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).

 

[Architeuthis Dux]

for reaching out for help with your math contest preparation. I understand the importance of being able to find reasonable solutions and provide clear explanations. Here are some tips that can help you approach these questions:

1. Read the question carefully: Make sure you understand what the question is asking for. Pay attention to any keywords or specific information given in the question.

2. Identify the relevant concepts: Look for key concepts or formulas that you have learned in your math class that can be applied to the question.

3. Work through the problem step by step: Start by writing down what you know and what you are trying to find. Then, break the problem down into smaller steps and solve each step using the appropriate method or formula.

4. Show your work: It's important to show your work and the steps you took to arrive at your solution. This not only helps you check your work, but it also allows others to understand your thought process and see where you may have made a mistake.

5. Check your answer: Once you have a solution, make sure to check your answer to see if it makes sense in the context of the question. If it doesn't, go back and review your work to find any errors.

Remember, practice makes perfect. Keep practicing and seeking help when needed, and you'll be well-prepared for your math contest. Good luck!