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2022 CMWMC Guts Round 3/8 - Carnegie Mellon University Womens' Competition

Source:

August 12, 2023
CMWMCalgebrageometrycombinatoricsnumber theory

Problem Statement

Set 3
p7. On unit square ABCDABCD, a point PP is selected on segment CDCD such that DP=14DP =\frac14 . The segment BPBP is drawn and its intersection with diagonal ACAC is marked as EE. What is the area of triangle AEPAEP?
p8. Five distinct points are arranged on a plane, creating ten pairs of distinct points. Seven pairs of points are distance 11 apart, two pairs of points are distance 3\sqrt3 apart, and one pair of points is distance 22 apart. Draw a line segment from one of these points to the midpoint of a pair of these points. What is the longest this line segment can be?
p9. The inhabitants of Mars use a base 88 system. Mandrew Mellon is competing in the annual Martian College Interesting Competition of Math (MCICM). The first question asks to compute the product of the base 88 numerals 124541581245415_8, 756326587563265_8, and 63214738 6321473_8. Mandrew correctly computed the product in his scratch work, but when he looked back he realized he smudged the middle digit. He knows that the product is 1014133027■276620411381014133027\blacksquare 27662041138. What is the missing digit?
PS. You should use hide for answers.
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