Today's Topics (June 2020)

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.

debbie

@RZ923 Yes, I replied to your other post in "Early Bird Question"! Sorry that you weren't there, but at least you can watch the replay!

debbie

Topics from the Wed 6/24/20

Three points are chosen randomly and independently on a circle. What is the probability that all three pairwise distances between the points are less than the radius of the circle?
I am an odd number. Take away a letter and I become even. What number am I?
I add $5$ to $9$ to get to $2.$ This is correct, so how is this possible? (Answer: On a clock.)
Is there any math in games such as Minecraft?
Find three positive rational numbers $a, b,$ and $c$ such that $a + b + c = a \times b \times c = 7.$
What three numbers, none of which is zero, give the same result no matter whether they're added or multiplied?
At what points on Earth can you travel in a perfectly straight line and go due east at the same time?
Can you arrange four $9$'s to make it equal to $100?$

debbie

Topics from the Friday 6/26/20 live stream

If $f(0.25) = 0.5 $ and $f(0.5) = 0.5,$ how can we find $f(x)?$ (Lagrange Interpolation)
Po the cat is catching fish! He has a $\frac{1}{3}$ chance of catching a salmon, $\frac{1}{8}$ change of catching a trout, and $\frac{1}{10}$ chance of catching a hallibut. How much chance does he have of catching a shrimp?
What is $x + x - x^{-2}$ if $x = 5?$
How many people do we need to watch the stream for Ask Math Anything to continue?
If $a + b = \frac{a}{b} + \frac{b}{a},$ find $a^2 + b^2.$
What fractions are equivalent to $ \frac{1}{2}?$ Can you list all of them?
Suppose you have $3$ variables; is it possible to have only $2$ equations with a definite answer for all variables?
What is a real number?
How is it possible to have an irrational measurement?
We know that in a right-angled triangle, $ \text{hypotenuse}^2 = \text{base}^2 + \text{perpendicular}^2.$ What will be the relation if we cube them?

debbie

Topics from the Saturday 6/27/20 live stream

What professions are fit for math majors?
Are people who are blind better at hearing?
Do you recommend looking at the MATHCOUNTS School Handbook every day?
Can we talk or yell something in ultrasound?
What types of things should we memorize?
What do you teach at CMU?
What is an undergraduate student?
Who invented your theme song?
Do you like China's education system, with lots of homework, memorization, and competition?

debbie

Topics from the Monday 6/29/20 live stream

What is the easiest way to solve this question? "A printer users 300 digits to number the pages of a book. How many pages are in the book? (The book starts at Page 1.)
The product of two prime numbers is 9 more than their sum. What is their absolute difference?
What is the difference between $ \pi $ and $ \phi?$ (Relates to the golden ratio!)
In a sequence the terms are $2, 6, 12, 72, \ldots$ where, starting with the 3rd term, every term is the product of the last two terms. If the $9^{\text{th}}$ term is $ 2^a * 3^b,$ what is $a + b?$
What's the difference between the Po-Shen Loh ratio and the Daily Challenge ratio?
If you had to eat one of these, which one would you eat? Crispy Apple, Savory Popcorn, or Sweet Energy Bar?
How can you make the following equation true by drawing only one straight line? $ 5 + 5 + 5 = 550$
If a truck goes at a speed of $2^x $ mph on the $x$th minute, then how many miles does it go in $1$ hour?
Use eight $8$'s and the operations of addition, subtraction, multiplication, division, exponentiation, factorials and square roots to make $1000.$
What is the max distance between two watchers of today's AMA (Ask Math Anything)?

debbie

Topics from the Tuesday 6/30/20 live stream

The graph of $y = 3x^2 - 2x + k$ has an x-intercept of $2.$ What is the value of $k?$
We wish to make a 3-Senator committee in which no two members are from the same state. How many ways can we choose three states to be represented?
A duck was given $\$9,$ a spider was given $\$36,$ a bee was given $\$27.$ Based on this information, how much money would be given to a cat?
How to lay out $51$ stars on the U.S. flag
How does the vector dot product relate to the area of a paralellogram?
What are the chances that your water will taste like bananas?
How can you find all the dogs and cats in America using math?
Move one number in $123 + 252 = 748$ in order to get a true math statement.
Do you agree that mosquitoes should be wiped out of existence?
Where have the old Daily Challenge videos gone?
There's an $8$-digit number that is divisible by $1, 2, 3, 4, 5, 6, 7,$ and $8.$ The number is a perfect number and all the digits are different. What is this number?
What if it literally rained cats and dogs?
Let $P(x)$ be a polynomial with nonnegative integer coefficients. If $P(1)=5$ and $P(4) = 83,$ then what is $P(3)?$
I am bad at math; what is the point of listening if I can't understand?
If $66=2, 999=3, 8=2, 0=1, 9696=4, 8123=2, 98=3,$ and $88=4,$ what does $816982$ equal?
My friends learned python already and I don't want to be the odd one out so I'm learning Python on the internet by myself. I don't go to any courses, so it's kind of hard. Any advice?

RZ923

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

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?)

This problem is mine.
I am watching the recordings now
37:06: “This is from Vera Weng...”
I didn’t even attend the livestream lol
(Also another huge thanks to @debbie for logging it in!)
(I was meant to say: all digits are unique, but I forgot to say this while logging it in)
PS I posted a comment below the livestream as Vera Weng.

debbie

@RZ923 Yes, I definitely know who you are by now. Same for many others Even still, I would like to protect people's privacy by not providing their usernames. It is so great to see you as well as many other familiar faces at the live streams!