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Note. I asked the question below on Math Stack Exchange (link), but didn't get a really satisfactory answer there, so I'm posting it here too.

I am looking for high school algebra/mathematics textbooks targeted at talented students, as preparation for fully rigorous calculus à la Spivak. I am interested in the best materials available in English, French, German or Hebrew.

Ideally, the book(s) should provide a comprehensive introduction to algebra at this level, starting from the most basic operations on polynomials. It should include necessary theory (e.g., Bezout's remainder theorem on polynomials, proof of the fundamental theorem of arithmetic, Euclid's algorithm, a more honest discussion of real numbers than usual, proofs of the properties of rational exponents, etc., and a general attitude that all statements are to be proved, with few exceptions). It should also have problems that range from exercises acquainting students with the basic algebraic manipulations on polynomials to much more difficult ones.

Specifically, I am looking for something similar in spirit to a series of excellent Russian books by Vilenkin for students in so-called "mathematical schools" from grades 8 to 11, although I am only looking for the equivalent of the grade 8 and 9 books, which are at precalculus level. To give you an idea, here are a sample of typical problems from the grade-8 book.

  1. Perform the indicated operations. $\frac{3p^2mq}{2a^2 b^2} \cdot \frac{3abc}{8x^2 y^2} : \frac{9a^2 b^2 c^3}{28pxy}$

  2. Prove that when $a \ne 0$, the polynomial $x^{2n} + a^{2n}$ is divisible neither by $x + a$ nor by $x - a$.

  3. Prove that if $a + b + c = 0$, then $a^3 + b^3 + c^3 + 3(a + b)(a + c) (b + c) = 0$.

  4. Prove that if $a > 1$, then $a^4 + 4$ is a composite number.

  5. Prove that if $n$ is relatively prime to $6$, then $n^2 - 1$ is divisible by 24.

  6. Simplify $\sqrt{36x^2}$.

  7. Simplify $\sqrt{12 + \sqrt{63}}$.

  8. Prove that the difference of the roots of the equation $5x^2 -2(5a + 3)x + 5a^2 + 6a + 1 = 0$ does not depend on $a$.

  9. Solve the inequality $|x - 6| \leq |x^2 - 5x + 2|$.

And here are the chapter titles for the grade 8 and 9 books.

Grade 8: Fractions. Polynomials. Divisibility; prime and composite numbers. Real numbers. Quadratic equations; systems of nonlinear equations; resolution of inequalities.

Grade 9: Elements of set theory. Functions. Powers and roots. Equations and inequalities, and systems thereof. Sequences. Elements of trigonometry. Elements of combinatorics and probability theory.

Broadly similar questions have been asked elsewhere, however the suggestions made there are not satisfactory for my purposes.

  1. The English translations of Gelfand's books are good; however they are not a sufficiently broad introduction to high school algebra, and do not have enough material on computational technique. They are more in the nature of supplements to an ordinary textbook.

  2. Some 19th century books like Hall and Knight have been suggested. On conceptual material, these tend to be too old in language and outlook.

  3. Basic Mathematics by Serge Lang seems more to dabble in various topics than to provide a thorough introduction to algebra.

  4. I am not inclined towards books with a very strong "New Math" orientation (1971-1983 France, for example). I don't think a student should need to understand the group of affine transformations of $\mathbb{R}$ to know what a line is.

Also, previous questions have perhaps focused implicitly on material in English. I have in mind a student who can also easily read French, German or Hebrew if something better can be found in those languages.

Edit. I'd like to clarify that I'm not asking for something identical to these books, just something as close as possible to their spirit. Fundamentally, this means: 1. It is a substitute for, rather than just a complement to, a regular school algebra textbook. 2. It is directed at the most able students. 3. It conveys the message that proofs and creative problem-solving are central to mathematics.

