs only on House.  :)

I'm actually curious whether biologically, sharing one parent is less "dangerous" than two. (probably is, but I'm not very good in biology, so...)

Yes, certainly. The risk is that a child gets two copies of the same chromosome with a bad gene- one good copy is enough to avoid trouble, while two bad copies is a disorder.
With two shared parents, there is a $ \frac14$ chance per chromosome pair of a duplication. With one shared parent, it's $ \frac18$ per chromosome pair.

And the chances of simply carrying a bad (recessive) gene is higher as well?

REAL LIFE EXAMPLE FTW

SAT ESSAY FTW

My mom is a carrier of... well it is something that involves strep. I think. Not sure.

When I was born, I had that same disease. Fortunately, they cured it, or I didn't die from it. Look at me now! :D

I also have a little brother, but before he was born, mom was diagnosed for the disease, so Joshie didn't have it.

hmm... I think I will answer all of feliz's last post.

[quote="feliz"]Hm... I prefer FibbonacciFan's question. I also think "I have hardly any experience with philosophy or religion so the term 'strictly moral objections' doesn't really mean much to me."

First, what is moral? Are we trying to decide whether incest is wrong or right? Well, I don't think it makes sense. "some things are innately bad, we all know it" is false. I don't know it. I can discuss the question only if it is well defined.

Now, if we are restricted to the USA, first we need somebody who understands the USA laws... Kalle, maybe individual freedoms should be maximized, so people should be allowed to make incest. But if you think "an individual should be able to do anything that doesn't hurt anyone else, and offending peoples' moral codex cannot be seen as hurting someone", why can't nude people walk on the streets? It is against US laws, and I suspect this is something related to moral.[/quote]

Paragraph 1: Strictly moral objections means 'You think it's wrong based on your beliefs'(am I getting this right?) So for example, you would probably have strictly moral objections to killing.

Paragraph 2: Morals are basically your standard qualities or qualities you wish to obtain, such as bravery, honesty, kindness, mercifulness etc. but not being good at mathematics or anything like that. It's not physical.

Paragraph 3: More often than not, if someone (A) says to someone else (B) not to do something, then (B) would want to do it even more. But if (A) gives the freedom to (B), then (B) feels less inclined to do it in the long run. I would agree to your point there. As for the last two sentences, mainstream people like to wear clothes, and as a child we are trained to keep them on( :) ) and be ashamed when they are off and a stranger is near. Also, I have a very tiny suspicion that walking nude might suggest sexual intercourse, which mothers strongly but secretly discourage to their young ones.

@Kalle, who posted at Wed Jan 07, 2009 9:50 am: 9 year old and 19 year old. HUGE age difference(and height!). And as for the blaming, I was just saying that it was the parents fault. They shouldn't be arrested for it, but they should take the responsibility for it. If the parents knew about the disease and didn't want a child that had it, but decided to do it anyway, that is completely irresponsible. If they knew about the disease and didn't care, then it's not irresponsible, but I am not completely sure if it IS responsible. If they didn't know about the disease and had the child anyway, it was a foolish act on their part not to check for/remember the disease. Either way, they should bring up the child. If the parents didn't have any diseases, but the child did, that's another story. IMO a complicated one. So let's stick with this one for now. :)

In general, I don't think it's WRONG to have incest. The only thing that's wrong is not bearing the responsibility of the affects of incest.

Well, then I don't have moral objections. About the third paragraph, the point is that moral codex [i]is[/i] considered when we decide if people should be able to do something or not. So there is a moral cost/issue.

Our "study" of incest can be divided in two categories: incest without having sons and sons of a couple of relatives (I am assuming our contraceptive methods are efficient). The problem of incest without sons is just a particular case of the following question:

How much does moral should be considered in our social organization?

Well, there is no a unique answer, I think. It depends on what we want to do. As I said, maybe we want to maximize individual freedoms without causing disorder (if somebody thinks it is a good idea to try to maximize other thing, other points of view are welcome!). In this case, allow incest is give more freedom and does not cause problems in large scale, so I would allow incest if it the decision were mine. It would be good to almost everything that does not generate big problems. If we want to maximize happiness, we can observe that, allowing incest, some couples will be perfectly happy. Maybe some couples will be marginalized by people with several moral "axioms", but in general the permission would increase happiness.

Is this what you meant, Kalle? There is no problem in allowing incest because it offends just some moral codes, but allow people to walk without clothes in the streets would create a horrible confusion.

We can be observe that moral is irrational. Many people advocate rational thinking. So, if we agree moral is irrational and does not help many times (like in the incest case), we could "just" try to make people become gradually more rational (society is already becoming more rational, isn't it? But the process is very slow...). They would accept more and more things, then. At some moment, some(not harmful)thing can be not well received, but the rationality will make it be accepted.

Now, I'd like to highlight the following (which is connected with incest with sons):

[quote="Kalle"]@HilbertThm90: It is really dangerous to start discussing that it should be banned because of the poor genetic material of the offspring. This opens a whole can of worms about eugenics type thinking. Why is not ok (anymore) to discuss eugenics in other contexts? Is it ok to fornicate with a person from a culture with, say, low intelligence? (However controversial it may be, there are areas where the average IQ is like 70). Should we allow children with down's syndrome? Thinking about some examples, clearly there are more variables to consider than just poor genetic material (and the offspring may be fine in many cases), so your argument is simply not good.[/quote]

[quote="Sunny"]That raises another interesting, and different question. If a parent carries a gene for a disease and there is a high chance of it being passed off to children...should they not have children? In particular, I'm thinking of Huntington's, which doesn't manifest itself until the individual is older.[/quote]

Indeed, it seems there are not many arguments against incest not based on genetic (I don't consider "it is imoral for me" an argument). Now our more general question is "should parents be allowed to do have a son if they knew their son would have some desease?". Similar questions are: should parents have the right to choose if they want a son without a disease (or a gene that could make very probable for the children to have a disease) by artificial selection (in this case, it would be look to some zygotes and choose one without the disease genes)? $ (*)$ Should the mother have the right to abort if she was carrying a baby/fetus who would have a disease? I would really answer the first question positively. By the way, there is a couple who recently answered "yes", and their son/daughter doesn't have a gene that probably would give him/her a cancer. In general, it would be dumb to need an answer to the second question, wouldn't it? The couple could have chose the child genes in a lab... I don't know much about the down's syndrome case... It happens when the zygote has three chromosomes of the same kind or something, right? Well, this must be easily seen too. So, I don't know if there is a case in which a disease could be identified by the first time in the pregnancy period. But maybe there is. In a couple without much money, it could happen (I think the cost to detect the cancer gene was about 5000 dollars)... I wouldn't see any problem, too. And I don't want to discuss abort (is it discussed somewhere in this forum?).

Anyway, the question here is very interesting... What are the acceptable limits to genetic? We could choose genes which would not cause some disease. If we allow this... What more could we allow? A better immune system? More intelligence? More physical force or agility? If it is bad to don't allow people without such attributes, then is it bad to don't allow people with some disease? A priori, I'd disagree. But... A ill person is a person. This person may has problems, as every person. This person will need the support of society, as every person. Maybe more than usual. If we can't avoid some disease... Why don't allow the person to exist? Everybody is allowed! And everybody has problems! If we can say "you can't have a son because your son would have down syndrome", we could say "you can't have a son because he would have IQ under the avarage" or "because he would has weak arms" or whatever. And I suppose there are know perfect persons. In the end, the conclusion is: what is the problem? Everybody is a cost to our society.

