Today's Topics (June 2020)

Mon 6/1/20
inteGIRLS Puzzle Hunt:
We covered these puzzles:
"User's Handbook to Mathematics", "Contagion," and "Grocery Shopping" What math contests do you recommend for 7th graders?
 What is the maximum amount of equal pieces you can obtain by cutting a sphere 10 times?
 Should I grind AIME or MATHCOUNTS?
 What was the programming behind NOVID? Was it difficult at first?
 Do you have any advice on how to prepare for precalc, or do you have a video every day at 4?
Tue 6/2/20

What is the least number when divided by 10 that leaves a remainder of 9, which leaves a remainder of 8 when divided by 9 and continuing in this way leaves a remainder of 1 when divided by 2?

How can you add eight 8's to get the number 1,000? (Use only addition)

Can you give an example of a surface other than a plane that can be superimposed on itself in such a way that you can take any one point to another given point?

If \(n\) is any positive integer, what is the greatest positive integer that must be a factor of \( n(n+1)(n+2)(n+3)?\)

How many people do you need in a room in order to have more than \(\frac{1}{2}\) the chance two or more people have the same birthday?

If a hen and a half lay an egg and a half in a day and a half, how many eggs will half a dozen hens lay in half a dozen days?

Can you use math to cut your nails?
Wed 6/3/20
 What is the tens digit of \( 2015^{2016}  2017?\)
 What type of mathematician are you? Platonist, Logicist, Formalist, Constructivist, or QuasiEmpiricist?
 If it takes 3 volts for 1 wire, and it acts as a sine and the wire gives an output of 5 volts, how many wires do you need in order to save energy and keep the wires' output as high and consistent as possible?
 What is the math behind harmonies in music? (The thirds, sixths, and octaves.)
 Could there ever be a curved vector?
 How can I develop intuition for math problems and also for the logical world?
Thu 6/4/20
 What's your favorite type of math, and why?
 You have a school with 1,000 students and 1,000 lockers. The first student to go opens all of them, the second closes ever second one, the next one goes to every third and either opens or closes it, etc. How many are open at the end?
 You are given a 3by3 grid made out of dots. How do you connect the dots to fit two squares without the corners touching?
 How many pairs \((a,b)\) where \(a\) and \(b\) are positive integers, satisfy the equation \( a^2  b^2 = 105?\)
 Which two shapes are more similar, a ball, donut, or tetrahedron?
 The number 15 can be written as the sum of two consecutive integers, and as the sum of five consecutive integers. What is the positive difference between the products of the numbers in each set?
 If you start with 2 and count by 3's until you reach 449, you will get the sequence 2, 5, 8, 11, ..., 449. If 449 is the nth number, what is the value of N?
 Are video games a waste of time?
 How many different ways are there to deduce the equation of a circle?
 Alan lives on the evennumbered side of his street. If he totals the numbers of the houses on either side of him, the total is the same. Find the number(s) of Alan's house.
 What are you drinking?
 If there was only one person here, would Ask Math Anything still be held?
Friday, June 5

Why does order of operations have to be so complicated and not just "parentheses first?" (Plus Reverse Polish Notation, Infix, Postfix, and Prefix Notation)

Joke about the reliability of landline telephones compared with the internet: "The five nines of reliability" for landlines, the "five eights of reliability for the internet."

What is the smallest positive integer that is 7 times the sum of its digits?

A word problem: My twin lives at the reverse of my house number. The difference between our house numbers ends in 2. What are the lowest possible numbers of our houses?

Coronavirus question: What is the smallest radius of a circle that contains 11 people such that each pair of them is at least 1 meter apart? (This was way more complex than what Prof. Loh could cover!)

Simplify \( x^2 + 3x + \frac{5}{x} + 3\) as \(x + c + \frac{r}{x} + 3.\) (Division of polynomials!)

Do you watch Youtube?

Do you approve of Ted Ed?

Let \(n\) be a positive integer. What are all possible remainders when \(n^8\) is divided by \(7?\) (Finding remainders, mod notation, and a tiny bit of Fermat's Little Theorem!)

I still wonder why don't they use infinity nines of reliability.
It is 99.9 recurring % which is basically 100%. 
Sat 6/6/20 Emory Math Circle Collaboration
 Josephus problem
 Is it true that some people are just good at math and some people are just bad at math?
 Would this math be relevant to computer science?
 What's the difference between a doctor and a professor?
 What's your favorite flavor of ice cream?
 What is your favorite thing to study that isn't math?
 Are you funny?
 Is there anything faster than light?
 Do you like computer science or math more?
 How did you make your intro (with equals, square root, division, times, pi, greater than)
 Have you showcased your technology setup?

