Weekly Contest #55
(Ended)
Standings
Q1.
A cubical cake with edge length 
inches is iced on the sides and the top. It is cut vertically into three pieces as shown in this top view, where 
is the midpoint of a top edge. The piece whose top is triangle 
contains 
cubic inches of cake and 
square inches of icing. What is 
?
![[asy] unitsize(1cm); defaultpen(linewidth(.8pt)+fontsize(8pt)); draw((-1,-1)--(1,-1)--(1,1)--(-1,1)--cycle); draw((1,1)--(-1,0)); pair P=foot((1,-1),(1,1),(-1,0)); draw((1,-1)--P); draw(rightanglemark((-1,0),P,(1,-1),4)); label("$M$",(-1,0),W); label("$C$",(-0.1,-0.3)); label("$A$",(-0.4,0.7)); label("$B$",(0.7,0.4)); [/asy]](https://latex.artofproblemsolving.com/b/9/a/b9a0b0d2b46c6d8034bbea244b0985f0df2a098d.png)
Q2.
Rachel and Robert run on a circular track. Rachel runs counterclockwise and completes a lap every 90 seconds, and Robert runs clockwise and completes a lap every 80 seconds. Both start from the same line at the same time. At some random time between 10 minutes and 11 minutes after they begin to run, a photographer standing inside the track takes a picture that shows one-fourth of the track, centered on the starting line. What is the probability that both Rachel and Robert are in the picture?
Q3.
A rectangular yard contains two flower beds in the shape of congruent isosceles right triangles. The remainder of the yard has a trapezoidal shape, as shown. The parallel sides of the trapezoid have lengths 
and 
meters. What fraction of the yard is occupied by the flower beds?
![[asy] unitsize(2mm); defaultpen(linewidth(.8pt)); fill((0,0)--(0,5)--(5,5)--cycle,gray); fill((25,0)--(25,5)--(20,5)--cycle,gray); draw((0,0)--(0,5)--(25,5)--(25,0)--cycle); draw((0,0)--(5,5)); draw((20,5)--(25,0)); [/asy]](https://latex.artofproblemsolving.com/3/e/e/3ee29f78e4a8eecdb8faec4092f2f1c361a869e4.png)
. What is the units digit of 
?