Hm... Now refuting myself: everybody is a cost, but some people are not able to [i]help[/i] the society. But anyway... If somebody is able to help the society, this person can be helped too, I think. This seems to be reasonable, at least.

I end it here... I need to do some other not so import things, like eating.

The example with walking naked on the streets is not that great. The streets are (usually) public property. In anything privately owned you should be able to walk naked if the owner approves. This is of course not how it works in reality (compare with smoking bans and pornography moral hysteria). Discussing the right to give birth to a child doesn't seem to fit in the public/private discussion, but in my view that a woman has the right to do whatever she wants with her body, and that extends to babies. This is also the main thing in the abortion debate, but there the issue is whether the baby has its own right, while the moral tangent in this incest debate is that people are disgusted by the baby's existence, siblings' love and/or some other hang-up.

Having sex in public domain is already illegal (usually), so there is no need to have such a law for incestuous couples. So while it is not anti-liberal to forbid people from walking naked on the streets, it is anti-liberal to forbid incest based purely on disgust of a notion that you don't have to see or experience in any way.

Most western societies (both the US and my country included) are very anti-liberal with tons of moral laws everywhere, so in practice you can forbid incest for whatever reason. I would like to see a stronger constitution so that people can't just restrict the lives of sexual deviants (e.g. homosexuals) based on a simple popular vote. As usual, I know nothing and what I say probably has poor basis in both theory and reality ;)

I would just like to clarify that I strongly oppose banning incest based on [b]normative|/b] concerns, there could be other valid reasons to do it.

Thank you all for the insightful responses. I won't inject my own opinions mainly because, as I observed right off the bat, I don't really have any yet. I will say that I'm not really viewing this as a religious discussion, since while religion does consider incest immoral, so do other cultural stigmas that need not have anything to do with religion. I also wasn't planning on making any attempt to tie this to homosexuality or any other sexual behaviour, since that gets very general very quickly (which is why I restricted from "incest" to "non-child producing incest between adults of similar age"). Perhaps mistakenly, I also didn't think this would be a discussion on morality in general, since it seems unjustified murder can be morally said to be wrong in a somewhat well constructed framework of morality (for example, by game theoretic arguments, one can say society as a whole suffers from this immoral behaviour).

I am interested in how one here differentiates between "immoral" and "disgusting". A couple posters don't necessarily find the situation I posited immoral, but yet still find it "disgusting". What do you mean by this? This question intrigued me because I found that I reasoned myself to essentially the same conclusion, and yet once there, I could no longer explain why I found such a practice disgusting if it was not immoral. Is it possible to differentiate between the two, or is the appropriate response to simply update one's catalog of emotional responses to the world so as to no longer think of such as disgusting ?

by the way, I find the "factoring in the moral cost to society" pretty interesting. I would never consciously do so and hence was very surprised to learn that one of my friends didn't necessarily think gay marriage should be allowed in America at this point since although he was entirely for it, he didn't think the country would be better off as a result due to the views of a substantial amount of its populace, and that's ultimately the purpose that lawmakers should be working towards in his view. of course, there's nothing special about gay marriage and it is interesting to see the same argument come up again here in the case of incest.

another edit: it looks like I may also be confusing morality with ethics. I really just sort of meant bad in some provable sense working from more obvious axioms (like the overall good of society) instead of more arbitrary axioms like just setting incest = bad. sorry, everybody-who-knows-more-philosophy-than-I.

(also, sweden is anti-liberal ??)

@mystic:

When I was younger, I thought it was important to think logically and live logically, and to able to justify everything I thought or did (at least in theory).  As I got older, I decided that this was just silly.

Case in point:  I find incest disgusting.  I find it disgusting because of a history and a set of circumstances over which I had no control.  I suspect that you find it disgusting for the same reason.  This is not a particularly good reason for feeling a certain way, but rational or not, I have no reason to actively fight this feeling, so I don't.  It's vaguely similar to finding a certain food disgusting.  Why should you have to justify why you don't like something, as long as it doesn't harm anyone?

(This is slightly less on point:  While it's okay to find incest disgusting on a personal level, it would be wrong to tear apart the completely hypothetical loving incestuous couple.)

I realize not everything in life can or should be placed in a strictly logical and quasi-mathematical framework, and thus I often act in my own life according to certain behavioural templates that others might find illogical; however, if I find a certain food disgusting, I'm not hurting that food's feelings or anything silly like that. That is simply my own personal preference. When I find incest disgusting, I am implicitly also passing some normative judgment on any couples that may practice this behaviour, and I do not think it is appropriate to do so without a good reason, at least based on the part of me that has undergone a fairly liberal upbringing. Saying "bleh gays make me feel bad inside" (or replace "gay" with "black" or "non-Aryan" or whatever) is not only an insufficient platform from which to direct public policy as you note, such an illogical and potentially dangerous "feeling" should be excised completely.

I tried to sneak in that parenthetical at the end rather than explaining myself more fully, but I guess I need to clarify.

When *I * say that I find incest personally disgusting, I mean that I'm disgusted by the idea of myself engaging in it.  It may even be the case that I am disgusted by the mere contemplation of others doing it.  But I don't see this as being the same as judging others or harming them in any way.  For example, I am disgusted by the thought of myself engaging in homosexual acts.  I may even be a little disgusted by watching or imagining certain homosexual acts being performed by others.  However, I'm about as pro-gay rights as possible, and I don't consider myself a homophobe.  To me, being "grossed out" by gay sex is no different from being grossed out by the thought of two unattractive (to use a polite word) heterosexuals getting it on.  Or going back to my food example, I think it's disgusting to eat cockroaches, but for argument's sake let's say I befriended someone from some other country who enjoys eating cockroach-like creatures.  I don't think I'd want to watch him eat the bugs, but if I did, I would at least try to hide my disgust out of respect, and I certainly wouldn't pass judgment on him or try to stop him from doing it.

Aside:  I would put this entire discussion, including what I just wrote, under the category of thinking too logically and rationally about why we do things or think things.  Of course, you are correct that dangerous or wrongheaded feelings ought to be excised, but I doubt it ever works out that way.  On basic questions of right and wrong, we almost always use logic to justify our already-held beliefs and almost never to discover them.  For reason and logic work to function properly, we have to have some degree of emotional detachment.  This incest discussion is sort of like a "sandbox" because it so rarely comes up in real life.


If cosx + (cosx)^2 + (cosx)^3 = 1 then find the minimum value of tanx.

thank you :)

let $ y=\tan x$, then we have that $ \frac{1}{\cos^{2}x}=1+y^{2}$.

Squaring the relation gives: $ (\cos^{2}x-1)^{2}=\cos^{2}x*(1+\cos^{2}x)^{2}$.

$ \left(\frac{1}{y^{2}+1}-1\right)^{2}=\frac{1}{y^{2}+1}*\left(1+\frac{1}{y^{2}+1}\right)^{2}$.

This simplifies into: $ y^{6}-4y^{2}-4=0$. I don't think that this has nice roots.


calculus
integration
function
real analysis
real analysis unsolved

Are difficult these problems ?! I think no ... You can solve these with the usual technics ...

