Pangolin14

Look for patterns!

$$99 \cdot 99=9801$$

$$ 999 \cdot 999 = 998001$$

When looking at the patterns, we see that the formula for the square a number consisting of $n$ $9's$ is:

Number of 9s in the number = Add $n-1$ 9s

Then add an 8

Number of 0s in the number = Add $n-1$ 0s

Then add a 1.

There are 16 digits in the number $999999999999999999.$ So, $$\boxed{999999999999999999^2 = 999999999999999998000000000000000001}$$

a) PEMDAS gives 23

b) 15/3 = 5

c) 3 by PEMDAS

Let the sides of the triangle be $a$, $b$, and $c.$ 

Then by the triangle inequality, 

$$a+b>c$$ $$b+c>a$$ $$c+a>b$$

We also have that $a+b+c=15.$

The longest side of this triangle must be shorter than $15/2=7.5$. However, we have to round this up to get an integer, so the longest side must be shorter than $8.$ This is because the remaning sides must sum to less than $8,$ or $7,$ and we won't have a triangle! 

Now, to focus on how short the longest side might be, we know that if $a=b=c$ then all of the sides must be the longest side. This is possible if we have side lengths of $5, 5, 5.$ However, if the longest side is $4,$ the remaining sides must sum to $15-4=11.$ We must have more than a 4 for the longest side, as $4+4+4 = 12.$ Let the longest side be l. So, $5\le l \le 7.$

Listing out the possibilities with 5 as the longest side, $$(5,5,5).$$ With 6, $$(6,3,6)~(6,4,5)$$ And with 7, $$(7,1,7) ~ (7,2,6) ~ (7,3,5) ~ (7,4,4)$$  we have a grand total of $$\boxed{7}$$ possibilities for non-congruent triangles.

I'm not a perfect translator, but I did my best at asking my various teachers for words.

How do you find the speed of an object if the mass of the object is 4352ton 34cen 423456kg 263g (I don't know what that is), and the acceleration of the object is 28745km 342m 5432cm per second, and the area of the object is 3km by 343m (the object is flat)? 3485. Google passed in 2552st century. 1000th anniversary, 6456 minutes divided by 1.5 and 2534662346345sec? Yet, you need to multiply all this by 834 and divide by 1.2!

This doesn't make sense! Sorry, I can't answer it!

I wiil do this without a calculator:

$4456 \cdot 4455 = 4455^2+4455 = 4450\cdot4460 + 25 + 4455.$

$4450\cdot4460  = 100\cdot(445\cdot446)  = 10\cdot((445^2)+445) = 100*((100*(44\cdot45) + 25) + 445)$

Yay, $44 \cdot 45$ is easy! Everyone knows it's $1980!$ So, $1980\cdot100=198000,$ and plus $25$ yields $198025.$ Next, we add $445$ getting $198470.$ Then times $100$ yields $19847000!$ Now, we can add this to $25$ and $4455,$ easy addition gets $19851480!$ That's your answer, WITHOUT A CALC!

Probability of drawing a card from spades: 13/52=1/4

1/4^4=1/256

1/256

Be careful guest, 5<6!

d is $\sqrt{\frac{293}{3}}$

Firstly, we factor this equation as $a(x + 6) = -x^2.$  Because $-x^2 \ne (-x)^2,$ we have $a(x+6)=-1(x^2),$ so $a(-x-6)=x^2.$ 

There are 18 integer solutions. Only 1 is positive for a, and two are non negative.