Math COntest Help

  • #1
Hey guys im 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
 

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Answers and Replies

  • #2
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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 [itex]kx^2-bx+k=(3x-5)q(x)[/itex] 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: [itex]ax^2+bx+c=a(x-r_1)(x-r_2)[/itex] where r1 and r2 are the two roots of the quadratic. Thus, you can write [itex]kx^2-bx+k=k(x-5/3)(x-r_2)[/itex]. 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 [itex]y=x^2+2[/itex]. Then, you can rewrite [itex]g(x^2+2)[/itex] in terms of y, so you would have g(y) = some function of y. Notice then that [itex]g(x^2-1)[/itex] is merely g(y-3).
 

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