Skip to main content
PPrime
← Back to archive

Problem #49 · September 14, 2026

Absolute value

Find the minimum value of the expression ∣x−1∣+∣x−2∣+∣x−3∣+∣x−4∣+∣x−5∣|x - 1| + |x - 2| + |x - 3| + |x - 4| + |x - 5|.
  1. A0
  2. B4
  3. C5
  4. D6
  5. E10

WORKED EXAMPLE

Solution

The expression is the sum of the distances from the point xx to the points 1,2,3,4,51, 2, 3, 4, 5. Such a sum is smallest at the middle point (the median), that is, at x=3x = 3: the outer pairs satisfy ∣x−1∣+∣x−5∣≥4|x-1|+|x-5| \ge 4 and ∣x−2∣+∣x−4∣≥2|x-2|+|x-4| \ge 2, with equality in both at x=3x=3. At x=3x = 3, the value is 2+1+0+1+2=62 + 1 + 0 + 1 + 2 = 6.
Solve today's problem