Voldemort

Here is a different solution:

Root transformation!

2x^2 - 7x + 2 = x^2 - 11x + 1

x^2 + 4x - 1 = 0

If a, b are the solutions. Then a-1, b-1 are the solutions to the equation below:

(x+1)^2 + 4(x+1) - 1 = 0 

x^2 + 6x + 4 = 0

The reciprocal of the roots of this equation would be reversed:

4x^2 + 6x + 1 = 0.

The sum of the roots is -6 by vietas. 

Use roots of unity!

z^7 = 1 x (-1)^7

\(z_k = -1 \cdot e^{\frac{2k\pi}{n}i}\)

The sum of the roots of unity is 0 by vieta's formula. So -1(0) = 0. 

Suppose O is the center of the circle. I am guessing that the shaded region refers to the area inside the sector BOA but outside of triangle ABO. But I am not sure if I am correct with what the shdaed region is equal to. If you get this wrong, don't blame it on me. 

We know that BO = AO = 10, as they are all radii of the circle. Because AB is 10, we know that tringle ABO is equilateral, which also means that angle BOA is 60 degrees. This means that the sector is 1/6 of the WHOLE circle. The area of the whole circle is 100pi so the area of the sector is 50/3pi. We have to then subtract by the area of triangle ABO which is 25sqrt(3). Therefore a + b = 50/3pi + 25sqrt(3). 

3x + xy + 2y = 117

x(3+y) + 2y = 117

x(3+y) + 2y + 6 = 123

x(3+y) + 2(y+3) = 123

(x+2)(y+3) = 123

123 = 41 x 3

x + 2 = 3 

y + 3 = 41

x = 1, y = 38

x + y = 39. 

Make sure you credit where the original problem came from! This, I believe, is from MATHCOUNTS States.

In the graph of y = f(x), we know that two coordinates are:

(10, f(10))

(0, f(0)).

Finding the slope of these two points gives us:

(f(10)-f(0))/10 = 1

Therefore, f(10)-f(0) = 10. 

There are infinite amount of values for x and y we can choose, suggesting the use of geometric probability. The area of (x, y) we can choose is a rectangle with the x width being 3 and the y length being 6. Let's the draw the line x + y = 5. Note that this creates similar triangles. 2/5 = x/5 which results in x = 2. There area of the big triangle - the area of the small triangle would be the area where we would get succesfful outcomes but still inside the rectangle.

This is:

25/2 - 4/2 = 21/2

Because the total area for the x and y values is 18. Our probability is 7/12. 

Disregard my message as well.