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of Problem the Week 2, Fall 2008 Evaluate r 111 .{z. 111 222 .{z. 222. . . } | } | 2000 digits 1000 digits Your answer should be a number in decimal notation without square roots or any other algebraic operation. Solution by organizers. Write x = 111 .{z. 111. Then . } | 1000 digits 2000 digits 1000 digits 1000 digits and 111 .{z. 111 = 111 .{z. 111000 .{z. 000 + 111 .{z. 111 = 101000 x + x . . . . } | }| } | } | 1000 digits We want the value of s p x 1000 + x 2x = 1000 1) x = 10 (10 (999 .{z. 999)x . } | 1000 digits 1000 digits 222 .{z. 222 = 2x. . } | 1000 digits The same works if we replace 1000 by an arbitrary positive integer n. Write x = 111 .{z. 111. Then . } | n digits p (9x) x = 9x2 = 3x = 333 .{z. 333. . = } | s r 111 .{z. 111 222 .{z. 222 = 10n x + x 2x = (999 .{z. 999)x = 9x2 = 3x = 333 .{z. 333. . . . . } | } } } | | | 2n digits n digits n digits n digits 1

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