NotThatSmart

We can write an equation. 

The measure of an interior angle is \( (n - 2)(180^\circ) / n \) where n is the sidelengths. Since, the exterior angle is 180  minues the interior angle, we have

\( (n - 2)(180) / n = 2 · ( 180 - (n - 2)(180) / n ) \\ (n - 2)(180) / n = 360 - 2 · (n - 2)(180) / n ) \\ 3 · (n - 2)(180) / n ) = 360 \\(n-2)(180)/n=120 \\(n-2)/n=2/3 \\n-2=2n/3 \\3n-6=2n \\n=6\)

So our answer is 6. 

We can check our work.

A 6-sided polygon has an interior of 120 and exterior of 60.

Thank! :)

Thanks! :)

There are actually more 2-digit numbers that are cozy than not. 

We can just list them out!

\(11,13,15,17,19 \\ 31,33,35,27,39 \\ 51,53,55,57,59 \\ 71,73,75,77,79 \\ 91,93,95,97,99\)

These numbers are those who are not cozy. Counting through them, there are 25. 

In total, there are 90 two-digit numbers, so we have

90 - 25 = 65. 

So 65 two-digit numbers are cozy. 

Thanks! :)

Although I don't really recommend it, we could just list out all the numbers since groups 5 and 4 are really not that big.

We have 

Group 4 - 26, 28, 30, 32, 34, 36

Group 5 - 38, 40, 42, 44, 46, 48, 50

Adding the numbers in group up, we get 308, which is our answer. 

We could also generate a polynomial equation to help us. 

We could put it in the form of a polynomial with 3rd degree in ax^3+bx^2+cx+d

From the problems above, we can get the system

\( a + b + c + d = 12\\ 8a + 4b + 2c + d = 44\\ 27a + 9b + 3c + d = 100\\ 64a + 16b + 4c + d = 186\\\)

Solving this, we get

\(a =1 \\ b= 6 \\ c= 7 \\ d= -2\)

This means we have a polynomial \(n^3 + 6n^2 + 7n - 2\). 

Plugging in 5, we get 308.

Thanks! :)

Well, notice that LCM[6, 3], is 6, so every factor of 6 between 0 and 100 is a factor. 

We have \(100/6=16.67\), rounding down means there are 16 factors. 

However, this is not including 0, so we add 1 to get 17. 

So both Sarah and Amos say 17 numbers out loud. 

Thanks! :)

This triangle is impossible to exist. 

\(144^\circ + 66^\circ = 210^\circ\), which is more than 180 degrees, what the triangle SHOULD add up to. 

Thanks! :)

Let

A =(0,0)

B =(6,0)

Construct a circle centered at A with a radius of 5   and a circle centered at  B with a radius of 7

If we let the x coordinate of D  = -1  we can  find the y coordinate as

(-1)^2 + y^2  = 25

y^2 = 24

y = 2sqrt (6)

Letting  the x coordinate of C = 11, we can  find the y coordinate as

( 11 - 6)^2 + y^2  = 49

5^2 + y^2   =49

y^2 = 24

y = 2sqrt (6)    ( which we would expect )

Since CP / PD = 1

Let  P  =  ([11 -1) /2, 2sqrt 6)  = (5, 2sqrt 6 )

The slope of the line through AD  = [ 2sqrt 6] / -1  =  -2sqrt 6

And the equation of this  line is  y = (-2sqrt 6)x

In standard form we have (2sqrt6)x + y  =  0        (1)

The slope of the line  through  BC  =  [ 2sqrt 6 - 0] / [ 11 - 6] =  (2/5)sqrt 6

And the equation of the line through BC  is  y  = (2/5)sqrt 6 ( x - 6) =

y =(2/5)sqrt (6)x - (12/5)sqrt 6

In standard from we have   (2/5)sqrt (6)x -y - (12/5)sqrt (6)  = 0     (2)

The  formula for the distance  between a point and a  line is  given  by

abs [ Ax + By + C ]  / sqrt [ A^2 + B^2 ]       where (x,y)  are the coordinates of the point

So   using (1) , the distance from P to the line through AD  = PX  is

abs [ (2sqrt (6) (5) + 2sqrt (6) ] /  sqrt [ (2sqrt (6))^2 + 1^2 ] =  [12sqrt (6)]  / 5

Likewise, the distance between P and the line through BC  = PY   is

abs [ (2/5)sqrt (6) (5) - 2sqrt (6) - (12/5)sqrt (6) ]  / sqrt  [ [(2/5)sqrt (6) ]^2 + 1^2 ]   =

[ (12/5)sqrt 6] / (7/5)  =  [12sqrt (6)]  / 7  

Thanks! :)

We have to note something important though. 

Let's look at the graph.

As you can see, the value gets smaller and smaller the bigger x gets, meaning there is no minimum value. 

Thanks! :)

\(floor (t) = -t^2 + 3t + 4 \)

We can graph to solve this, where the x value is t. 

The highlighted are solutions.

Thanks! :)

This problem is actually really really easy!

First off, let's look at the first few terms. \(f(x) = \left\lfloor\frac{2 - 3x}{3x + 1}\right\rfloor \)

f(1) = -1, f(2) = -1, f(3) = -1, f(x) = -1, even f(1000)=-1. 

Why is this? This is because the floor symbol means round down no matter what. Since \(2-3x<3x+1\) for \(x>1/6\). 

This means we will always have a fraction less than 0 and greater than -1,, meaning it always rounds down to -1. I hope this is a clear explenation. 

In the end, we just have \((-1)(1000)=-1000\)

This might be wrong, but I'm not sure. 

Thanks! :)