[quote="Virgil Nicula"]$ 1\blacktriangleright$ ([color=darkred][b]Virgil Nicula[/b][/color]) Let $ f\ : [0,1]\rightarrow\mathcal R$ be a continue function for which $ \int_0^1f(x)\ \mathrm {dx} \equal{} 0\ .$
Prove that for any $ \lambda\ne 0$ exists $ c_{\lambda}\in (0,1)$ so that $ c_{\lambda}^2\cdot f(c_{\lambda}) \plus{} |\lambda|\int_0^{c_{\lambda}}f(x)\ \mathrm {dx} \equal{} 0\ .$

$ 2\blacktriangleright$ Let $ f\ : [a,b]\rightarrow\mathcal R$ be a continue function for which $ f(a) \equal{} f(b)\ .$ Then

$ 2.1\blacktriangleright$ ([color=darkred][b]T.M. Flatt[/b][/color]). Prove that $ (\exists )\ c\in (a,b)$ so that $ \int_a^cf(x)\ \mathrm {dx} \equal{} (c \minus{} a)f(c)\ .$

$ 2.2\blacktriangleright$ ([color=darkred][b]Virgil Nicula[/b][/color]). Prove that  $ (\exists )\ c\in (a,b)$ so that

$ (b \minus{} c)\cdot\left[f(c) \minus{} \frac {1}{c \minus{} a}\cdot\int_a^cf(x)\ \mathrm {dx}\right] \equal{} (c \minus{} a)\cdot\left[f(c) \minus{} \frac {1}{b \minus{} c}\cdot\int_c^bf(x)\ \mathrm {dx}\right]\ .$[/quote]

I am not able to solve 1 and 2.2.Can you supply some hint.
Thank you in advance

For Problem 1, I defined the function $ g$ by
\[ g(t) \equal{} e^{\minus{}|\lambda| / t} \int_0^t f(x) \, dx
\]
for all $ 0 < t \le 1$, with $ g(0) \equal{} 0$.  Then apply Rolle's theorem to $ g$.

For 2.1 I think it is called Flett's theorem :?:


Find all the positive integer P,that 2*p^4 - p^2 + 16 is a square.Thanks!

Of course I can solve it!

Or in fact no, I cannot. Because it has no solutions.

(hint: substitute a = p/4, and work out from there)

Giving it a try like this may help :)

3x!
I know we can slove it by what you have said.But the one who ask me this problem is too young to know it.So I want to konw is there any simple method.

look modulo 5 and you ill get a contradicition!

To"pascual2004" and "Peter VDD":can you slove it and write down?


function
induction
real analysis
real analysis unsolved

Let 0<x<1 and n a natural number. Let f_n(x)=x+x^2/n. Find as a function of x the limit when n tends to infinity of the sequence f_n(f_n(...(f_n(x)...) (n times composition).

You are right Harazi, this problem is beautiful. The answer should be $x/(1-x)$, but I didn't find any elementary solution. What I did is to define $x_{n,k}$ as the k-th iteration of $f_{n}$ starting with $x_{n.0}=x$. Writing the recurrence as
\[\frac{x_{n,k+1}-x_{n,k}}{\frac{1}{n}}=x_{n,k}^{2},\]
it is clear that this is a discrete scheme (with time step equal to $1/n$) for the differential equation $u'(t)=u^{2}(t)$ with initial condition $u(0)=x$, whose solution is $u(t)=x/(1-tx)$. Therefore $x_{n,n}$ tends to $u(1)$. This clarifies also that the limit is $+\infty$ if $x\geq 1$.

Is there a more elementary solution?

Yes, there is a very elementary solution! Try to find it!

Xamog's answer has lead me to the following approach:

Let $h^n(x)$ denote $h$ composed $n$ times.

Let $g_n(x) = \frac{x}{1-\frac{x}{n}}$.  Then by a trivial induction, $g_n^k(x) = \frac{x}{1-\frac{kx}{n}}$ so that $g_n^n (x) = \frac{x}{1-x}$.
Finally, note that $g_n(x)=f_n (x) + O(n^{-2})$ implying that $g_n^n(x)=f_n^n(x) + O(n^{-1})$.  Therefore, $lim f_n^n(x) = \frac{x}{1-x}$.

Very nice problem.  Is this the approach you had in mind, Harazi?

No, my approach is different from both these beautiful problems. I fixed n and took the sequence $ a_0=x,a_{k+1}=a_k+\frac{a_k^2}{n} $. Then one can easily see that it is increasing and $\frac{1}{a_k}-\frac{1}{a_{k+1}}=\frac{1}{n+a_k} $. Sum up these relations from k=0 to n-1 and you will find that $\frac{1}{x}-\frac{1}{a_n}=\sum{k=0}^n-1\frac{1}{n+a_k} $. Thus, it is clear that $ \frac{1}{x}-\frac{1}{a_n}<1 $ and consequently $ a_n<\frac{x}{1-x} $ . Since the sequence is increasing using again the same identity we find that $ \frac{1}{x}-\frac{1}{a_n}>\frac{n}{n+\frac{x}{1-x}}$ which suffices to conclude that
 $\frac{nx+\frac{x^2}{1-x}}{n(1-x)+\frac{x}{1-x}}<a_n < \frac{x}{1-x} $. The conclusion is immediate.

Very nice solution, Harazi.

Is there a similar approach to the case where
\[f_{n}(x)=x+\frac{x^{a}}{n},\]
where $a$ is a fixed exponent?

By the way: in another topic you ask a question about a sequence "inspired by the previous problem", but now it is no more clear to which sequence you refer. Please add some explanations.

O, sorry Xamog, but, you know, Ive posted quite a lot, and I dont know exactly what I posted in that post. But I remember to have posted something related to a problem of Laurentiu Panaitopol and after that I put another post calling  inspired by previous problem. In that post I think I put the following problem:

    In 1994 at the national mathematical Olympiad, Laurentiu Panaitopol proposed the following problem: define the sequence $ x_n $ by $\sum_{k=1}^n\frac{1}{\sqrt{nx+k}}
 =\sqrt{n} $. Find the limit of this sequence.
   It is not hard to find that this limit is $\frac{9}{16} $. My problem was to show that $ n^a\cdot (\frac{9}{16}-x_n) $ tends to 0 whenever $ a<1 $. And the open problem was to show that $ n(\frac{9}{16}-x_n) $ converges. Ive managed only to show that this sequence is bounded.
   Refering to the more general problem you posted, I think that my method works for $ a\geq 1$. For other values, it doesnt work, or maybe Im idiot and I dont see how it could work.

I guess that your bound is $0\leq n(9/16-x_n)\leq 9/16$, which can be obtained by approximating the given sum with two telescopic sums. I think that the limit is 1/2, but I'll continue the discussion under the appropriate thread.


If $ (a_n)$ is a sequence of non-negative terms and $ \sum a_n$ diverges show that $ \sum \frac{a_n}{1\plus{}a_n}$ diverges.

Is it true that $ \lim_{n\to\infty}a_n\equal{}0?$ Either it is or it isn't.

Try using that to make cases.

If $ \lim_{n \to \infty}a_{n}\neq 0 \Rightarrow \lim_{n \to \infty}\frac{a_{n}}{1\plus{}a_{n}} \neq 0 \Rightarrow$ divergence.

If $ \lim_{n \to \infty}a_{n}\equal{} 0$, then there exist $ N$, such that $ a_{n}<\frac{1}{2}$ and $ \frac{a_{n}}{1\plus{}a_{n}}>\frac{2}{3} a_{n}$ for $ n\geq N$. Therefore $ \sum_{n\equal{}N}^{\infty}\frac{a_{n}}{1\plus{}a_{n}}>\frac{2}{3}\sum_{n\equal{}N}^{\infty} {a_{n}}$.

