Today's Topics (June 2020)

debbie

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?

debbie

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?

debbie

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?
Po-Shen 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 Po-Shen 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?

debbie

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 ?

debbie

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 (non-prime) 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?

debbie

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 $153?$

debbie

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 $40$-frame-per-second 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?

RZ923

@debbie
—According to Wikipedia

debbie

@RZ923 Awesome, thank you for looking this up.

debbie

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? (Helicopter-like 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?

RZ923

@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)
$a^2\times b^2\times c^2=4\times 9\times 25=900$?

debbie

@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.

debbie

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 five-point 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?

divinedolphin

@debbie said in Today's Topics (June 2020):

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

haha

sqwishy

@divinedolphin haha

The Blade Dancer

Ooh can I go when you do infiltrate it

debbie

Topics from the 06/22/20 live stream

Find the number of three-digit numbers such that the digits are in strictly increasing order from left to right. For example, $123, 489,$ etc.
We have a suggested expression, $9 \times 9 \times 8,$ for the above question, which isn't actually correct. Let's try to find a question that would give $9 \times 9 \times 8$ as an answer!
What would happen if all the water in a rainstorm came down in one huge drop? (It was raining hard )
In the right triangle $\bigtriangleup ABC, AC = 12, BC = 5,$ and angle $\angle C$ is a right angle. A semicircle is inscribed in the triangle on side $AC$ at the right angle. What is the radius of the semicircle?
Why is it named the Pythagorean Theorem?
How many $0$'s does the base ten numeral $58!$ end with when written in base $12?$
Which award is considered to be like an Oscar for math?
How to find the volume of a banana
What is Avogadro's Number?
The area of a cuboid is $150 \text{ } \text{m}^2.$ What is the height of the cuboid if its volume is $1800 \text{ } \text{m}^3?$

debbie

Topics from the Tues 6/23/20 live stream

If $k$ and $n$ are integers and $ \left( 3^{2006} + 2006 \right)^2 - \left( 3^{2006} - 2006 \right)^2 = k \times 3^n,$ where $k$ is not divisible by $3,$ compute $\frac{n+k}{2006}.$
What is the value of $\frac{2020^2 - 2012^2}{2017^2 - 2015^2}?$
Do you consider yourself a mathematician, mathemagician, or math teacher/coach?
The positive integer $n$ is between $1$ and $20.$ Milly adds up all the integers from $1$ to $n$ inclusive. Billy adds up all the integers from $n+1$ to $20$ inclusive. Their totals are the same. What is the value of $n?$
Are eggs good for you?
On an algebra quiz, $10\%$ of the kids scored $70$ points, $35\%$ got $80$ points, $30\%$ got $90$ points and the rest scored $100\%.$ What is the difference between the mean and the median of the students' scores?

debbie

Topics from the Wed 6/24/20 live stream

The lengths of the sides of a triangle in inches are three consecutive integers. The length of the shortest side is $30\%$ of the perimeter. What is the length of the longest side?
The base-3 repeating decimal $0.\overline{12}_3$ is equivalent to what base ten common fraction?
How do you build a tank out of wood?
How many bananas do you need to make enough radiation for a nuclear missile?
How many bananas do you need in order to die of radiation?
Can you do percents with binary?
Did you guys know that $64$ is the smallest positive integer that is both a perfect square and perfect cube besides $1?$
There is a ten-digit number with the property: the top digit is divisible by $1,$ the $2$ top digits are divisible by $2,$ and so on, until the top-10 digits are divisible by $10.$ What is the number? (Variation: What if each digit was only used once?)
Let $ \bigtriangleup ABC $ be an equilateral triangle of length $3$ that has mirrors as sides. What is the shortest distance a light beam, starting from point $A,$ would have to travel to end at point $B?$ (From the 2015 Knights of Pi Math Tournament, run by the Newport Math Circle.)
If you go to the movies and you're paying, is it cheaper to take one friend to the movies twice, or two friends to the movies at the same time?
How do you push yourself to solve more problems? Of late, I've been acting a bit lazy, but I've realized that I can do more, but I'm unable to push myself to complete a problem in the stipulated time.

RZ923

The one with the ten digit number was my question I logged.
Sadly I couldn’t come today.