2021 WSMO Team Round
Contents
Problem 1
How many ways are there to pick a three-letter word using only letters from "Winter Solstice" such that the word is a capital letter followed by two lowercase letters? (A word does not have to be an English word, and the word can only use a letter in "Winter Solstice" as many times as it appears)
Problem 2
Bobby has some pencils. When he tries to split them into 5 equal groups, he has 2 left over. When he tries to split them into groups of 8, he has 6 left over. What is the second smallest number of pencils that Bobby could have?
Problem 3
Farmer Sam has dollars. He knows that this is exactly enough to buy either 50 pounds of grass and 32 ounces of hay or 96 ounces of grass and 24 pounds of hay. However, he must save 4 dollars for tax. After some quick calculations, he finds that he has exactly enough to buy 18 pounds of grass and 16 pounds of hay (and still have money left over for tax!). Find
Problem 4
Consider a triangle satisfying
. For all successive triangles
, we have
and
, where
is outside of
. Find the value of
where
is the area of
.
Problem 5
Two runners are running at different speeds. The first runner runs at a consistent 12 miles per hour. The second runner runs at miles per hour, where
is the number of hours that have passed. After
hours, the runners have run the same distance, where
is positive. Find
.
Problem 6
Suppose that regular dodecagon has side length
The area of the shaded region can be expressed as
where
is not divisible by the square of any prime. Find
.
Problem 7
A frog makes one hop every minute on the first quadrant of the coordinate plane (this means that the frog's and
coordinates are positive). The frog can hop up one unit, right one unit, left one unit, down one unit, or it can stay in place, and will always randomly choose a valid hop from these 5 directions (a valid hop is a hop that does not place the frog outside the first quadrant). Given that the frog starts at
, the expected number of minutes until the frog reaches the line
can be expressed as
, where
and
are relatively prime positive integers. Find
.
Problem 8
Isaac, Gottfried, Carl, Euclid, Albert, Srinivasa, René, Adihaya, and Euler sit around a round table (not necessarily in that order). Then, Hypatia takes a seat. There are possible seatings where Euler doesn't sit next to Hypatia and Isaac doesn't sit next to Gottfried, where
is maximized. Find
. (Rotations are not distinct, but reflections are).
Problem 9
In triangle points
and
trisect side
such that
is closer to
than
If
and
then find
where
is the area of
.
Problem 10
The minimum possible value of can be expressed as
Find
Problem 11
Find the remainder when is divided by
. (
)
Problem 12
Choose three integers randomly and independently from the nonnegative integers. The probability that the sum of the factors of
is divisible by
is
, where
and
are relatively prime positive integers. Find
.
Problem 13
Square is drawn outside of equilateral triangle
Regular hexagon
is drawn outside of square
If the area of triangle
is 3, then the area of triangle
can be expressed as
where
is not divisible by the square of any prime. Find
Problem 14
Suppose that is a complex number such that
and the imaginary part of
is nonnegative. Find the sum of the five smallest nonnegative integers
such that
is an integer.
Problem 15
Let and
(vertices labelled clockwise) be squares that intersect exactly once and with areas
and
respectively. There exists a constant
such that
where
is maximized. Find