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Keith
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    If by "gifted/talented" you truly mean "gifted", then I suggest that any "systematic approach" is too plodding, and that "every statement must be proven" is too stodgy. (E.g., "must be provable" connotes something else...) If they're really gifted, there's no reason to spend nine months on "middle school algebra", rather than move on to something that has happened within the last 200 years. I'm not being sarcastic or mean, just suggesting that standard curriculum specifications are crazily wrong for genuinely interested+gifted kids. – paul garrett Jun 17 '15 at 22:22
  • @paulgarrett Have a look at this interesting report from a panel of mathematicians and mathematics teachers in England: http://education.lms.ac.uk/wp-content/uploads/2012/02/Acceleration_or_Enrichment_15Aug12.pdf It distinguishes between acceleration, enrichment for breadth and enrichment for depth. It emphasizes that for top students, enrichment for depth is the most important thing, and acceleration is appropriate only in limited circumstances. At the end of the report, Tim Gowers writes a piece mentioning cases where acceleration may have been counterproductive. I'm not saying that... – Keith Jun 18 '15 at 02:02
  • (cont'd) everything needs to be proved. The exceptions are those things whose technical difficulty is excessive for the level of development of the student. Otherwise, I think an approach which routinely leaves assertions unproved which could reasonably be proved, sends the wrong message about what mathematics is. And if "middle school algebra" means solving problems that challenge a student (without ever being too far above his ability) and contribute to his problem-solving skills, then I don't see what's wrong with spending time on that. The goal is to build up to the point where moving... – Keith Jun 18 '15 at 02:08
  • (cont'd) on to difficult higher math with proofs, (e.g., Spivak's book, where essentially everything is proved), is not a dramatic step up in sophistication. – Keith Jun 18 '15 at 02:11
  • I'm well acquainted with such recommendations, but I disagree. I'm not in favor of "acceleration through the curriculum", either, because the curriculum itself is (in my opinion) skewed in the direction of fussiness over small, not-so-interesting things, up through the standard undergrad stuff and into beginning graduate-level material (in the U.S. system). Teaching people to worry about plausibly innocent things is not good... Not room for an essay in this comment box... – paul garrett Jun 18 '15 at 14:30
  • @paulgarrett When I say studying the regular curriculum in greater depth, I'm talking about things like arithmetic, algebra and geometry, not the fluff and bloat that's been added over the years on peripheral topics. You just need to look at some of the questions that have been asked by teenagers on Math SE about field extensions or multiple integrals and the like to see that they are sometimes accelerated into advanced courses when their understanding of lower-level material (and their approach to proofs) is still shaky. The top 1% isn't necessarily like the top 0.1%, or they the top 0.01%. – Keith Jun 22 '15 at 03:48
  • (1) Not clear to me that enrichment at the time is right approach for gifted math students versus acceleration of a more typical curriculum. (2) Not clear to me that a more theoretically rigorous algebra course is key preparation for a rigorous calc class. Perhaps just a very strong manipulative course is the more important (e.g. since epsilon delta involves a lot of manipulation. (3) My experience (80s US, good public school) was that there was a year in between algebra 2/trig and calculus, that we called "functions and analytic geometry" that gave extra prep. – guest Apr 04 '18 at 17:32
  • I think there are early 20th century books that come close to what you want, although they sometimes use old-fashioned language and notation, and thus would not be a complete substitute for modern books. For example, I have a copy of C. Smith's 'A Treatise on Algebra' that has plenty of difficult exercises similar to those you mention. – A. Goodier Apr 19 '20 at 10:16

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Check out the Art of Problem Solving books. They are designed to prepare students for mathematics competitions.

SlyPuppy
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    Can you comment about which ones you'd suggest and why? For reference, here are some other cursory mentions of AoPS on MESE: 1, 2, 3, 4, 5. – Benjamin Dickman Jun 20 '15 at 13:24
  • I've had a look at the sample material on their website. It would appear to be a bit contest-oriented, at least in the way it's marketed. For example, they mention that many problems are taken from math competitions. I would note that this material was written by people known for being former Olympiad participants, rather than for being mathematicians (like Vilenkin). Problems and material chosen with a view towards competitions will not necessarily be the same as that aimed at preparing students for (and attracting them to) higher math, although there will undoubtedly be some overlap. – Keith Jun 22 '15 at 03:37
  • That being said, it's not clear what alternatives there are in English. – Keith Jun 22 '15 at 04:05
  • @Keith Actually maybe the contest math is a better way to attract kids to math than something a research mathematician would want. Realize you are dealing with motivations and psychology, not just explication. After all, Andrew Wiles was motivated by the romance of Fermat's Last Theorem. – guest Oct 12 '18 at 05:04