First Steps for Math Olympians: Using the American by J. Douglas Faires

By J. Douglas Faires

Any highschool pupil getting ready for the yank arithmetic Competitions should still get their arms on a duplicate of this publication! an enormous point of mathematical education and its gain to society is the facility to take advantage of good judgment to resolve difficulties. the yankee arithmetic Competitions (AMC) were given for greater than fifty years to hundreds of thousands of highschool scholars. This ebook considers the fundamental principles at the back of the strategies to nearly all of those difficulties, and provides examples and workouts from earlier checks to demonstrate the innovations. someone taking the AMC tests or supporting scholars organize for them will locate many beneficial rules the following. yet humans regularly drawn to logical challenge fixing must also locate the issues and their recommendations attention-grabbing. This booklet will advertise curiosity in arithmetic by way of offering scholars with the instruments to assault difficulties that happen on mathematical problem-solving checks, and in particular to point the enjoying box if you would not have entry to the enrichment courses which are universal on the best educational excessive faculties. The e-book can be utilized both for self-study or to provide those that are looking to aid scholars arrange for arithmetic assessments quick access to topic-oriented fabric and samples of difficulties in line with that fabric. this can be precious for academics who are looking to carry specified classes for college kids, however it is both worthy for folks who've young ones with mathematical curiosity and talent. As scholars' challenge fixing skills increase, they are going to be capable of understand tougher thoughts requiring larger mathematical ingenuity. they are going to be taking their first steps in the direction of changing into math Olympians!

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What is x1 + x2 ? h h b b (A) − (B) (C) (D) 2b (E) − 3 3 3 3 Exercise 3 (A) 0 What is the remainder when x 51 + 51 is divided by x + 1? (B) 1 (C) 49 (D) 50 (E) 51 Exercise 4 What is the maximum number of points of intersection of the graphs of two different fourth-degree polynomial functions y = P(x) and y = Q(x), each with leading coefficient 1? (A) 1 (B) 2 (C) 3 (D) 4 (E) 8 18 First Steps for Math Olympians Exercise 5 The parabola with equation y(x) = ax 2 + bx + c and vertex (h, k) is reflected about the line y = k.

R ESULT 1 Suppose that a > 0 and b > 0, then for every pair of real numbers x and y: 21 Exponentials and Radicals • a0 = 1 • • a x · a y = a x+y ax = a x−y ay • • • 1 ax x y (a ) = a x y a x ax = x b b a −x = The following properties of radicals follow directly from the exponential properties. R ESULT 2 • • If a > 0 and b > 0, then for each positive integer n we have √ √ √ n ab = n a n b • √ √ a = mn a • m n √ n a a = √ n b b √ n m m/n a =a n There are certain special situations that require care when working with exponentials and radicals.

Consider the function f that cubes numbers, that is, f (x) = x 3 . Suppose that we are asked to determine the number of values of x for which f (x 2 + 1) = ( f (x))2 + 1. This object of the problem is simply to see if functional notation can be correctly applied. Since f is the function that cubes, we have f (x 2 + 1) = (x 2 + 1)3 = (x 2 )3 + 3(x 2 )2 + 3x 2 + 1 = x 6 + 3x 4 + 3x 2 + 1, and ( f (x))2 + 1 = (x 3 )2 + 1 = x 6 + 1. So we need to determine how many real numbers satisfy x 6 + 3x 4 + 3x 2 + 1 = x 6 + 1, that is 0 = 3x 4 + 3x 2 = 3x 2 (x 2 + 1), There is only one real solution to this final equation, the number x = 0.

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