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7.3. Inequalities 223 (a) Prove by induction on n that, if zi 2 -2 and xi all have the same sign, then When does equality hold ...
224 7. Approximations and Inequalities (e) The results of (c) and (d) can be generalized to u,-ru,+r < uz for r = 2,3,..., n ...
7.4. Problems on Inequalities 225 7.4 Problems on Inequalities Find all real triples (x, y, z) for which (l-~)~+(x-y)~+(y-~)~+z ...
226 7. Approximations and Inequalities Determine u so that the zeros of the polynomial t2 - (3u + 1)t + (2u2 - 3u - 2) are real ...
7.5. Other Problems 227 (b) (u - byyu - c) ’ (b - ai;b - c) ’ (c - uy;c - b) = (a + b + c)~ - 2(a + b + c)(bc + cu + ub) + abc. ...
(k = 0,1,2,... , n) is given for x # zk for any k by P(X) = www where N(x) = g[(-l)kyk sin uk/(X - xk)] k=O and D(x) = &(-I) ...
7.5. Other Problems 229 4.4. Apply the AGM inequality to {rf}. 4.5, 4.6. Take the difference of the two sides. 4.7. If a # 0, le ...
230 7. Approximations and Inequalities 5.6. Begin by a careful study of the quadratic case. What happens if all the xi have the ...
8 Miscellaneous Problems Solve x(3y - 5) = y2 + 1 in integers. Solve a3 - b3 - c3 = 3abc a2 = 2(b + c) simultaneously in positi ...
232 8. Miscellaneous Problems If x, y, z are rational numbers for which x3 + 3y3 + 9z3 - 9xyt. = 0, prove that x = y = z = 0. S ...
Miscellaneous Problems 233 (a) Let nl, 122,... , nd be a set of positive integers exceeding 1, with any pair relatively prime. ...
234 8. Miscellaneous Problems Determine necessary and sufficient conditions on a, b, c such that az+by+cz=O and a~=+b&-&am ...
Miscellaneous Problems 235 Show that the set of real numbers x which satisfy the inequality 70 k c- x-k >2 - 4 k=l is a uni ...
236 8. Miscellaneous Problems If a/(bc - u2) + b/(ca - b2) + c/(ab - c2) = 0, prove that a/(bc - a2)2 + b/( ca - b2)2 + c/(ab - ...
Miscellaneous Problems 237 A triangle has sides of length 29, 29, 40. Find all other triangles with integer sides with the same ...
238 8. Miscellaneous Problems property. Do there exist polynomials of higher degree which, along with all their derivatives, hav ...
Miscellaneous Problems 239 the points of the Mandelbrot set on the complex plane. There is, of course, the problem of deciding w ...
240 8. Miscellaneous Problems E.69. Quaternions. A polynomial of degree n over a field has at most n zeros. This result may no l ...
Hints 241 Multiply by 4 and complete the square on both sides. Find a quadratic equation for y in terms of z. Multiply by a and ...
242 8. Miscellaneous Problems (a) 1 - p(x) can be written as a product. We want fi + 6 + fi to be a zero of p(x) - (x - &)q ...
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