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R3

Part of 2022 CMWMC

Problems(2)

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

Source:

8/12/2023
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.
CMWMCalgebrageometrycombinatoricsnumber theory
2022 CMWMC Relay Round 3/4 - Carnegie Mellon University Womens' Competition

Source:

8/12/2023
Set 3
3.1 Annie has 2424 letter tiles in a bag; 88 C’s, 88 M’s, and 88 W’s. She blindly draws tiles from the bag until she has enough to spell “CMWMC.” What is the maximum number of tiles she may have to draw?
3.2 Let TT be the answer from the previous problem. Charlotte is initially standing at (0,0)(0, 0) in the coordinate plane. She takes TT steps, each of which moves her by 11 unit in either the +x+x, −x-x, +y+y, or −y-y direction (e.g. her first step takes her to (1,0)(1, 0), (1,0)(1, 0), (0,1)(0, 1) or (0,−1)(0, -1)). After the T steps, how many possibilities are there for Charlotte’s location?
3.3 Let TT be the answer from the previous problem, and let SS be the sum of the digits of TT. Francesca has an unfair coin with an unknown probability pp of landing heads on a given flip. If she flips the coin SS times, the probability she gets exactly one head is equal to the probability she gets exactly two heads. Compute the probability pp.
PS. You should use hide for answers.
CMWMCcombinatorics