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Part of 2000 Putnam

Problems(2)

Putnam 2000 A1

Source:

9/6/2011
Let AA be a positive real number. What are the possible values of ∑j=0∞xj2,\displaystyle\sum_{j=0}^{\infty} x_j^2, given that x0,x1,⋯x_0, x_1, \cdots are positive numbers for which ∑j=0∞xj=A\displaystyle\sum_{j=0}^{\infty} x_j = A?
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Putnam 2000 B1

Source:

9/6/2011
Let aja_j, bjb_j, cjc_j be integers for 1≤j≤N1 \le j \le N. Assume for each jj, at least one of aja_j, bjb_j, cjc_j is odd. Show that there exists integers r,s,tr, s, t such that raj+sbj+tcjra_j+sb_j+tc_j is odd for at least 4N7\tfrac{4N}{7} values of jj, 1≤j≤N1 \le j \le N.
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