Math Contest Help: Get Reasonable Solutions & Explanations

[ballaholic8]

Hey guys I am preparing for a math contest and this is a review set that we have recieved, however i have no idea how to answer these questions,

Can someone please help with some reasonable solutions with some explanation.

Thanks

 

[Tedjn]

18. As in all geometry problems, draw a diagram. Draw in the angles you need to find. See if you can go from there.

19. What does it mean for 3x-5 to divide kx²-bx+k? It means that $kx^2-bx+k=(3x-5)q(x)$ for some polynomial q(x). It also means that if we set 3x-5=0, then the original quadratic will be 0 too. That is, when x = 5/3, kx²-bx+k=0, so 5/3 is a root of the quadratic. Every quadratic can be factored as follows: $ax^2+bx+c=a(x-r_1)(x-r_2)$ where r₁ and r₂ are the two roots of the quadratic. Thus, you can write $kx^2-bx+k=k(x-5/3)(x-r_2)$. Multiply out and find out what r₂ must be. You then can find k/b.

20. Which triangles are similar in that picture?

24. Again, draw a picture (this is a common theme in geometry problems), and try to draw a triangle using the information you have available.

25. Suppose $y=x^2+2$. Then, you can rewrite $g(x^2+2)$ in terms of y, so you would have g(y) = some function of y. Notice then that $g(x^2-1)$ is merely g(y-3).