Problem:
Let A,B and C be three distinct points on the graph of y=x2 such that line AB is parallel to the x-axis and △ABC is a right triangle with area 2008. What is the sum of the digits of the y-coordinate of C ?
Answer Choices:
A. 16
B. 17
C. 18
D. 19
E. 20
Solution:
Let A=(a,a2) and C=(c,c2). Then B=(−a,a2). If either ∠A or ∠B is 90∘, then c=±a, but this is impossible because A,B, and C must have distinct x-coordinates. Thus ∠C=90∘, so AC⊥BC. Consequently
c−ac2−a2⋅c+ac2−a2=−1
from which 1=a2−c2, which is the length of the altitude from C to AB. Because △ABC has area 2008, it follows that AB=4016,∣a∣=2008 and a2=20082=4032064. Therefore c2=a2−1=4032063 and the sum of the digits of c2 is 18 .
The problems on this page are the property of the MAA's American Mathematics Competitions