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2016 | Buch

University of Toronto Mathematics Competition (2001–2015)

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This text records the problems given for the first 15 annual undergraduate mathematics competitions, held in March each year since 2001 at the University of Toronto. Problems cover areas of single-variable differential and integral calculus, linear algebra, advanced algebra, analytic geometry, combinatorics, basic group theory, and number theory. The problems of the competitions are given in chronological order as presented to the students. The solutions appear in subsequent chapters according to subject matter. Appendices recall some background material and list the names of students who did well.

The University of Toronto Undergraduate Competition was founded to provide additional competition experience for undergraduates preparing for the Putnam competition, and is particularly useful for the freshman or sophomore undergraduate. Lecturers, instructors, and coaches for mathematics competitions will find this presentation useful. Many of the problems are of intermediate difficulty and relate to the first two years of the undergraduate curriculum. The problems presented may be particularly useful for regular class assignments. Moreover, this text contains problems that lie outside the regular syllabus and may interest students who are eager to learn beyond the classroom.

Inhaltsverzeichnis

Frontmatter
Chapter 1. Problems of the Contests
Abstract
Except for the first contest, for which 3 h was allotted, the time allowed for each contest was \(3\frac{1} {2}\) h.The chapter in which the solution appears is given in parentheses at the beginning of the problem.
Edward J. Barbeau
Chapter 2. Algebra
Abstract
2001:5. Let n be a positive integer and x a real number not equal to a nonnegative integer. Prove that
Edward J. Barbeau
Chapter 3. Inequalities
Abstract
2001:1. Let a, b, c > 0, a < bc and \(1 + a^{3} = b^{3} + c^{3}\). Prove that \(1 + a < b + c\).
Edward J. Barbeau
Chapter 4. Sequences and Series
Abstract
2001:6. Prove that, for each positive integer n, the series
Edward J. Barbeau
Chapter 5. Calculus and its Applications
Abstract
2001:2. Let O = (0, 0) and Q = (1, 0). Find the point P on the line with equation \(y = x + 1\) for which the angle OPQ is a maximum.
Edward J. Barbeau
Chapter 6. Other Topics in Analysis
Abstract
2001:7. Suppose that x ≥ 1 and that \(x = \lfloor x\rfloor +\{ x\}\), where ⌊x⌋ is the greatest integer not exceeding x and the fractional part {x} satisfies 0 ≤ { x} < 1. Define
$$\displaystyle{f(x) = \frac{\sqrt{\lfloor x\rfloor } + \sqrt{\{x\}}} {\sqrt{x}}.}$$
Edward J. Barbeau
Chapter 7. Linear Algebra
Abstract
2001:4. Let V be the vector space of all continuous real-valued functions defined on the open interval \((-\pi /2,\pi /2)\), with the sum of two functions and the product of a function and a real scalar defined in the usual way.
Edward J. Barbeau
Chapter 8. Geometry
Abstract
2001:2. Let O = (0, 0) and Q = (1, 0). Find the point P on the line with equation \(y = x + 1\) for which the angle OPQ is a maximum.
Edward J. Barbeau
Chapter 9. Group Theory
Abstract
2003:10. Let G be a finite group of order n. Show that n is odd if and only if each element of G is a square.
Edward J. Barbeau
Chapter 10. Combinatorics and Finite Mathematics
Abstract
The integer points on the line and the edges between them can be coloured \( 1 --(3) --2 --(1) --3 --(2) --1 \) and so on, where the edge colouring is in parentheses. Form a plane by stacking these lines unit distance apart, making sure that each vertex has a different coloured vertex above and below it; use colours 4 and 5 judiciously to colour the vertical edges. Now go to three dimensions; stack up planar lattices and struts unit distance apart, colouring each with the colours 1, 2, 3, 4, 5, while making sure that vertically adjacent vertices have separate colours, and use the colours 6 and 7 for vertical struts. Continue on.
Edward J. Barbeau
Chapter 11. Number Theory
Abstract
2002:3. In how many ways can the rational 2002∕2001 be written as the product of two rationals of the form \((n + 1)/n\), where n is a positive integer?
Edward J. Barbeau
Backmatter
Metadaten
Titel
University of Toronto Mathematics Competition (2001–2015)
verfasst von
Edward J. Barbeau
Copyright-Jahr
2016
Electronic ISBN
978-3-319-28106-3
Print ISBN
978-3-319-28104-9
DOI
https://doi.org/10.1007/978-3-319-28106-3