Numbers and Sets - exercises for enthusiasts 1. W. T. G. 1. Let A be the sum of the digits of 4444 4444 , and let B be the sum of the digits of A . What is the sum of the digits of B ? 2. Let x 1 ,. .. ,x n be real numbers such that ∑ n i =1 x i = 0 and ∑ n i =1 x 2 i = 1. How large can x 1 x 2 + x 2 x 3 + .. . + x n-1 x n + x n x 1 be? (If you cannot solve this problem, try it for small values of n , or just experiment to see how large you can make the sum.) 3. Let R be a rectangle which can be divided into smaller rectangles, each of which has at least one side of integer length. Prove that R has at least one side of integer length. 4. Does there exist a real number c > 0 such that for every positive integer n it is possible to choose points ( x 1 ,y 1 ) ,. .. , ( x n ,y n ) in the unit square [0 , 1] 2 with the property that | x i-x j || y i-y j | > cn-1 whenever i 6 = j ? 5. Does there exist a cycle in Z 3 (i.e., a path consisting of line segments going from
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