thedudemanguyperson

We claim that it's $\boxed{x=1}$.

Notice that $f(1)=f(1)$, and $f(1+k)=f(1-k)$, implying the answer.

The minimum value of $x^2$ is $0$, the maximum is $81$. The answer is just $\boxed{81}$.

I think you wrote $x=2x+5$? 

If so, subtract $2x$ from both sides to get $-x=5$. From here, $\boxed{x=-5}$.

Yep, sorry about that.

$19:8^2=64$

$p:3^2=9$

We are multiplying by $\frac{9}{64}$, and the inverse is $\frac{64}{9}$.

So $p=\boxed{\frac{1216}{9}}$.

Let's have him get his first $4$ pairs from white, red, blue, rainbow. Then we have to pick as many socks as possible without getting another pair. I will take a green sock, a purple sock, and another white, red, blue, and rainbow sock. Now, the next sock I pick, I am guaranteed to get another pair. So far, I have picked $4*2+6=14$ socks. The answer is $14+1=\boxed{15}$ by the pigeonhole principle. 

I like $y=mx+b$ form better, so I'll use that. We have:

$11=m+b, 5=-4m+b$. Subtracting, $5m=6\implies m=\frac{6}{5}$

And $b=11-m=\frac{49}{5}$.

So $y=\frac{6}{5}x+\frac{49}{5}\implies -\frac{6}{5}x+y=\frac{49}{5}\implies -6x+5y=49\implies -\frac{6}{49}x+\frac{5}{49}y=1$. So $a=\boxed{-\frac{49}{6}}$. 

There is no diagram, so I am assuming $E\in ABCD$. 

We have $\angle PAE=60-\angle BAC=60-45=15$

And $\angle APE=180-60-15=180-75=105$.

By Vertical angles, $\angle APE=\angle BPC=\boxed{105^{\circ}}$.

Putting $x=-5$, $50-5p-15=0\implies 5p=35\implies p=7$. 

So $7(x^2+x)+k=0$ has equal roots. Let's expand: $7x^2+7x+k=0$ has equal roots. The discriminant is, therefore, $0$, so we have $7^2-4*7*k=0 \implies 49=28k\implies k=\boxed{\frac{7}{4}}$.