You can try to show that:

1. $ \sum_{n\equal{}1}^{\infty} {a_{n}}$ converges if and only if $ \sum_{n\equal{}1}^{\infty}\frac{a_{n}}{1\plus{}a_{n}}$ converges.

2. If $ \sum_{n\equal{}1}^{\infty} {a_{n}}$ converges, then $ \sum_{n\equal{}1}^{\infty}\frac{a^{2}_{n}}{1\plus{}a^{2}_{n}}$ converges.

3. Find the example of diverging $ \sum_{n\equal{}1}^{\infty} {a_{n}}$ but converging $ \sum_{n\equal{}1}^{\infty}\frac{a^{2}_{n}}{1\plus{}a^{2}_{n}}$.


geometry
conics
hyperbola
geometric transformation
reflection
ellipse
parallelogram

Let $\alpha$ be the circle escribed to quadrilateral $ABCD$, $S$ be it's center. $AC$ intersects $\alpha$, diagonals of $ABCD$ intersect at $E$. Line through $E$ perpendicular to $AC$ intersects lines $BS$ and $DS$ at points $P$, $Q$. Prove that $EP=EQ$.

This is analogous to [url=http://www.mathlinks.ro/Forum/viewtopic.php?t=20538]"Hard geometry from Polish MO"[/url].

Since the quadrilateral $ABCD$ is tangential with an excircle and the diagonal segment $AC$ intersects this excircle, the following sums of adjacent sides are equal: $DA+AB = BC+CD.$ Then $AB-BC = CD-DA,$ so that $B,\ D$ lie on different branches of a hyperbola $\mathcal H$ with focal points $A,\ C.$ Since $BS,\ DS$ bisect the angles $\angle ABC,\ \angle CDA$ internally, $BS,\ DS$ are hyperbola tangents at $B,\ D.$ Let $BS,\ DS$ meet $AC$ at $X,\ Y.$ Reflect $B,\ D$ in the line $AC$ into $B',\ D'.$ Since the hyperbola $\mathcal H$ is symmetrical with respect to its major axis $AC,$ the reflected points $B',\ D'$ also lie on this hyperbola and $B'X,\ D'Y$ are hyperbola tangents at $B',\ D'.$ Let $BS \equiv BX,\ D'Y$ meet at $P$ and let $DS \equiv DY,\ B'X$ meet at $Q.$ The quadrilateral $XPYQ$ with a diagonal intersection $E \equiv XY \cap PQ$ is symmetrical with respect to $AC \equiv XY,$ hence $PQ \perp AC$ and $PE = QE.$ Since $XPYQ$ is tangential to the hyperbola $\mathcal H,$ it has the same diagonal intersection as its contact quadrilateral $BB'D'D,$ hence $E \equiv BD \cap B'D' \equiv XY \cap PQ \equiv AC \cap PQ.$ (For a quadrilateral tangential to a circle, this can be proved by a repeated use of Pascal's theorem, hence it holds for a quadrilateral tangential to any conic.)

[b]Another solution by polarity transformation.[/b] It seems more complicated thant the one with a hyperbola (if the diagonal segment BD intersected the excircle, it would be an ellipse), but it works without change for any tangential quadrilateral, convex or concave, with an incircle or excircle (S). Besides, the hyperbola solution relies on the hyperbola pedal circle with respect to either focus (to show that BS, DS are hyperbola tangents) and I can prove this only by analytical geometry (see [url=http://www.mathlinks.ro/Forum/viewtopic.php?t=39890]Ellipse[/url]).

Let K, L, M, N be tangency points of AB, BC, CD, DA with the circle (S), forming the cyclic contact quadrilateral KLMN. These 2 quadrilaterals have a common diagonal intersection $E \equiv AC \cap BD \equiv KM \cap LN$ and moreover, the opposite sides of KLMN at $V \equiv KL \cap MN \in AC,$ $U \equiv LM \cap NK \in BD.$ Since NK, KL, LM, MN are polars of A, B, C, D with respect to the circle (S), U, V are poles of the diagonals AC, BD and UV is a polar of the diagonal intersection E. Let a normal to AC at E meet the appropriate (internal or external) bisectors BS, DS of $\angle B,\ \angle D$ at P, Q. Similarly, let a normal to BD at E meets the appropriate (internal or external) bisectors AS, CS of $\angle A,\ \angle C$ at Y, X.

In polarity transformation with respect to the circle (S), the points P, X, Q, Y are carried into their polars p, x, q, y, forming a quadrilateral P'X'Q'Y' with sides the $p = Y'P',\ x = P'X',$ $q = X'Q',\ y = Q'Y'.$ Since the points A, P are collinear with the center of the polar circle (S), their polars are parallel, $NK \parallel Y'P'$ and similarly, $KL \parallel P'X',$ $LM \parallel X'Q',$ $MN \parallel Q'Y'.$ Thus the quadrilaterals KLMN and P'X'Q'Y have the corresponding sides parallel. Let $V' \equiv p \cap q,$ $U' \equiv x \cap y$ be intersections of the opposite sides of P'X'Q'Y'. The points V', U' are poles of the diagonals $PQ \perp AC, XY \perp BD$ of PXQY. Since the lines AC, BD, PQ, XY all intersect at the point E, their poles U, V, U', V' are all collinear, $U',\ V' \in UV.$ The triangles pairs $\triangle UVK \sim \triangle U'V'P',$ $\triangle UVL \sim \triangle U'V'X',$ $\triangle UVQ \sim \triangle U'V'M',$ $\triangle UVY \sim \triangle U'V'N'$ with the common side line $UV \equiv U'V'$ and parallel sides are then all similar with the same similarity center and coefficient, which means that the quadrilaterals $KLMN \sim P'X'Q'Y'$ are also centrally centrally similar with the same similarity center and coefficient.

The lines $PQ \perp AC,$ $XY \perp BD$ form the same angles as the diagonals BD, AC of tangential quadrilateral ABCD. The lines U'S, V'S connecting the poles U', V' of XY, PQ with the center of the polar circle (S) are perpendicular to XY, PQ. Consequently, $U'S \parallel BD \equiv UE$ and $V'S \parallel AC \equiv VE.$ Since these parallel lines pass through the corresponding pairs of points U', U and V', V of the centrally similar quadrilaterals $P'X'Q'Y' \sim KLMN,$ they are centrally similar with the same similarity center and coefficient. Since BD, AC meet at the diagonal intersection E, it follows that $S \equiv KS \cap LS,$ the center of the polar circle (S), is the diagonal intersection of the quadrilateral P'X'Q'Y'. Since P'X'Q'Y' is an image of PXQY in the polarity transformation in (S), P', X', Q', Y' are poles of YP, PX, XQ, QY, respectively. Since P', Q' are collinear with the center of the polar circle (S), their polars are parallel, $YP \parallel XQ.$ Likewise, since X', Y' are collinear with the center of the polar circle (S), their polars are also parallel, $PX \parallel QY.$ As a result, the quadrilateral PXQY is a parallelogram with the diagonal intersection E, which means that its diagonals cut each other in half at E, hence PE = QE and XE = YE.


quadratics
number theory proposed
number theory

Prove that for a positive integer $ n$, all prime divisors of $ 2^n\plus{}1$ are not of the form $ 8k\minus{}1$ $ (k\in\mathbb N)$

[hide="Solution"]
Assume to the contrary, that there exists a prime divisor, $ p$, of $ 2^n \plus{} 1$ so that $ p\equiv \minus{}1\bmod 8$.  This means that $ \frac{p^2 \minus{} 1}{8}$ is even, so by Quadratic Reciprocity, we have that $ 2$ is a quadratic residue $ \bmod p$.  Say that $ x^2 \equiv 2\bmod p$, so $ x^{2n}\equiv \minus{}1\bmod p$, so $ x^{4n}\equiv 1\bmod p$.  Let the order of $ x\bmod p$ be $ k$.  We have that $ k|4n$, but $ k$ is not a factor of $ 2n$, so $ 4|k$.  We then have that $ 4|k|(p\minus{}1)$, so $ p\equiv 1\bmod 4$, which contradicts the fact that $ p\equiv \minus{}1\bmod 8$.  Thus, no such prime exists. [/hide]

[quote="The QuattoMaster 6000"]... but $ k$ is not a factor of $ 2n$, [/quote]
Why's that?

