By Akiva Moiseevič Âglom; Isaak Moiseevič Âglom; James McCawley; Basil Gordon
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Extra info for Challenging mathematical problems with elementary solutions [Vol. II]
Example text
B. C, D, E, F, comprise the nine points. As the nine lines we take the six diagonals, the horizontal lines AE and DF, and the line MN. which lies halfway between them. It is easy to see that these nine points and nine lines satisfy all the conditions of the problem. b. Suppose we could arrange seven points AI> A z, A a, A" As, Ae, A7 and seven lines Pb P2, Pa. p" Ps, Pe, P7 in a configuration satisfying the conditions of the problem. We show first that in this case any line joining two of the points AI> As, Aa, A 4, As.
From the identity of 142c deduce Leibniz's9 formula 7T 1 1 1 -=1--+---+···. 4 3 5 7 b. What is the sum of the infinite series 111 1+-+-+-+"'? 32 52 72 147. ~ ... 2 133 5 5 7 8 Fran~is Vieta (1540--1603), a French mathematician, one of the creators of modem algebraic notation. • Gottfried Wilhelm Leibniz (1646-1716), a German mathematician. one of the inventors of the differential and integral calculus. 10 John Wallis (1616-1703), an English mathematician. 25 XI. Areas of regions bounded by curves XI.
1. + ... + !. 2 3 5 7 p 11 In In N tends to a limit {J as N -- 00. It follows from part a that {J {J is approximately t· We thus have the approximate equality ~ 15; in fact, -1+ -1+ -1+ 1 - + -1+ ... + -1~ In In N + {J. 2 3 5 7 11 P 174. Mertens' third theorem. Let 2,3,5, 7, II, ... ,p be the primes not exceeding the integer N. Show that as N -- 00, the product In N ( 1- ~) (1 - j) (1 - ~) (1 - ~) (1 - 111) ... ) tends to a limit c. We can also write Mertens' third theorem in the form (1 - ~) (1 - j) (1 - ~) (1- ~) (1 - /J ...