Problem by Raymond Tao. Difficulty: Medium

A research laboratory is working on a cure to COVID-19 through analysis of the coronavirus’s genome. As the scientists work around the clock, they fail to notice that a group of monkeys has broken free of their holding cells. The monkeys find their way to a researcher’s workstation, where they manage to turn on the computer and start mashing keys on the keyboard at random. Which is more likely to happen first: that they produce a 100% accurate genomic sequencing of SARS-CoV-2 (an RNA virus with 30,000 bases, each one of G, U, A, or C), or that they produce a 100% accurate, 10,000 character technical report on the cure using English letters with correct capitalization and digits, but not punctuation? (Hint: a calculator might be helpful - or it might not!)

Problem by Adithya Kalyanam. Difficulty: Medium

You are a computational epidemiologist for an actuarial firm working with the US government*. Your team has been tasked with designing a statistical model to predict the spread of SARS-CoV-2. In the latest virus transmission simulation iteration, there is a group of 10 people and 5 rooms. Out of these 10 people, 2 are infected with SARS-CoV-2, but have not begun showing symptoms, so it is impossible to say who’s who. The 10 people are randomly divided into 2 groups of 5 people each. Each member of the first group enters a different one of the rooms, possibly introducing the virus into the air. Then, each member of the second group enters a different one of the rooms, possibly exposing themselves to the virus. What is the probability that a total of 3 individuals will be infected with SARS-CoV-2 at the end of the simulation?

* This problem not endorsed by the US government.

Problem by Raymond Tao. Difficulty: Medium

Your architectural firm has been commissioned to design meeting rooms for a business. Each room should fit 6 employees. Market research indicates that equilateral triangle-shaped rooms are the latest and greatest. It is imperative that employees can follow all CDC-recommended* coronavirus guidelines, which includes social distancing of 6 feet. The local fire marshal reminds you that occupants should not be forced to within 3 feet of the room's walls.

Find the minimum amount of flooring for the room in square feet. Remember: it is your moral duty to conserve resources during times of national crisis!

* This problem is not endorsed by the US government.

Problem by Adithya Kalyanam. Difficulty: Easy

Ron is the leader of a convention for the popular video game “Stick Fighter”. Unfortunately, this year, concerns of COVID-19 forced Ron to move the convention online, using Yoom*. Participants had the option of joining the red group or the blue group. 100 people joined the virtual convention, and all of them chose their corresponding group correctly. Tragically, during the conference, Ron was forced to disconnect and rehost the conference three times, as he suffers from bad WiFi. Additionally, many of the people at the convention did not know how to use Yoom properly, so every time Ron rehosted, exactly 50 percent of the red group switched to the blue group and exactly 25 percent of the blue group switched to the red group. At the end of the convention, Ron forgot how many people were initially in the red group and initially in the blue group. All he remembers is that the majority of people were initially in the red group.

Using the information provided, can you help Ron figure out how many people were initially in the red group? (Hint: despite the theme of the convention, Ron’s group advocates non-violence. Thus, there cannot be a fractional number of people.)

*All names, characters, organizations, places, events, and incidents are either products of the author's imagination or are used fictitiously.

Elmer draws a square with side length of two inches. He decides to fix one point on a corner A, and move a second point P by intervals of 0.1 inches on side AB. This means that the distance from A to P can be 0.1 inches, 0.2 inches, … up to 2 inches. Elmer then draws a third point D on one of the other three sides, so that ADP creates a triangle. What is the probability that the triangle ADP is obtuse, if point P is randomly selected on AB and point D is randomly fixed on one of the other three sides?

Square

Note: Because this is a multi-part problem, when you are submitting your answer, the answer title will be Tao's Trifecta, Part (1, 2, or 3).

Raymond likes to make Tao symbols. His Tao symbol consists of equal amounts of shaded (black) and unshaded (white) parts. If a line is drawn straight down the Tao symbol, the arc from A to B is a semicircle with its center at the center of the small black circle. B is the center of the largest circle. Similarly, the arc from B to C is also a semicircle at the center of the small white circle.

Tao Diagram

1) Raymond wants to draw a Tao symbol, where the distance between the center of the small white circle and the center of the small black circle is 6 inches. How much black paint will Raymond need if 1 liter of paint covers π square inches? Express your answer in liters.

Two Tao Diagram

2) Raymond takes a second Tao symbol (with the same dimensions as described in problem 1) and flips it over as shown above. He then places this second Tao symbol on top of the first one, and adds up the area of the new symbol that is black. What is this area? Express your answer in square inches.

Red Marker Tao

3) Raymond draws another symbol identical to the first one, where the distance between the centers of the small circles is 6 inches. He then uses a red marker to completely trace the border of black part of the Tao symbol, as shown above. If the small circles have a radius of one inch, what is the total length (in inches) of the border that was traced by the red marker?

Problem by Adithya Kalyanam. Reviewed by Raymond Tao. Difficulty: Medium

The esteemed scientist Dr. Mike Mence has created the new Viruskiller 9000 to kill viruses within solid items. It shines a cone-shaped virucidal beam that can penetrate and kill viruses 2 inches below any surface it touches. While underneath the surface, the beam continues to move in the same conical shape as above the surface. To test his device, Dr. Mence shines the light 4 inches above the center of one side of a cylindrical object. The cylindrical object has radius 3 inches and height 6 inches. Dr. Mence measures the beam on the surface of the cylinder and finds that it has radius 2 inches. If the virus has an equal chance of being anywhere within the cylinder, what is the probability that Dr. Mence will kill the virus?

You may wish to consult the patent diagram for the Viruskiller 9000.