If $ k|2n$, then $ 1\equiv x^{2n}\equiv \minus{}1\bmod p$, so $ p \equal{} 2$, which is a contradiction (recall that $ p$ is odd since the $ 2^n \plus{} 1$ is odd).


algebra
function
domain
calculus
derivative
integration
absolute value

I was wondering, is proving the validity of differentiation and integration in complex domain an extremely difficult task to do or is it something trivial?

(I asked this question because I just started using complex differentiation, along with Euler's formula, in analyzing signals and it makes things so elegentaly simpler!)  :)

[quote="Xevarion"]There's a big difference between differentiating a function $f: R\rightarrow C$ and a function $g: C \rightarrow C$. Which one are you trying to do?[/quote]
I am talking about $g: C \rightarrow C$.

(I guess I am specially surprised about the fact that $\frac{d}{dt} e^{i\omega t} = i\omega e^{i\omega t}$)

Well, the function you described is actually from $R$ to $C$. You probably only want to stick real values of $t$ in there -- in many physical questions, that'll be time.

If your variable is always real, everything is just like when the range is also in the reals. To work with the limits and show all the same rules for derivatives and such are true, just look at the modulus (abs value) of everything.

If you want to do stuff with complex variables, see http://www.artofproblemsolving.com/Forum/viewtopic.php?t=88930

[quote="Xevarion"]Well, the function you described is actually from $R$ to $C$. [/quote]
I am sure you are right, but how's my function from $R$ to $C$? Isn't $e^{i\omega t}$ a complex number?

[quote="Xevarion"]If your variable is always real, everything is just like when the range is also in the reals. To work with the limits and show all the same rules for derivatives and such are true, just look at the modulus (abs value) of everything. [/quote]
Does this mean that the PROOF for differentiation/integration for a function $g: R \rightarrow C$ is trivial more or less?

$e^{i\omega t}$ is indeed a complex number. But that's the value the function takes, which is in the range. The domain is where the values for $t$ come from. In this example it's $R$.

Differentiation and integration are exactly the same in this case as in functions from teh reals to the reals. The proofs are exactly the same, except that you want to use the absolute value (modulus) of the function. I guess you might have to make minor adjustments of wording and stuff too. But the ideas are basically the same. The idea of a limit existing is the same in any dimension -- $C$ is just like $R^2$.

[quote="Xevarion"]$e^{i\omega t}$ is indeed a complex number. But that's the value the function takes, which is in the range. The domain is where the values for $t$ come from. In this example it's $R$. [/quote]Oh yeah! I guess I was think of the derivative as the function. How silly of me. :blush:

[quote="Xevarion"]Differentiation and integration are exactly the same in this case as in functions from teh reals to the reals. The proofs are exactly the same, except that you want to use the absolute value (modulus) of the function. I guess you might have to make minor adjustments of wording and stuff too. But the ideas are basically the same. The idea of a limit existing is the same in any dimension -- $C$ is just like $R^2$.[/quote]Makes sense. Thanks a lot Xevarion. :)


If $ f(x) \equal{} \frac{4x\plus{}1}{3}$ what is the value of $ \left(f^{\minus{}1}(1)\right)^{\minus{}1}$?

Hmm... I think this is how you do it:

$ f(1)^{\minus{}1} \equal{} \frac{3}{4(1) \plus{} 1}$

Then, take the reciprocal of that (because of the -1) to get $ \boxed{\frac{5}{3}}$.

Wait... that's the same thing though...

hello, we have $ y\equal{}\frac{4x\plus{}1}{3}$ and from here we get $ 3y\minus{}1\equal{}4x$ or $ x\equal{}\frac {3y\minus{}1}{4}$ so we get $ f^{\minus{}1}(x)\equal{}\frac{3x\minus{}1}{4}$ and $ \left(f^{\minus{}1}(1)\right)^{\minus{}1}\equal{}\left(\frac{1}{2}\right)^{\minus{}1}\equal{}2$.
Sonnhard.


function
trigonometry
real analysis
real analysis unsolved

There  are functions $f: {\mathbb R}\to{\mathbb R}$ such that $f(x+a)=-f(x-a),\forall x$, where $a \in {\mathbb R}$ is a fixed real number ??? Give an example.

How about sine or cosine functions with period $4\alpha$?


Prove or disprove: The denominator in the fraction for the formula in terms of $n$ of $\sum_{i=1}^n{i^x}$ is always divisible by $x+1$

[quote="tetrahedr0n"]I have some work, but I'm not done. I think it's true.

We note:

$\sum_{i=1}^{n} (i+1)^{x+1}-i^{x+1} = \sum_{i=1}^{n} (x+1)i^x + ...$
$(n+1)^{(x+1)} - 1 - ... = \sum_{i=1}^{n}(x+1)i^x$

$\sum_{i=1}^{n}i^x = ((n+1)^x - 1 - ....)/(x+1).$

Of course, I need to show that stuff on the top cannot cancel out with the denominator.[/quote]
you're on the right track, but I also don't knowhow to prove that the numerator could be divisible by x+1, so I came to you guys


function
real analysis
real analysis unsolved

Let $f_n$ be a function defined : $f_n(x)=x^n(1-x)^n$.
For $p\in \mathbb{N}$ find $f_{n}^{(p)}(0)$

Thanks! :)

[b]Lemma.[/b] $P(x)\equiv \sum\limits_{k=0}^n a_kx^k=\sum\limits_{k=0}^n \frac{P^{(k)} (0)}{k!}\cdot x^k\Longrightarrow P^k(0)=\{ \begin{array}{ccc} 0 & if & k>n\\\ & & \\ k!\cdot a_k & if & 0\le k\le n\ \end{array}$

[b]The answer. [/b]$P(x)\equiv x^n(1-x)^n\Longrightarrow P^{(k)} (0)=\{ \begin{array}{ccc} 0 & if & k\not\in \overline {n,2n}\\\ & & \\(-1)^{k-n}\binom {n}{k-n} k! & if & k\in \overline {n,2n}\\ \end{array}$


[quote="citistarz"]I have kind of a stupid question about mass point but how do you add the weights proportionally? Thats the only thing I'm having trouble with say you have a figure like this:

[img]http://www.mfdabbs.pwp.blueyonder.co.uk/Maths_Pages/SketchPad_Files/Triangles/Triangle_Gifs/Sketch15.gif[/img]

AB'=3, B'C=8, BA'=x, A'C=y, BP=a, PB'=b and say that C'C did not even exist. What is $\frac{a}{b}$ and $\frac{x}{y}$.