Sun 6/7/20
 Does NOVID interfere with the Apple privacy policy?
 Can this app pose a security risk to your data?
 Can you calculate the number of people who will die from being "protected" (physical/economic stress, inability to ge medical tests or procedures done, etc., vs. the small number who might die from COVID19?
 Will the graph obtained always be a connected graph?
 How did you grow your Youtube channel so well?
 Has your life changed due to this virus?
 What's the environment at CMU like?
 How to relax and do well, and how important are grades?
 Using APs to place out of college classes
 Professor, what do you do for selfdevelopment? Who do you look up to or go to or read
 Is there a math curriculum you recommend for children?
 Aren't math curriculi from Eastern Europe superior to the ones in North America?
 Are there any experiences you loved from MOP?

Mon 6/8/20
 If \(p\) is a prime larger than \(3,\) show that \(2p + 1\) and \(4p + 1\) cannot both be primes simultaneously
 How to do well in math competitions
 If you dig a deep hole from the North Pole, will you reach the South Pole?
 How many sides does a circle have?
 Find the minimum positive value of \(1  2  3  4  5  6  7  8  9 \) where each operation "" is either a plus or a minus (Greedy Algorithm)
 An angle is \(45^{\circ}.\) What will it look like under a microscope that magnifies everything 10 times its size before?
 Can anyone prove that this is wrong? \( \frac{x+x}{2} = \sqrt{x \times x}\) if \(x\) is positive
 Why we don't use Roman Numerals (arithmetic is a pain)
 NOVID is making audible noises; is that supposed to happen?
 Can any two decimals multiply to make a whole number?
 What's the name of the physics book that your mom bought for you when you were a child? (The Cartoon Guide to Physics, by Larry Gonick)
 If \( (a + c)\times b = 24\) and \(b\) is an even number, \(\frac{b}{a} = 0.8,\) and all of them are positive integers, what are the values of \(a, b,\) and \(c?\)
 How many times would everybody in the world have to spam "e" in this chat in order to disable Youtube?
 \( a^2 b= 30, b^2 c=40, c^2 a = 45.\) What are the values of \(a, b,\) and \(c?\)

Tue 6/9/20
 How do I solve two linear equations with 2 variables each if all the decimals are linear, faster than just doing substitution and elimination followed by checking?
 Do you think that algebra is the most useful area of math? Why or why not?
 Parabolic mirrors and cooking eggs, using parabolic mirrors to burn enemy ships (Archimedes story)
 The curved building in London that melted a Jaguar car  The "Walkie Talkie"
 If \(20\) diagonals are drawn in a regular octagon, at how many distinct points in the interior of the octagon (not on the boundary) do two or more diagonals intersect?
 If you had to give up one forever, which one would you give up on, math or programming? Why?
 Please find out the units digit of \( 1^{2^{3}} \times 2^{3^{4}} \times \ldots \times 98^{99^{100}},\) can anyone solve it?
 Do you know Richard from AoPS? The owner of AoPS?
 Can a solution to a math problem be copyrighted? The difference between copyrighting and patenting
 Are imaginary numbers real?
 What is tough in math for you, and how do you learn it?
 Can you talk about your Ted Talk?

Wed 6/10/20
 How many triangles are there in a pentagram? (Prof. Loh drew a pentagram inside a pentagon, and proceeded to solve that harder version instead )
 Can we write a program to obtain the answer, professor?
 Are you hosting MOP this year?
 Which platform or programming language can be used to develop such a program?
 I drew a picture of you on Saturday; did you receive it?
 The arithmetic mean of Josh's four test scores is 80. If three of Josh's test scores have a mean of 87, what is Josh's score for the fourth test? Express your answer to the nearest whole number.
 What is a good age to start learning programming?
 There's a broken calculator. When you put in 86 you get 44; when you put in 14 you get 5; when you put in 53 you get 18. What do you get when you put in 49?
 Why do we say that the triangle is the strongest shape?
 Can I have a free spot in your CMU class?
 What is the minimum possible product of three different numbers of the set { 8, 6, 4, 0, 3, 5, 7}, A, 336. B, 280. C, 210. D, 192. E, 0.
 Why did you get involved with MATHCOUNTS?
 How to take preparation for competitive math when you get overwhelmed with all the resources and topics?
 How many plus signs should we put between the digits of 987,654,321 to get a total of 99, and where?
 Do computers have emotions?
 Are problems better or lessons better problems better or lessons better problems better or lessons better
 Have you ever been in a debate tournament?
 How long should you study math per day?

