Problem:
In parallelogram ABCD, point M is on AB so that ABAM​=100017​, and point N is on AD so that ADAN​=200917​. Let P be the point of intersection of AC and MN. Find APAC​.
Solution:
Let point S be on AC such that NS is parallel to AB. Because △ASN is similar to △ACD,ACAS​=ACAP+PS​=ADAN​=200917​. Because △PSN is similar to △PAM,APPS​=AMSN​=100017​AB200917​CD​=20091000​, and so APPS​+1= 20093009​. Hence AP200917​AC​=20093009​, and APAC​=177​.
The problems on this page are the property of the MAA's American Mathematics Competitions