Please explain.[/quote]

you want to balance the lines about a point, if you know the fulcrum principle, the distance from the center multiplied by the weight of the weight on that side is constant, (this is balance of torque for physics people), so if the distance from A to B' is 8, and B' to C is 8,3 respectively, a weight of 3 at A ($w_A=3$), and a weight of 8 at C ($w_C=8$) balances it, also you add the masses of the 2 ends to get the mass of B' ($w_{B'}=11$), this comes from the fact that since A and C push down, there needs to be equal and opposite force up to maintain the balance, if you don't know physics, forget this, but it is the way that i remember

using this, to balance BC about A', $w_B=\frac{8y}{x}$, see how $w_B* (BA')=w_C*(CA')$

then using the sum thing, $w_{A'}=8(1+\frac{y}{x})$, using mass points,
$w_B*BP=w_{B'}*(B'P)$

$\frac{BP}{B'P}=\frac{8y}{11x}$

so what is the weight of B and how do you find it?

[quote="Altheman"]
[b]see how $w_B* (BA')=w_C*(CA')$[/b]
[/quote]

(the above it the basic balancing equation, explained already)

you have the ratio of BA' and CA', and the weight of C, so you can find the weight of B

i need more help though. i still dont understand how to put the weights on properly.


Euler
function
abstract algebra
modular arithmetic
number theory
totient function

A permutation $ p \equal{} (p_1, p_2, \ldots, p_n)$ of $ {\bf Z}/n\bf Z$ is said to be [i]good[/i] if the first differences $ (p_2 \minus{} p_1, p_3 \minus{} p_2, \ldots, p_n \minus{} p_{n \minus{} 1})$ are a permutation of $ \left({\bf Z}/n{\bf Z}\right) \setminus \{0\}$.

For $ n$ even, compute the number $ a_n$ of good permutations of $ {\bf Z}/n\bf Z$.

So far as I can tell, this is open.  For $ n \equal{} 0, 2, 4, \ldots, 16$, the associated values are $ 1, 2, 8, 24, 192, 2880, 46272, 1250592, 44095488, 103653120\times18$ and $ 5074931072\times20$.  (Most terms courtesy of Po-Ru Loh.)  Sequence [url=http://www.research.att.com/~njas/sequences/A067601]A067601[/url] in the [url=http://www.research.att.com/~njas/sequences/]OEIS[/url] is related.

Easy exercises associated with this question:
1) Show that if $ n > 1$ is odd then there are no good permutations of $ {\bf Z}/n\bf Z$.
2) Show that if $ n \geq 2$ is even then $ n$ divides $ a_n$.
3) Show that if $ n \geq 4$ is even then $ 2n$ divides $ a_n$.
4) Show that if $ n \geq 2$ is even then $ n\cdot \phi(n)$ divides $ a_n$.  (Here $ \phi$ is the Euler totient function, i.e. the number of positive integers less than and relatively prime to $ n$.)
5) Show that if $ n \equal{} 2m$ then any good permutation $ p$ satisfies $ p_1 \minus{} p_n \equiv m$.

Generalization: for an arbitrary finite group $ G \equal{} \{g_1, g_2, \ldots, g_n\}$ with identity $ e$, how many permutations $ p \equal{} (p_1, \ldots, p_n)$ of $ G$ are there so that the first differences $ (p_2p_1^{ \minus{} 1}, p_3p_2^{ \minus{} 1}, \ldots, p_np_{n \minus{} 1}^{ \minus{} 1})$ are a permutation of $ G \setminus \{e\}$?

By analogy with exercise (1), the answer is $ 0$ if $ G$ is abelian and the product of the elements of $ G$ is $ e$ (and so in particular if $ G$ has odd order).  This suggests the following, simpler question: find necessary conditions on a group $ G$ for there to be $ 0$ such permutations.

Comment: For any group, the enumerative problem is independent of whether we take first differences as indicated or in the reverse order, i.e. $ (p_1^{ \minus{} 1}p_2, p_2^{ \minus{} 1}p_3, \ldots, p_{n \minus{} 1}^{ \minus{} 1}p_n)$, since reversing permutations and inverting them element-wise is a bijection between good permutations in one case with good permutations in the other.

(Reposted with minor modifications from http://www.artofproblemsolving.com/Forum/viewtopic.php?t=198963 on suggestion of t0rajir0u.)

I am currently mentoring a student in [[RSI]] and gave him this problem to work on.  As of today, I believe he has resolved the question of which Abelian groups permit good permutations.  If he is willing, I (or he) may post some or all of his proof here for those who are interested.

On the enumeration front, he has managed to improve on exercises 3 and 4 to show that for $ n \geq 6$, $ 2n \phi(n)$ divides $ a_n$.

(In the extremely unlikely case that this should be relevant in the future, I'd like to make this an official claim of priority on behalf my mentee.)

My student is Hyunsub Hwang.  With his permission, I am posting the following outline of his major results, and some comments about possible future directions:

For a finite abelian group $ G \equal{} \prod_i {\bf Z}/c_i {\bf Z}$, Hyunsub was able to show that $ G$ admits a good permutation if and only if exactly one of the $ c_i$ is even.  To show that no other finite abelian group admits a good permutation $ (p_1, p_2, \ldots, p_n)$, it's enough to consider the difference $ p_n \minus{} p_1 \equal{} (p_n \minus{} p_{n \minus{} 1}) \plus{} (p_{n \minus{} 1} \minus{} p_{n \minus{} 2}) \plus{} \ldots \plus{} (p_2 \minus{} p_1)$ under the assumption that these differences include all nonzero elements of $ G$.  (This is the argument I mentioned as "by analogy with exercise (1)" in the original post.)  To show that every abelian group of this sort does admit a good permutation, Hyunsub used a clever recursive construction based on a particular expression for $ G$ as a product of cyclic groups.

(Incidentally, the problem of determining for just the cyclic groups which groups admit a good permutation was ARMO 1995, grade 11, second day: see http://www.artofproblemsolving.com/Forum/viewtopic.php?t=202477 .  The "expected" construction for that problem was used in the construction of good permutations for more general groups.)

For the problem of calculating $ a_n$, Hyunsub was able to improve on the exercises of the original post to show that $ 2n \phi(n) | a_n$ when $ n \geq 6$.  In order to do this, he used a pairing argument by introducing two operations on good permutations.  The first is reversal, $ (p_1, \ldots, p_n) \mapsto (p_n, \ldots, p_1)$.  The second is more sophisticated: we know that there is some $ i$ such that $ p_{i \plus{} 1} \minus{} p_{i} \equiv \frac {n}{2} \pmod n$.  Then the permutation $ (p_{i \plus{} 1}, p_{i \plus{} 2}, \ldots, p_n, p_1, p_2, \ldots, p_i)$ is also a good permutation.  Neither of these operations alone is enough to provide the necessary pairing, but the two operations together work nicely.

Hyunsub has also pursued bounds on $ a_n$.  So far, his results are nontrivial but also unsatisfactorily far apart: $ c_1 n\cdot 2^{n/4} < a_n < c_2\cdot n!$ for some constants $ c_1,c_2$.  Numerically, the true answer looks to be something more like $ a_n \approx n!/c^n$ for some constant $ c$.  Other possible extensions of his research are to "nice" nonabelian groups (dihedral groups or symmetric groups, for example) or infinite finitely generated abelian groups (in particular, to $ \bf Z$).  Also of potential interest are other countable abelian groups, especially $ \bf Q$.  Since the existence question is now settled for all finite abelian groups, one can also ask the enumeration question in this more general setting.

