Problem:
The harmonic mean of a collection of numbers is the reciprocal of the arithmetic mean of the reciprocals of the numbers in the collection. For example, the harmonic mean of 4,4,5 is
31(41+41+51)1=730.
What is the harmonic mean of all the real roots of the 4050th degree polynomial
k=1∏2025(kx2−4x−3)=(x2−4x−3)(2x2−4x−3)⋯(2025x2−4x−3)?
Answer Choices:
A. −35
B. −23
C. −56
D. −65
E. −32
Solution:
The sum of the reciprocals of the roots of kx2−4x−3 for some k is r11+r21=r1r2r1+r2=k−3k4=−34.
Therefore, the sum of all the reciprocals of the roots of the polynomial is −34∗2025 and the average is then 4050−34∗2025=3−2.
Reciprocating gives our answer of −23→(B)
The problems on this page are the property of the MAA's American Mathematics Competitions