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Mathematics Magazine Contents—October 2013

Several of the articles in this issue are about integrals—mainly, how to avoid them. You don't need them to find antiderivatives, for example; and if you want to find areas, you can follow the 17th-century practice and use infinitesimals. This issue mentions the 1943 volume of the magazine, Charlemagne's education adviser, and Hardy's taxicab number, and has all of the solutions for the USAMO, USAJMO, and IMO. Walter Stromquist

Vol. 86, No. 4, pp.238-316.


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Indivisibles, Infinitesimals and a Tale of Seventeenth-Century Mathematics

Maureen T. Carroll, Steven T. Dougherty, and David Perkins

In this article, we describe clever arguments by Torricelli and Roberval that employ indivisibles to find the volume of Gabriel’s trumpet and the area under the cycloid. We detail 17th-century objections to these non-rigorous but highly intuitive techniques, as well as the controversy surrounding indivisibles. After reviewing the fundamentals of infinitesimal calculus and its rigorous footing provided by Robinson in the 1960s, we are able to revisit the 17th-century solutions. In changing from indivisible to infinitesimal-based arguments, we salvage the beautiful intuition found in these works.

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Lebesgue’s Road to Antiderivatives

Ádám Besenyei

The traditional way of proving the existence of antiderivatives of continuous functions is through the concept of definite integrals. In the years 1904–1905, H. Lebesgue provided an alternative proof of this result not relying on the theory of integrals. His method is based on piecewise linear approximations of continuous functions, which also yields the mean value inequality as a by-product. In this note we recall Lebesgue’s ideas.

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Unified Proofs of the Error Estimates for the Midpoint, Trapezoidal, and Simpson’s Rules

Frank Sandomierski

Unified, elementary proofs are given for the error estimates associated with the midpoint rule, the trapezoidal rule, and Simpson’s rule.

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Rational Ratios and Concurrent Cevians

Antonio M. Oller-Marcén

Euler, in a paper published in 1812, studied in detail the situation when three cevians of a triangle were concurrent. In particular, he found a relation between the ratios in which the common point divided each of the cevians. This relation can be seen as an equation in three unknowns (the three ratios). In this paper we find the integer and rational solutions of this equation, providing explicit constructions and geometric interpretations in the positive cases.

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The Volume of the Unit $$n$$-Ball

Harold R. Parks

Two simple proofs are given for the fact that the volume of the unit ball in n-dimensional Euclidean space approaches 0 as n approaches ∞. (Some authors use the term “unit sphere” for what is here called the unit ball.) One argument involves covering the unit ball by simplices. The other argument involves covering the unit ball by rectangular solids.

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Ramanujan, Quadratic Forms, and the Sum of Three Cubes

James D. Harper

This article shows how, using only basic precalculus tools and a seed such as 33 + 43 + 53 = 63, to generate quadratic-form formulas for the Diophantine equation A3 + B3 + C3 = D3.

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Finite Sums of the Alcuin Numbers

Raymond A. Beauregard and Vladimir A. Dobrushkin

The Alcuin number $$t(n)$$ is equal to the number of incongruent integer triangles having perimeter $$n$$, where $$n$$ is an integer. Using generating functions, we give a derivation of well-known formulas for the Alcuin sequence $$\{t(n)\}$$ that involves the closest integer function $$\|x\|$$, and floor function $$\lfloor x \rfloor$$. These formulas do not lend themselves very easily to operations such as summation. To find the number of incongruent integer triangles having perimeter at most $$n$$, we must evaluate the sum $$\sum_{k=0}^{n}t(k)$$, or to find those with even perimeter up to $$2n$$, we must evaluate $$\sum_{k=0}^{n}t(2k)$$. These computations are theoretically awkward. In this article, we develop formulas in both closed form and abbreviated form for these sums using generating functions. In the process, we exploit the relationship between the Alcuin numbers and partitions of integers.

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Proposals 1926-1930
Quickies 1033 & 1034
Solutions 1901-1905

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Primal progress and the "mathematical mind"

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42nd USA Mathematical Olympiad, 4th USA Junior Mathematical Olympiad

Jacek Fabrykowski and Steven R. Dunbar

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54th International Mathematical Olympiad

John Berman, Zuming Feng, and Yi Sun

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2013 Carl N. Allendoerfer Awards

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