^{2}/4 + y

^{2}= 1 and C(r) be the circle with center (1,0) and radius r. For which values of r do the curves E and C(r) have point(s) of tangency?

This is fairly routine, but still a bit challenging to find all solutions.

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Peter Ash's Thoughts on Math and Education

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A13. Points of tangency of an ellipse and a circle

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A12. Fifth powers final digit - generalized.

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E32. Fifth powers final digit

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Hippasus & The Discovery of Irrational Numbers

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E32. Chimes

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Embarrasing mistake

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A simple (?) number theory conjecture

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Musings on doing and teaching mathematics, book reviews, math problems. Information about my math education business, Cambridge Math Learning.

Let E be the ellipse with equation x^{2}/4 + y^{2} = 1 and C(r) be the circle with center (1,0) and radius r. For which values of r do the curves E and C(r) have point(s) of tangency?

This is fairly routine, but still a bit challenging to find all solutions.

This is fairly routine, but still a bit challenging to find all solutions.

If numbers are expressed in base b, for which b is it true that n^{5} and n end in the same digit for all positive integers n?

This is an obvious generalization of Problem E32.

This is an obvious generalization of Problem E32.

The following rather neat problem occurs in *Challenging Problems in Algebra* by Alfred S. Posamentier and Charles T. Salkind. I think it is suitable for a bright high school student or, with some hints, even for average high school students.

Prove that n and n^{5} always end in the same digit (in ordinary base-10 representation).

Prove that n and n

In November I gave a speculative talk to the New England Section of the Mathematical Association of America on the discovery of irrational numbers by the Pythagorean Hippasus through an examination of the mystic pentagram, the sacred symbol of the Pythagoreans. I have expanded it a bit, adding a method of recursively computing the golden ratio, phi. If you want to see how these topics are related, please check my paper on Scribd: https://www.scribd.com/doc/294006886/Irrational-Numbers-the-Mystic-Pentagram-and-Eigenvectors.

The following problem is given in Jacobs' *Geometry*:
If it takes a clock 3 seconds to ring 3:00 (3 chimes), how long does it take the same clock to ring 6:00 (6 chimes)? The answer is not 6 seconds.
The answer depends on making some assumptions, which I think are reasonable ones.

I just realized that my last post was far off the mark. There is a super-obvious example that shows that f(n) >= n/2, and so in fact f(n) = n/2. The example is the subset {n/2 + 1, n/2 + 2, ... , n} which contains n/2 elements and clearly has the non-divisible property. The problem has very little to do with prime numbers. The previously-published result was sharp.

Call a set S of positive integers non-divisible if, whenever a and b belong to S, it is not true that (a|b or b|a). For example, the set of prime numbers is non-divisible. Let S(n) be the set of the first n positive integers, and let f(n) be the cardinality of the largest non-divisible subset of S(n). Then clearly f(n) >= pi(n), where pi(n) denotes the number of prime numbers <= n. Furthermore, I know a pretty proof (published, but not by me) that shows f(n) <= n/2 (n even). That is, in any subset of n/2 + 1 positive integers all <= n, at least one integer divides another integer in the set. This proof involves the pigeonhole principle. So, pi(n) <= f(n) <= n/2. (n even)
My conjecture is that, in fact, f(n) = pi(n) for all n > 1. Does anyone have a counterexample or a proof? It seems like it should be very simple.

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From time to time I will post an elementary mathematics problem which I hope readers may enjoy. "Elementary" to me means that the problem does not require any specialized mathematical knowledge beyond high-school mathematics to solve. Some of these elementary problems will be very simple, others will require a great deal of cleverness. Elementary math problems will be denoted by the letter E followed by the problem number.

I will sometimes post a link to a solution. If I haven't yet posted a link, you may send me your answer and if it is correct I will credit you on the blog. To send answers, please mailto: peterash3@gmail.com.

Thanks,

Peter

I will sometimes post a link to a solution. If I haven't yet posted a link, you may send me your answer and if it is correct I will credit you on the blog. To send answers, please mailto: peterash3@gmail.com.

Thanks,

Peter

From time to time post problems that are somewhat more advanced than those in the Elementary Math Problems. These problems will require a knowledge of some college-level mathematics, either for their statement or for the solution that I know. Advanced math problems will be denoted by the letter A followed by the problem number.

I will sometimes post a link to a solution. If I haven't yet posted a link, you may send me your answer and if it is correct I will credit you on the blog. To send answers, please mailto: peterash3@gmail.com.

I will sometimes post a link to a solution. If I haven't yet posted a link, you may send me your answer and if it is correct I will credit you on the blog. To send answers, please mailto: peterash3@gmail.com.

Here are some recent reviews on mathematics, learning theory, education, and related technology:

The Number Sense: How the Mind Creates Mathematics by Stanislaus Dehaene, Oxford University Press 1997

Three Books on the Riemann Hypothesis

The King of Infinite Space: Donald Coxeter, the Man Who Saved Geometry by Siobhan Roberts, Walker and Company, 2006

The Poincare Conjecture: In Search of the Shape of the Universe, Donal O'Shea, Walker & Company, 2007

Where Mathematics Comes From: How the Embodied Mind Brings Mathematics into Being, George Lakoff and Rafael E. Nunez, Basic Books, 2000.

The Number Sense: How the Mind Creates Mathematics by Stanislaus Dehaene, Oxford University Press 1997

Three Books on the Riemann Hypothesis

The King of Infinite Space: Donald Coxeter, the Man Who Saved Geometry by Siobhan Roberts, Walker and Company, 2006

The Poincare Conjecture: In Search of the Shape of the Universe, Donal O'Shea, Walker & Company, 2007

Where Mathematics Comes From: How the Embodied Mind Brings Mathematics into Being, George Lakoff and Rafael E. Nunez, Basic Books, 2000.

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