The Forty-Seventh Annual William Lowell Putnam Competition
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Done: All A–1 Find, with explanation, the maximum value of f(x) = x−3x on the set of all real numbers x satisfying x +36 ≤ 13x. Solution: Write the constraint as (x − 13/2) + 36 ≤ 169/4 or (x − 13/2) ≤ 25/4. This becomes |x − 13/2| ≤ 5/2 or 4 ≤ x ≤ 9 which is the union of−3 ≤ x ≤ −2 and 2 ≤ x ≤ 3. The given cubic has derivative 3(x−1) which is positive for x > 1 and for x < −1. Thus the maximum over each of the two intervals occurs at the right hand end of that interval; that is, the maximum is either at x = −2 or at x = 3. The former gives the value −8 + 6 = −2 while the latter gives 27− 9 = 18. Thus the maximum value is 18 which occurs when x = 3. A–2 What is the units (i.e., rightmost) digit of ⌊ 1
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The Forty-Sixth Annual William Lowell Putnam Competition
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