HW 2025-03-14 WS

[BMMT 23.8]

There are two moles that live underground, and there are five circular holes that the moles can hop out of. The five holes are positioned as shown in the diagram below, where A, B, C, D, and E are the centers of the circles, AE = 30 cm, and congruent triangles △ABC, △CBD, and △CDE are equilateral. The two moles randomly choose exactly two of the five holes, hop out of the two chosen holes, and hop back in. What is the probability that the holes that the two moles hop out of have centers that are exactly 15 cm apart?

[BMMT 23.9]

Carson is planning a trip for n people. Let x be the number of cars that will be used and y be the number of people per car. What is the smallest value of n such that there are exactly 3 possibilities for x and y so that y is an integer, x < y, and exactly one person is left without a car?

[BMMT 23.10]

Iris is eating an ice cream cone, which consists of a hemisphere of ice cream with radius r > 0 on top of a cone with height 12 and also radius r. Iris is a slow eater, so after eating one-third of the ice cream, she notices that the rest of the ice cream has melted and completely filled the cone. Assuming the ice cream did not change volume after it melted, what is the value of r?

[BMMT 23.11]

As Natasha begins eating brunch between 11:30 AM and 12 PM, she notes that the smaller angle between the minute and hour hand of the clock is 27 degrees. What is the number of degrees in the smaller angle between the minute and hour hand when Natasha finishes eating brunch 20 minutes later?

[BMMT 23.11]

On a regular hexagon ABCDEF, Luke the frog starts at point A; there is food on points C and E and there are crocodiles on points B and D. When Luke is on a point, he hops to any of the five other vertices with equal probability. What is the probability that Luke will visit both of the points with food before visiting any of the crocodiles?

[BMMT 23.14]

Aditya’s favorite number is a positive two-digit integer. Aditya sums the integers from 5 to his favorite number, inclusive. Then, he sums the next 12 consecutive integers starting after his favorite number. If the two sums are consecutive integers and the second sum is greater than the first sum, what is Aditya’s favorite number?

[BMMT 23.15]

The 100th anniversary of BMT will fall in the year 2112, which is a palindromic year. Compute the sum of all years from 0000 to 9999, inclusive, that are palindromic when written out as four-digit numbers (including leading zeros). Examples include 2002, 1991, and 0110.