Download A Friendly Mathematics Competition: 35 Years of Teamwork in by Rick Gillman PDF

By Rick Gillman

"A pleasant arithmetic Competition" tells the tale of the Indiana collage arithmetic pageant (ICMC) by way of proposing the issues, recommendations, and result of the 1st 35 years of the ICMC. The ICMC was once equipped in response to the Putnam examination - its difficulties have been to be extra consultant of the undergraduate curriculum, and scholars may perhaps paintings on them in teams.
Originally participation was once initially constrained to the small, deepest schools and universities of the country, yet was once later unfolded to scholars from the entire colleges in Indiana. the contest used to be speedy nicknamed the "Friendly" festival as a result of its specialise in fixing mathematical difficulties, which introduced school and scholars jointly, instead of at the aggressive nature of successful. geared up through 12 months, the issues and options during this quantity current a great archive of data approximately what has been anticipated of an undergraduate arithmetic significant during the last 35 years. With greater than 245 difficulties and recommendations, the booklet is additionally a needs to purchase for college and scholars drawn to problem-solving.

The index of difficulties lists difficulties in: Algebraic constructions; Analytic Geometry, Arclength, Binomial Coefficients, Derangements, Differentiation, Differential Equations, Diophantine Equations, Enumeration, box and Ring idea, Fibonacci Sequences, Finite Sums, basic Theorem of Calculus Geometry, workforce idea, Inequalities, limitless sequence, Integration, restrict overview, common sense, Matrix Algebra, Maxima and Minima difficulties, Multivariable Calculus, quantity thought, variations, chance, Polar Coordinates, Polynomials, actual Valued features Riemann Sums, Sequences, platforms of Equations, information, man made Geometry, Taylor sequence, Trigonometry, and Volumes.

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A) Show that there exist infinitely many twosomes of consecutive positive integers each of which can be written as the sum of two nonzero perfect squares. For example, 17 = 12 +42 , 18 = 32 +32 and 25 = 32 +42 , 26 = 12 + 52 . (b) Find three consecutive positive integers each of which can be written as the sum of two nonzero perfect squares. Are there infinitely many such trios? (c) Do there exist four consecutive positive integers each of which can be written as the sum of two nonzero perfect squares?

Suppose you repeatedly toss a fair coin until you get two heads in a row. What is the probability that you stop on the 10th toss? P1983-3. Consider an isosceles right triangle with legs of fixed length a. Inscribe a rectangle and a circle inside the triangle as indicated in the figure below. Find the dimensions of the rectangle (and the radius of the circle) which make the total area of the rectangle and circle a maximum. a a 26 Exams P1983-4. Show that every positive real number is a sum (possibly infinite) of a subset of the numbers {1, 1/2, 1/3, 1/4, .

Pn (x) by x − xj pk (x) = , x ∈ R. xk − xj j=k Prove that n k=1 pk (x) = 1 for all x in R. P1983-9. Suppose that g : [0, 1] → [0, 1] is a continuous function and that gm (x) = x (g composed with itself m times) for all x and for some positive integer m. Prove that g2 (x) = x for all x. Exam #19–1984 This competition was held at Rose-Hulman and a team from Rose-Hulman consisting of Todd Fine, Byron Bishop, and Dan Johnson won. Teams from Wabash College and Valparaiso University came in second and third, respectively.

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