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# nasex2 - M 2003 NUMBERS AND SETS EXAMPLES SHEET 2 W T G 1...

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M 2003 NUMBERS AND SETS – EXAMPLES SHEET 2 W. T. G. 1. Prove by induction that the following two statements are true for every positive inte- ger n . (i) The number 2 n +2 + 3 2 n +1 is a multiple of 7. (ii) 1 3 + 3 3 + 5 3 + . . . + (2 n - 1) 3 = n 2 (2 n 2 - 1) . 2. Suppose that you have a 2 n × 2 n grid of squares (if n = 3 then you have a chessboard) and you remove one square. Prove that, wherever the removed square is, the remaining squares can be tiled with L-shaped tiles - that is, tiles consisting of three squares that form a 2 × 2 grid with one square removed. 3. By considering the equation (1 - 1) n = 0, give another proof that exactly half the subsets of { 1 , 2 , . . . , n } have even size. 4. Prove that k k + k + 1 k + . . . + n + k - 1 k = n + k k + 1 for any n > k . [Hint: for each set of size k + 1 consider its largest element.] 5. Prove that n 0 2 + n 1 2 + n 2 2 + . . . + n n 2 = 2 n n . [Hint: ( n k ) = ( n n - k ) .] 6. There are four primes between 0 and 10 and four between 10 and 20. Does it ever happen again that there are four primes between two consecutive multiples of 10?

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nasex2 - M 2003 NUMBERS AND SETS EXAMPLES SHEET 2 W T G 1...

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