I posted the problem in $ \bf Z$ here: http://www.artofproblemsolving.com/Forum/viewtopic.php?t=217875 .  JoeBlow responded, but I haven't had a chance to decide if what he wrote works/makes any sense.

I managed, with some difficulty, to produce a nested loop that can enumerate good permutations of small groups.  The smallest noncyclic abelian group for which there exist good permutations is $ {\bf Z}/3{\bf Z} \oplus {\bf Z}/6{\bf Z}$, with 18 elements, and it will strain my patience to count them there.  For nonabelian groups, we have the following results:

No good permutations in $ S_3 \equal{} D_{2\cdot 3}$.
No good permutations in $ D_{2 \cdot 4}$.
No good permutations in the quaternion group $ Q$.
$ 320 \times 10$ good permutations in $ D_{2\cdot 5}$.  (These first three were also computed previously by Hyunsub.)
$ 3072 \times 12$ good permutations in $ D_{2 \cdot 6}$.
$ 2304 \times 12$ good permutations in $ A_4$.
There is one other group of order 12 (what wikipedia calls the dicyclic group of order 12), and I haven't gotten around to figuring out its multiplication table, so I don't know if it has any good permutations.

For groups of odd order a related question is [url=http://garden.irmacs.sfu.ca/?q=op/snevilys_conjecture]Snevily's conjecture[/url].  (Taking $ A$ and $ B$ to be all of $ G$, with the elements of $ B$ written as the negative of the elements of $ A$, one arrives at a very similar statement.)

We can relate a good permutation in $ \mathbb{Z}/n\mathbb{Z}$ to a transversal on a latin square.

Let $ (p_1,p_2,p_3,\dots,p_n)$ be a good permutation.
Lets form the Cayley table of $ \mathbb{Z}/n\mathbb{Z}$.

We will mark all the pairs $ (p_i, p_j) \in \{(p_1,p_2), (p_2,p_3), \dots, (p_{n \minus{} 1},p_{n})\}$
to the table by choosing the number $ p_i$ from the column $ p_j$. The row of the chosen cell
is $ p_i \minus{} p_j \pmod n$. All of them are by definition different. In the end we are left with one
row and column empty. Those can be removed. We are left with a latin transversal in an
$ n \minus{} 1 \times n \minus{} 1$ latin square. Because every permutation with the same starting element
yield the same latin square. We obtain an upper bound for the number of good permutations
by multiplying the upper bound for latin transversals in a square of given size by $ n$.

Thus by results on latin squares we get $ a_n \le c^n n!$ with $ c$ approximately $ 0.6$.

An example on doing this with a permutation of length 6:

(0,1,5,2,4,3) ->
[code]
0  1  2  3  4  5      0  1  2  3  4  5
1  2  3  4  5  0      1  2  3  4* 5  0      2  3  4* 5  0
2  3  4  5  0  1  -\  2  3  4  5  0  1* -\  3  4  5  0  1*
3  4  5  0  1  2  -/  3  4  5* 0  1  2  -/  4  5* 0  1  2
4  5  0  1  2  3      4  5  0  1  2* 3      5  0  1  2* 3
5  0  1  2  3  4      5  0* 1  2  3  4      0* 1  2  3  4
[/code]

Note that the reverse doesn't hold. There are latin transversals that don't
correspond to good permutations.

Hey, that's very cool!  This also agrees with my eyeballed estimate (I think my muddling around with the first 20 or so terms suggested something like $ c \underset{\sim}{<} 0.43^n n!$), so "all" we need is a matching lower bound :).

I know almost nothing about Latin squares -- is the result you quoted easy?

Good permutations and Latin squares popped up in another thread on the forum, as well -- it's at http://www.artofproblemsolving.com/Forum/viewtopic.php?t=217639 if you're interested.

Actually, the upper bound $ a_n\le c^nn!$ with [b]some[/b] $ c<1$ is very easy. Just count the permutations the usual way adding two numbers a time. Assume that $ m$ different numbers have already been chosen and their differences are also different. The next number $ x$ can be chosen in $ n\minus{}m$ ways at most but, if $ S$ is a set of chosen $ m$ numbers and $ T$ is the set of their first differences (which are all different), we get only $ n\minus{}(2m\minus{}1)\plus{}|(x\plus{}T)\cap S|$ possibilities for the next number $ y$. Adding that up, we obtain only $ (n\minus{}m)[n\minus{}(2m\minus{}1)]\plus{}\sum_x|(x\plus{}T)\cap S|$ possibilities for the pair $ (x,y)$. Note now that the sum is at most $ m(m\minus{}1)$, so the resulting number is about $ n\minus{}3mn\plus{}3m^2$, which is noticeably less than the trivial bound $ (n\minus{}m)(n\minus{}m\plus{}1)$ for $ m<n/2$.

So, the point is to get a good constant $ c<1$, not just some. I haven't thought of the lower bound yet (and do not want to do it unless you are completely stuck at this point) ;).

[quote]JBL: I know almost nothing about Latin squares[/quote]
So do I, so I just quoted the result. In addition it seems the quotation is wrong!
I copied it from Wikipedia but when I looked at the original article the correct
upperbound would be $ c^n \sqrt{n} n!$ with $ c$ approximately 0.614.
Not so good as it seemed then :/

Anyways, I've done some calculations on my computer regarding number of
latin transversals compared to number of good permutations.

The results are as follows (note that I've only calculated permutations starting
with 0 and the number of transversals in the $ n\minus{}1 \times n\minus{}1$ latin square.)

[code]
n   T      P     T/P
2   1      1     1
4   2      2     1
6   8      4     2
8   64     24    2.6666
10  928    288   3.2222
12  17536  3856  4.5477
14  454016 89328 5.0825
[/code]

Haskell code I used:

[hide]
[code]
module Main where

import Data.Array
import Data.List

latinsquare n = array ((1,1),(n,n)) [((a,b),(a+b) `mod` n) | a <- [1..n], b <- [1..n]]
cutlatinsquare n = array ((1,1), (n-1,n-1)) [((a,b), res) | a <- [1..n-1], b <- [1..n-1], let res = (a+b) `mod` n]

printlatin sq = do
  let n = size sq
  mapM_ (\r -> putStrLn (show r)) [row | a <- [1..n], let row = [sq ! (a,b) | b <- [1..n]]]

size sq = let (_,(n,_)) = bounds sq in n

rows sq =
  let n = size sq in
  [row | a <- [1..n], let row = [sq ! (a,b) | b <- [1..n]]]

transversals sq =
  transversals' [] [] (rows sq)
  where transversals' used usedcols [] = [reverse used]
        transversals' used usedcols (r:rs) =
          concat [transversals' (x:used) (c:usedcols) rs | (x,c) <- zip r [1..length r], x `notElem` used, c `notElem` usedcols]

main = do
  print $ length $ transversals $ cutlatinsquare 14
[/code]
[/hide]


Prove that if $a,b,c$ are the sides of  a triangle
then $\frac{a}{a+1}<\frac{b}{b+1}+\frac{c}{c+1}$.

 :)  :)

Yes, but there is a nicer and more direct solution, too............
!!!! :D  :)

Is there? I can't see any. Seems there's no way to deal with this one rather than expanding.