Thu 6/11/20
 What is \( 59303451 \times 99999999 \) ? (You cannot use a calculator!)
 The \(n^{\text{th}}\) number of the Fibonacci sequence if the square of \(n.\) Find the value of \(n.\) Also, an approximation for Fibonacci numbers as \( e^{\frac{n}{2}}\) rounded up. (That's because \(e\) is close to the golden ratio, but it will fail at some point.)
 How did you meet your wife?
 Do you like her?
 PoShen Loh said that this livestream will finish in 10 minutes, but someone asks him a Lexington Math Tournament question. If \(n\) is how much overtime PoShen Loh is, what are the chances that \(n\) is 0?
 Who is your technical team?
 Consider the sequence \( 2, 3, 5, 6, 7, 8, 10, 11, \ldots\) of all positive integers that are not perfect squares. Can I find the \(2020^{\text{th}}\) term of the sequence?
 When will you retire from CMU (Carnegie Mellon University)?
 Find the sum of all positive integers \(x\) such that \( 3 < \sqrt{x} < 3.6 \)
 If 3 cats can catch 3 bunnies in 3 minutes, how long will it take 100 cats to catch 100 bunnies?
 If \( \frac{1}{a}  \frac{1}{b} = \frac{1}{a + b},\) what is \( \frac{a}{b} + \frac{b}{a} ?\)
 What are the next three letters in this combination? OTTFFSS
 How can you stay calm under pressure and do mathematics?

Fri 6/12/20 Live stream topics
 The long diagonal of a rhombus is three times the length of the other diagonal. The perimeter of the rhombus is 40 cm. What's its area?
 How many rotations does a little circle make when it rolls around a big circle?
 Do you eat bruised bananas?
 Will we have to call Evan Chen Dr. when he gets his Ph.D?
 How do computer crops work? (Don't forget the circle and special shape crops!)
 What can you hold in your right hand but never in your left hand?
 Is there a thing called the 2.5th root?
 Without a calculator, what are the first three digits of \( 123123123 \times 333333333 \) ?
 Have you ever played Turing Tumble?
 Why are the solutions of a quadratic its \(y\)intercept?
 Your parents have six sons including you, and each son has one sister. How many people are in the family?
 How old is Po?
 Please explain what are directed angles
 What do you do on the weekends?
 Solve \( \frac{x}{a} + \frac{a}{x} = b\) for \(x.\)
 How can I 1.) manage my time more efficiently; 2.) get distracted less; 3.) study efficiently; 4.) go the extra mile and do what's right even when it's difficult ?

Mon 6/15/20
 How many integers \(n\) between \(1\) and \(100\) have the property that \(n^n\) is a square number?
 Prove that in the sequence \(1, 31, 331, 3331, 33331, \ldots \) there are infinitely many composite (nonprime) numbers.
 What is the smallest prime number whose digits sum to \(19?\)
 When \(9!\) is expressed as an integer in base \(9,\) the result ends in \(m\) zeroes, and the last nonzzero digit immediately preceding the zeroes is \(n.\) What is the value of the ordered pair \((m,n)?\)
 With the numbers \(123456789,\) make them add up to \(100.\) They must stay in the same order. You can use addition, subtraction, multiplication, and division. Remember, they have to stay in the same order.
 Bob the cat is eating mice. He has a \(\frac{2}{3}\) chance of catching a mouse when he pounces. How many times does he have to pounce to get a greater than \(\frac{1}{2}\) chance of catching \(10\) mice? (Application to binomial distribution.)
 Tips for productivity in the summer
 The banana produce industry and why there is only one type of banana
 Where can I find info about theorems that can be useful?
 What is the graphical difference between convex and strictly convex functions?

Tue 6/16/20
 Show that all the terms in the sequence \( 10001, 100010001, 1000100010001, \ldots \) are composite.
 A quadrilateral is inscribed in a circle of radius \(200 \sqrt{2}\). Three of the sides of this quadrilateral have length \(200.\) What is the length of the fourth side?
 I couldn't control my other broken calculator, so I bought a new one. I typed too many numbers in it, and it broke. \(36\) gives \(105, 82\) gives \(486, \) and \(29 \) gives \(196.\) What number did I put in if the output is \(153?\)

Topics Wed 6/17/20
 \(111111111^2\) ; also, the version with eleven \(1'\)s
 Roman Numerals
 What's the name of a \(360\)sided shape? (A triacosiahexecontagon, apparently, according to our audience!)
 "How old is your son?" asked a man to his neighbor. "My son is five times as old as my daughter, and my wife is five times as old as my son. I am twice as old as my wife, whereas my grandmother, who is celebrating her \(81\)st birthday, is as old as all of us put together." How old is the man's son?
 My eye sees things at \(30\) frames per second, so why is my \(40\)framepersecond video game laggy?
 What frame rate is our life going at?
 What frame rate is this video?
 How did you get the idea to start this livestream?
 "I am about \(79.2\)% sure that Prof. Loh won't see this.
 Today is my birthday, and I have a problem about it: If there are \(200\) people watching this live stream, what is the chance that at least \(2\) of them have the same birthday?

