2021 WSMO Speed Round Problems
Contents
Problem 1
Let , and let . Find the value of .
[i]Proposed by pinkpig[i]
Problem 2
A square with side length of units is rotated around one of its sides by . If the volume the square sweeps out can be expressed as , find .
[i]Proposed by mahaler[i]
Problem 3
Let . Find the value of .
[i]Proposed by asimov[i]
Problem 4
A square with side length is placed inside of a right isosceles triangle with such that and are on , is on , and is on . Find the area of .
Problem 5
The number of ways to arrange the characters in "delicious greenbeans" into two separate strings of letters can be expressed as where is maximized and both and are positive integers. Find (A string of letters is defined as a group of consecutive letters with no spaces between them.)
[i]Proposed by pinkpig[i]
Problem 6
A bag weighs 1 pound and can hold 16 pounds of food at maximum. Danny buys 100 packages of tomatoes and 300 packages of potatoes. Tomatoes come in packages that are ounces each and potatoes come in packages that are ounces each. If all of Danny's food must go in bags, how many pounds does Danny's total luggage weigh, including the bags? (Note that Danny will use only as many bags as he needs and that packages have to stay together).
[i]Proposed by pinkpig[i]
Problem 7
Consider triangle with side lengths and incircle . A second circle is drawn which is tangent to and externally tangent to . The radius of can be expressed as , where and is not divisible by the square of any prime. Find .
[i]Proposed by pinkpig[i] f
Problem 8
Let be the number of ways to seat distinguishable people around a regular hexagon such that rotations do not matter (but reflections do), and two people are seated on each side (the order in which they are seated matters). Find the number of divisors of .
[i]Proposed by captainnobody[i]
Problem 9
Bobby is going to throw 20 darts at the dartboard shown below. It is formed by 4 concentric circles, with radii of and , with the largest circle being inscribed in a square. Each point on the dartboard has an equally likely chance of being hit by a dart, and Bobby is guaranteed to hit the dartboard. Each region is labeled with its point value (the number of points Bobby will get if he hits that region). The expected number of points Bobby will get after throwing the 20 darts can be expressed as where . Find \newline
[i]Proposed by pinkpig[i]
Problem 10
Find the remainder when is divided by 11.
[i]Proposed by pinkpig[i]