Challenge questions for extremely bright kids

Question

I suppose this is the place for my questions as much as any place is:

I'm a math student coming on my 3rd year of undergrad, and I am working as a counselor at a Summer math camp. The camp is for 12-15 year old kids. About 40 are taken from across Texas each year. The kids take 38 hours of math each week for three weeks, by the end of it completing algebra 1, geometry, algebra 2, or pre-calculus. Some kids will finish the class they had initially been placed in before the three weeks are up, and they might finish two or three different courses in those three weeks.

The point is that these kids are bright, and by bright I mean bright. A handful of them will most likely become mathematicians themselves. I actually did the camp when I was 15, and it paved the way for me to enter university two years early, as well as made me want to pursue math.

What I want to do is give these kids a problem of the day or week, depending on what I can come up with. These should be problems the higher-performing kids will be able to do, but not easily. They should take a good amount of thought, but still be doable for very gifted 12-15 year olds. I'd rather it wasn't highly technical, i.e. that they don't have to use some calculus method or something to solve it; I also want them to seem like challenge questions, rather than just more homework. Moreover, I'd rather it were fairly tough to Google.

I am, however, not an educator, so I was hoping some instructors with a history of dealing with precocious and intelligent students could offer some advice. Thanks in advance.

Answer 1

Here are some suggestions for problem sources in English. Some of them are appropriate for very bright students studying geometry or Algebra II, but might nonetheless prove too difficult for students accelerated to this extent.

-Mathematical Circles, Fomin et al.

-Mathematical Problems: An Anthology, Dynkin et al.

-Problems in Elementary Mathematics, Lidsky et al.

-A Problem Book in Algebra, Krechmar.

-Many of the books in the New Mathematical Library. (See the link for book descriptions.)

-Selected Problems and Theorems of Elementary Mathematics, Shklyarsky et al.

-Challenging mathematical problems with elementary solutions, Yaglom and Yaglom.

-Problems in Plane and Solid Geometry, Prasolov.

-The Stanford Mathematics Problem Book, Polya and Kilpatrick.

-Five Hundred Mathematical Challenges, Barbeau et al.

-One Hundred Problems in Elementary Mathematics, Steinhaus.

-Mathematical Scholarship Problems, Burkill and Cundy. (It's fascinating to see just how far the standard geometry curriculum in England could take you at one time. Géométrie: Classe de Mathématiques by Hémery and Lebossé is equally telling in this regard, with respect to France in the 1950s.)

-Lessons in Geometry, Vols. 1 and 2, Hadamard (based on the 1902 French curriculum).

-Geometry, Vols 1, 2, Kiselev (English translation by Givental).

See the bibliography here, as well as the one in the Mathematical Olympiad Handbook by Gardiner.

The books Functions and Graphs, Algebra and Sequences, Combinations and Limits by Gelfand et al. have some interesting problems in them as well, usually moderately, but not extremely, hard.

Here are some high-school level books (translated into English from Russian) which, without focusing on "olympiad-level" problems, do aim for much higher technical proficiency than typical American precalculus books would, and they devote appropriate attention to proofs. The first two are textbooks in their own right. Sometimes the Spanish translations are easier to locate than the English ones.

-Algebra and Analysis of Elementary Functions, Potapov et al. (Despite the name, there is no calculus in it except for one chapter on limits. If I'd been in a summer camp learning pre-calculus, I'd much rather have used this book than, say, Larson or some similar American book.) Here is a link to the Spanish translation.

-Elementary Mathematics: A Review Course, Skanavi et al.

-Elementary Mathematics: Selected Topics and Problem-solving, Dorofeev et al.

There is an excellent textbook by Boltyanski, Sidorov and Shabunin focusing on "the most difficult topics in the school curriculum of algebra and elementary functions", particularly those topics "not sufficiently elucidated in textbooks." Unfortunately, this book appears never to have been translated. Neither have any of the books specifically written for mathematical schools, including the excellent algebra textbooks for grades 8-11 by Vilenkin et al., the problem book by Galitski et al. for grades 8 and 9, Algebra and Number Theory for Mathematical Schools by Alfutova and Ustinov, and a number of other problem books.

Answer 2

I work with gifted elementary school students, but one of my favorite sites, nrich has challenging problems that you could use for older gifted students.

Try looking at secondary problems for stages 4 and 5.
Here are some suggested problems to see if you'll like the site:

You can also look at the Post-16 Curriculum on nrich here. See description below.

The links below take you to a selection of "rich tasks" from the NRICH collection, chosen because they are ideal for developing subject content knowledge as well as mathematical thinking and problem-solving (process) skills.

Answer 3

The website mathschallenge.net has pdfs of tiered problems with solutions. They are nicely posed problems, often with some humour, and are based on different stages of mathematical maturity. To begin with they have no real prerequisites, but eventually they require some of the 'canonical' mathematical knowledge. I just found them to be well organised, interesting and adaptable. From the outset, the solver is asked to generalise a particular problem, which lends itself to proof techniques. The site is from the same developers as projecteuler, to give you some idea as to its credentials. Just click on the 'problems' tab at the top of the page.