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Solutions to Problems; Chapter (^8 383) slightly exceeds any one of the integers 1,2,... ,70 and not satisfied when z is slightl ...
384 Answers to Exercises and Solutions to Problems Similarly, (Q + e)(a + f)(e + f) = -(b + c)(b + d)(c + d). Multiplying the co ...
Solutions to Problems; Chapter 8 385 so that (u + bi)2 = (c + di) + 2(u - bi)(cosO + isine) =+ I(u + bi)2 - (c + di)l = 21~ - bi ...
386 Answers to Exercises and Solutions to Problems The local maximum and minimum values are those values of k for which y = k i ...
Solutions to Problems; Chapter (^8 387) Let s(x) = C-ldf(+n) = 2” + u,-lx”-l + unun4x”-2 +.. - + u;-2u1x + uyuo. Then g(x) is a ...
388 Answers to Exercises and Solutions to Problems w2 + w + 1 is a positive real multiple of -(w2 - w + 1). If there is equality ...
Solutions to Problems; Chapter (^8 389) Prom the difference of the last two equations, we obtain that Y(l - Y2> + 4% - 1) + x ...
390 Answers to Exercises and Solutions to Problems (3) f’(x) = (x - k)g(x) where g(x) = 4x2 + 4kx + 2k(2k + 1) = (2x + k)2 + k(3 ...
Solutions to Problems; Chapter 8 391 (b) k=-; ” -2-43 Y’ (cl) k=T -2-g (e) 6 <kc-; Yb (f) k=-; ” ...
392 Answers to Exercises and Solutions to Problems (i) -+<k<-i t y (k) +k<(-2+&)/6 yt (ml I -2+@ J, 6 <k<Q ...
Solutions to Problems; Chapter (^8 393) Let a, b, c be the lengths of the sides. Then u+b+c = 98, and, by Heron’s formula for t ...
Notes on Explorations E.l. The identity u(ud + bc) + (2bd + c”) = c(c + ub) + d(2b + u”) is useful in picking the coefficients o ...
Notes on Explorations 395 The question of finding distinct sets with equal sums of powers pops up persistently in the literature ...
396 Notes on Explorations J.B. Roberts, Splitting consecutive integers into classes with equal power sums Amer. Math. Monthly 71 ...
Notes on Explorations 2y2 - %2 = 1. The positive solutions to equation (1) are given by where (Xk,Yk) = (l,l), (3,5), (11,19), ( ...
398 Notes on Explorations Taking note of the special case k = 3 of the identity (p2 + kq2)(r2 + ks2) = (pr + kqs)2 + k(ps - qr)2 ...
Notes on Explorations 399 and g have integer coefficients and the leading coefficient of g is 1, then it can be arranged that h ...
400 Notes on Explorations Thomas, Rational tabulated values of trigonometric functions, Amer. Math. Monthly 69 (1962), 789-793. ...
Notes on Explorations 401 An equivalent diophantine equation is X3 + Y3 = U3 + V3. One polyno- mial solution is given by (X, Y, ...
402 Notes on Explorations Similarly, we can discover that k(2k - 1) + k = k2 + k2. The number of integers which can be expressed ...
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