Problem by Raymond Tao. Difficulty: Medium

The New Amsterdam Stock Exchange is located among the perfectly parallel streets of Namhattan, which are all one-tenth of a mile apart. At 12:00 PM, a thief stole a Gloomberg terminal from the New Amsterdam Stock Exchange. New Amsterdam Police Department officers are swarming the scene, but due to Mayor Gloomberg’s incompetent policies, the NAPD must stop and physically search individuals to determine if they are in possession of the Gloomberg terminal. It is now 2:00 PM, and the NAPD has searched all streets and buildings within four blocks of the stock exchange. Meanwhile, the thief, burdened by the weight of the stolen goods, takes 15 minutes to travel one city block. Since all Namhattan buildings are connected, if a person enters a building, they could be anywhere within that block of buildings. (In other words, if a person reaches an intersection, they could immediately enter any of the four blocks that forms a corner of that intersection.)

How many square miles are covered by the thief’s possible location?

Problem by Adithya Kalyanam. Difficulty: Medium

Raymond is at the famous Baseton Market, where there are three different shops that sell different items. The first shop sells different items at every integer price, but in base 3 (i.e., items cost $1, $2, $10, $11, $12, $20, and so on). The second shop also has different items at every integer price, but in base 6. The third shop has different items at every integer price, but in base 7. Raymond signed a legally binding agreement that he will buy something at every shop and that he will start at the base 3 shop, go on to the base 6 shop, and end at the base 7 shop. If Raymond has $20 (in base 10) with him, in how many different ways can Raymond buy items at the market?

Problem by Adithya Kalyanam. Difficulty: Easy

Solved by Krishna Mukunda on 2/28/2020! Congratulations, Krishna!

The famous chemist Bartholomew Montesque is conducting experiments with some magic beans.

Every day, one magic bean splits into two magic beans, each of which splits again into two magic beans the next day. Montesque starts with one magic bean, allowing it to grow freely. After three days, he takes one bean and injects it with Montesque’s Madness growth serum. The serum causes this bean to split into x beans after a day, each of which also split into x beans the next day, and so on.

Montesque goes on vacation for a brief respite from the bean-sanity. When he returns to his lab, exactly ten days have passed since he started with the first magic bean. He painstakingly counts every bean by hand and finds 280,832 beans in his lab. Can you help Montesque find the value of x?

Problem by Adithya Kalyanam. Difficulty: Medium

This problem remains unsolved!

Garry Potter is underneath Gogwarts School of Witchcraft and Wizardry, searching for the hidden Warlock’s Stone. After descending into the depths of the unescapable labyrinth, he comes across a large door with the seal of the Warlock’s Stone on it. The seal consists of a circle, with a right isosceles triangle inscribed in the top half, which has a circle inscribed in it, which has a right isosceles triangle inscribed in it, and so on until there is an infinite amount of circles and triangles. As a voice asks for the magic number, Garry pulls out his wand and tries to decipher the number. However, Garry is not aware of the fact that his wand has been rigged by the evil Goldemort! While Garry is using the wand, it bursts into flames! Garry is able to protect himself from the flames, but he can no longer use his wand to find the magic number. Fortunately, Garry’s wand has given him two valuable pieces of information. Firstly, it has told him that the radius of the largest circle of the seal is 4 inches. Secondly, it has told him that the magic number is equal to the distance, in inches, between the center of the largest circle and the center of the smallest circle. Can you find the magic number to prevent Garry Potter from being trapped in the labyrinth for all of eternity?

Garry Potter casts "Drawus Diagramus" and gets this result.

Note: All triangles are isosceles right triangles.

Helpful formula:

Allow S to be the sum of the series, a to be the first term, and r to be the common ratio

For any geometric series,

S = a ÷ (1 – r)

Math by Adithya Kalyanam. Story by Raymond Tao. Difficulty: Medium

Solved by Mourad Zeynalov on 2/26/2020! Congratulations, Mourad!

Egbert works as a corporate spy for Taco Bell. Today, he is conducting an undercover investigation at a local Chipotle. His superiors need him to determine the size of a Chipotle burrito to prove that Taco Bell provides more bang for the consumer's buck. Unfortunately, Egbert has forgotten his handy burrito ruler. All he knows is that Chipotle burritos are 4 inches in diameter, according to their advertisements, and that Chipotle employees pride themselves on making perfectly cylindrical burritos.

As Egbert sits there, pondering his dilemma, a mosquito (named Hildebrandt) lands on the edge of his burrito on one end. Egbert picks the burrito up, about to shoo Hildebrandt away, when the little critter begins walking along the surface of the burrito at a 45 degree angle. To Egbert's delight, Hildebrandt makes exactly 3 complete revolutions around the burrito before reaching opposite end of the burrito. Armed with enough info to calculate the volume of the burrito, Egbert excitedly pulls out his phone to call his superior, when Chipotle's internal counterespionage division appears out of nowhere and arrests him.

Can you solve the mystery of the Chipotle burrito size to appease Taco Bell's shareholders and line the pockets of the company's executives? (Marketing requests that you express your answer in cubic inches.)

You may find it helpful to view a picture of the burrito Egbert captured before his untimely arrest.

Problem by Adithya Kalyanam. Difficulty: Hard

Solved by Krishna Mukunda on 2/25/2020! Congratulations, Krishna!

Jacob has a Minecraft house with five rooms labeled A, B, C, D, and E. Jacob enters room A, leaves, and enters room B. He continues this process alphabetically until he exits room E. However, an enderman is in one of the rooms, and whenever Jacob exits a room, the enderman teleports to a new room. The enderman will not teleport to a room that it has already been to, and has an equal chance of starting in any room. What is the probability that Jacob will go through the five rooms without encountering the enderman?