AnExtremelyLongName

Yes, I know this is a problem. I too, am a user on AoPS and I have seen so much cheating on this site. (I am not enrolled in classes though). The problem is, we can't really recongize which problems are from AoPS so we kind of need help from users like you. 

One question per post please.

1. This is mostly geometry believe it or not.

The value of K in the circle equation would be the blue line squared. Or in other words, the blue line is \(\sqrt{k}\)

The value of BC in the tangent equation would just be \(k\)

We know ABC is a 45 - 45 - 90 triangle (If the coefficients of x and y are 1 when graphed in standard form for the red line). The blue line is the perpendicular bisector. 

Based on pythagorean theorem: \(2 *\sqrt{k}^2 = k^2\) (two legs are \(\sqrt{k}\) and hypotenuse is BC)

\(2k=k^2\)

\(k(k-2)=0\)

\(\boxed{k=2}\)

1. Connect \(RG\)

2. Since \(FR=FG\), triangle \(FRG\) is isosceles with \(F\) as the vertex.

3. Angle \(GRF\) subtends arc \(GF\), which is 130, therefore angle \(GRF\) = 130/2 = \(65\)

4. Since \(FRG\) is isosceles, that means \(FRG=RGF\).

5. Angle \(GFR\) = \(50\) because all triangles have an angle sum of 180, and we know that two angles are 65 degrees. (180 - 65 * 2)

6. Angle \(PGF\) subtends arc \(FQ\), so angle \(PGF\) = 28/2 = \(14\)

7. Conside triangle \(PFG\), we know the measure of two angles \(GFR\) and \(PGF\) is  \(50\) and \(14\), respectively. That means the third angle \(GPF\) is 180 - 50 - 14 = \(116\)

8. By supplements angle \(RPG=180-116 = 64\)

Done

Just remember \(f^{-1}\) is just swapping \(x\) and \(y\). 

An alternative method is draw the altitude from C to AB. Then calculate the area, and I got 31.25 as well.

Angle A subtends to arc BC, which is equal to 180. Therefore angle A is 180/2 = 90. Knowing that it is a 45-45-90 triangle, can you figure out the area of the triangle? After that, just subtract it from the area of the semi-circle.

PAB is similar to PDC by AA. The ratio of sides AB:DC is 1:3. That means the ratio of the areas must be 1:9.

_____________________________________________________________________________________

Call the area of PAB \(x\).

 - The area of PDC is obviously \(x+224\)

Since the ratio is 1:9, then we have the following proportion.

 - \(\frac{x}{x+224}=\frac{1}{9}\)

\(x\) is the area of PAB. I leave the rest of the question up to you.

Where is point P?

If your ego and self-esteem comes from flexing your "superior intelligence" on a math forum website then you should be worried and seek help.

Hint: Check if the triangle is a right triangle by seeing if there are slopes that are negative reciprocals.

- Do you know distance formula?

I have cold-like symptoms. Stumbling upon this post makes me scared...

WOAH! If you look at the computer screen, there are beautiful mathematical curves there. If anyone with the knowledge on why the curves look the way they do and their color, it would be very appreciated. I am very curious right now .