Obviously a must be largest.  Then

$\frac{b}{b+1}+\frac{c}{c+1} > \frac{b+c}{a+1} > \frac{a}{a+1}$


probability
limit
logarithms
real analysis
real analysis unsolved

here is a question that appeared in the probability section. Would be nice to have another approach:

let $ u_1 \equal{} 1$ and $ u_n \equal{} \sum_{k \equal{} [\frac {n}{2}]}^{n \minus{} 1} \frac {u_k}{k}$. Study the convergence and the value of $ \lim_{n \to \infty} u_n$ (with $ [x]$ is the greatest iteger less or equal to $ x$)

I don't think that there exist finite nonzero  limit for $ \lim u_{n}$
since :
Suppose there exist finite limit $ 0 < \lim u_{n} \equal{} \alpha < \infty$ then choose sufficiently large $ n$ such that $ |u_{k} \minus{} \alpha| < \varepsilon$
for all $ k \geq [n/2]$  and also $ \sum_{n/2}^{n \minus{} 1} 1/k \approx \ln 2 \plus{} \varepsilon$.
then $ u_{n \plus{} 1} \equal{} \sum_{n/2}^{n} \frac {u_{k}}{k} \approx \alpha \ln 2 \plus{} const. \varepsilon$
so if limit exist, then  it must be zero :)

in fact the real question is to show that $ \lim_n n u_n$ exists, and is non trivial  :blush:
(and the same kind of argument of course does not work anymore since $ n \sum_{[\frac{n}{2}]}^{n\minus{}1} \frac{1}{k^2} \to 1$)

Do you mean that the coefficients should be $ \frac 1{k^2}$ rather than $ \frac 1k$? :? (Otherwise I cannot make sense out of your last remark).

I don't understand what you mean, Fedja. I just meant that $ v_n \equal{} n u_n$ satisfies $ v_n \equal{} n \sum_{[\frac{n}{2}]}^{n\minus{}1} \frac{v_k}{k^2}$, and because $ n \cdot \sum_{[\frac{n}{2}]}^n \frac{1}{k^2} \to 1$, the argument of Extremal does not work anymore  :maybe:


How can one box both sides of an equation when using \eqnarray ? (boxes like the \boxed command gives)

When I tried doing that, it gave me 100 errors, all of which being "missing } inserted".

It has nothing to do with using eqnarray as you see here
\begin{eqnarray}\boxed{1+1}&=&\boxed{2}\end{eqnarray}
so there must be another error. Please post your code to show what it is. Don't forget you must be using amsmath to use \boxed.

Finally, please don't use eqnarray as it is obsolete, replace it by align. See [url=http://www.artofproblemsolving.com/Forum/viewtopic.php?t=123690]LaTeX FAQ[/url] Q5

I think he wants to box the whole equation. If you do it on the site, it doesn't work. I haven't tried it locally.

He says [quote]When I tried doing that, it gave me 100 errors, all of which being "missing } inserted".[/quote]which I assume means in a document.

You're probably right about boxing the whole equation as that does give 100 errors. The problem is that once you've boxed the equation there's no way of lining up the equal signs using &. It is possible to box the whole of an align environment but you can't box one equation and expect the equal signs to line up.

OK. I have found a very clever solution to this problem thanks to a chap called Andreas Bühmann. He defines a new command called \aboxed and this is how it works:

Suppose you have the following:
[code]\begin{align}
1+1&=2\\
10+10&=20
\end{align}[/code] and you want to box only the 1+1=2 equation [i]and[/i] line it up with the second equation. Replace 1+1&=2 by \aboxed{1+1}{=2} so you separate the two sides of the equation.
The full code is this:
[code]\newcommand*{\aboxed}[2]{%
  \rlap{\boxed{#1#2}}%
  \phantom{\hskip\fboxrule\hskip\fboxsep #1}&\phantom{#2}%
}
\begin{align}
\aboxed{1+1}{=2}\\
10+10&=20
\end{align}[/code]which looks like this:

[quote="Multiple"]Hey stevem

Is there a way to use the command eqnarray*?

It works fine with align and align*, but not with eqnarray* :([/quote]eqnarray is deprecated and has been replaced by align. See [url=http://www.artofproblemsolving.com/Forum/viewtopic.php?t=123690]LaTeX FAQ 5[/url].

[quote="roadnottaken"]Hey stevem, what does each command in the definition of \aboxed do?[/quote]You'll find it explained [url=http://tinyurl.com/29e6sr]here[/url] though you may also need to consult [url=http://www.cs.utk.edu/~eijkhout/texbytopic-a4.pdf]Tex By Topic[/url]


1. An encyclopedia has 26 volumes.  The front and back covers of each volume are 1/8 inch thick, and the pages average 1 3/4 inches per book.  The volumes are numbered and placed consecutively from 1 to 26 on the shelf.  A bookworm begins at p. 1 of volume 2 and eats to the last page of volume 25.  Express as a mixed numeral the number of inches it travels horizontally.

[hide]Answer: 44 1/4[/hide]

2. A restuarant makes 7 burgers form one pound of ground beef and three-quarter pounds of filler.  One pound of ground beef costs 3.50 dollars, the buns, filler and other extras cost a combined 15 cents per burger, and the restaurant must make a 55% profit on each sale.  How many dollars to the nearest nickel should be charged per burger.

[hide]Answer:1.45 Dollars[/hide]

Hint #1

Most of the trick on this is to remember that books have front and back covers, and to keep in mind How you place books on a shelf.

[hide=" #1"]
to go through an entire volume the worm travels 2 vertical inches. He goes through 22 complete volumes. that's 44 inches. If you've peaked at the answer you're probably wondering how we only use another quarter inch to get through all but the front and back covers of volumes 2 and 34, but the thing is we don't have to. When you put a book on a shelf the first page is where the last page would be if you were reading the book, because the spine faces out. Now we only have to go throught the front cover of vol. 2 and the back cover of vol. 34.

(P.S. I've done this exact problem before. Don't be discouraged if you don't get it the first time.) [/hide]

Just wondering, was this a mathcounts question? it seems a little sly for them…

# 2 hints
s= sales price
i= ingridient price
p= disired profit %

The key to this problem is understanding proft basics. your profit margine isn't $ i*p\equal{}s$ it is $ {{s\minus{}i}\over p}\equal{}s$ Kinda annoying huh?

[hide="#2"]
using the hint I just gave $ {{s\minus{}.65}\over .55}\equal{}s$ then solving the equation $ s\minus{}.65\equal{}.55s$
$ .65\equal{}.45s$
$ {.65\over.45}\equal{}s$ and .65/.45= 1.44blablabla which rounds up to 1.45.
[/hide]

Wow, just to tell you guys I had a lot of outside help and looked at the answers to help get to these solutions. These problems are just on the edge of being more riddle with math content than math question. However they are fun to do so I like them.

TTYL

[hide="2"]the profit, since its

$ \frac{s\minus{}i}{p}$, is $ \frac{s\minus{}.65}{.55}$

solve for s

s-.65=.55s

.45s=.65

s=1.44something, rounded to the nearest 5 is 1.45[/hide]

1-im asuming the question is, how much does the bookwormeat.
[hide="soooooo......"]i like things as common fractions so we have it eats 15/8 for the first and last book. for the 22 inbetween, it eats 2 inches or 16/8 per book. so thats 382/8 or 47 3/4. i dont get the same answer. hmmmmm...i think i miscounted. too lazy to check. w/e  [/hide]
2-just do the arithamtic and stuff. im too lazy to explain


let $ ABCDEF$ a hexagone Circumscribed by a circle . and let  $ AB\equal{}31$ and  the others sides  $ BC\equal{}CD\equal{}DE\equal{}EF\equal{}FA\equal{}81$
find the  sum of  the  lenght of his three diagonals which fasten a with the others points