IMC 2012 Q22

The options given imply that the median cannot exceed 3.5, so we first look to see if the median can have this value. The median of 190 numbers is \(\tfrac{1}{2}(a + b)\) where, when the numbers are arranged in increasing order, \(a\) is the 95th number and \(b\) is the 96th number in the list. In this problem, \(a\) and \(b\) are positive integers with \(a \leqslant b\), so if \(\tfrac{1}{2}(a + b) = 3.5\) then \(a + b = 7\). The smallest total of all 190 numbers with median 3.5 occurs when \(a = 3\), \(b = 4\) and the numbers, in increasing order, are

In this case their total is 94 × 1 + 3 + 95 × 4 = 94 + 3 + 380 = 477, which is the total number of cups of coffee given in the question. So 3.5 is a possible value for the median. Note also, that, if the median were greater than 3.5, then at least one of the numbers in the above list would need to be larger and so the total would be larger than 477. So 3.5 really is the maximum possible value of the median.