Given 12 real numbers d

_{1}, ... , d

_{12}on the interval (1, 12), show that there exist distinct indices i, j, k such that there is an acute triangle with side lengths d

_{i}, d

_{j}, d

_{k}.

I will post a hint as a comment in a few days.

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

###
A8. Acute triangles with given side lengths

## Invitation

## Elementary Math Problems

## Advanced Math Problems

## Book Reviews

## Labels

## Contributors

This is a neat problem from last year's Putnam Examination. It was published (with answer given) in the most recent MAA Monthly.

Given 12 real numbers d_{1}, ... , d_{12} on the interval (1, 12), show that there exist distinct indices i, j, k such that there is an acute triangle with side lengths d_{i}, d_{j}, d_{k}.

I will post a hint as a comment in a few days.

Given 12 real numbers d

I will post a hint as a comment in a few days.

Subscribe to:
Post Comments (Atom)

If you'd like to follow my blog, click on the Follower button. You have your choice of making yourself known or not. In either case, you can receive all my future posts as I add them. You can unsubscribe at any time. If you think I might want to follow your blog, you can send me an email at peterash3@gmail.com.

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.

- advanced problems (17)
- algebra (3)
- books (13)
- business (6)
- educational technology (9)
- elementary problems (39)
- geometry (24)
- humor (4)
- mathematics anxiety (6)
- mathematics education (37)
- mathematics research questions (5)
- mathematics research results (6)
- meetings (3)
- number theory (1)
- obituaries (3)
- probability and statistics (1)
- random thoughts (21)
- teaching tips (13)
- tutoring (4)

## 1 comment:

Hint:

First solve the problem: Given a, b, c, such that 0 < a <= b <= c find a simple necessary and sufficient condition such that there is an acute triangle with side lengths a, b, and c.

Post a Comment