@debbie
—According to Wikipedia 
@RZ923 Awesome, thank you for looking this up.

Topics from the Friday 6/19/20 live stream
 Please explain the metallic ratios (gold, silver, etc.)
 Is the myth true that the golden ratio appears in the Mona Lisa?
 Didn't Da Vinci invent the Ornithopeter or whatever it's called? (Helicopterlike flying machine.)
 One fourth of the population of a newly discovered planet have 4 legs. The rest have two legs. There are 60 legs total. How big is the population of the newly discovered planet?
 If \( a + b + c = 10, a \times b \times c = 30,\) what is the value of \(a^2 \times b^2 \times c^2?\) (Many possible solutions.)
 Prove that if you square root any positive number (it doesn't have to be an integer, and it can't be 0,) an infinite number of times, you will get 1.
 There is a toilet roll whose radius is \(4\) and the "empty part" of the roll has a radius of \(1.\) What is the length of the paper if we unroll the toilet paper? (Audience contributed helpful numbers: 375 sheets per roll in Singapore, 276 layers in one roll (counted by hand), 0.0004 inches per sheet)
 How many bananas do you eat every day? (Answer: 1)
 Do you use bootstrap for your websites?
 How do you build a thermonuclear missile with a launch site using only metal and 10 pounds of plutonium? I need to know by Friday because that's when my geometry is due.
 Why do you love math?

@debbie
I was sorry I couldn’t attend today’s livestream.
For the question where:
\(a+b+c=10\)
\(a\times b\times c=30\)
\(a^2\times b^2\times c^2=?\)
I don’t get why you put “unsolvable”.
Shouldn’t it be:
\(a=2,b=3, c=5\)(or some other way around)
\(a^2\times b^2\times c^2=4\times 9\times 25=900\)? 
@RZ923 Thanks for your comment! I actually meant that there are many solutions. The set of equations doesn't uniquely determine a single solution. So "unsolvable" isn't exactly the right word. For example, another solution is \(a = \sqrt{3}i, b = \sqrt{3}i, c = 10.\) Then we would have
$$ a + b + c = \sqrt{3}i  \sqrt{3}i + 10 = \boxed{10} $$
$$ a \times b \times c = \sqrt{3}i \times \sqrt{3}i \times 10 = 3 \times 10 = \boxed{30} $$Thanks for pointing this out.

Newport Math Club Knights of Pi Math Tournament Special Livestream Saturday 6/20/20
Questions from Newport Math Club:

Find the sum of the digits of the sum \( 9 + 99 + 999 + 9999 + … + 999999999 \) (nine 9’s)

Marisa draws a fivepoint star by extending the sides of a convex pentagon with all angles less than 120°. What is the sum of the angles of the points of her star, in degrees?

How many solutions does the equation \(2^x = x^2 \) have for some real value of \(x?\)

If everyone is connected to all the people they’ve ever greeted, how many degrees of separation is the world away?

If we met an alien civilization and exchanged mathematics, what maths do you think would be similar, and what would be different? Would they have things like real numbers?

What does convex mean?

Compare CMU and Caltech

Can you describe your stack from Caltech's Ditch Day?

Would you rather win the Nobel Prize or "invent" math?

How can you predict when a popsicle will melt?

How does base \(\frac{1}{37}\) work?

Would you rather have cake, ice cream, or cupcakes for the rest of your life?

Have you ever participated in an Improv Troupe? If so, how was your experience?

If a jet goes 90 mph, and a monkey goes 5 mph, and if the monkey drops from a tree it goes 4 miles per hour. How fast does a banana go if the monkey jumps for 5 hours?

What would happen if someone used a sonic boom on NOVID? (Relates to this BBC documentary about pistol shrimps)

How do I make a nuclear reactor?

Can you give some advice on trading? (And some mention of how oil prices went negative in April...)

How do I infiltrate the ISS (International Space Station)?

Metric or Imperial system for measurements? (foot, inch, barley grain = smallest unit of weight in the troy and avoirdupois systems, equal to \(\frac{1}{5760}\) of a pound troy, and \(\frac{1}{7000}\) of a pound avoirdupois (approximately 0.0648 grams)

Besides math and programming, do you have other hobbies that you do in your free time?

Can someone hack NOVID?


@debbie said in Today's Topics (June 2020):
 How do I infiltrate the ISS (International Space Station)?
haha

@divinedolphin haha