By Fletcher C.

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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.