Competition that i need to pass

[tongos]

I'm hoping on making the United States of America Math Olympiad for Highschool Students this year and i need some help on these questions. Future help will be appreciated. Please make the solutions easy to understand and to follow, please. Hints would be the best.

how many primes, p, make 13p^4-888 a positive prime?
Thoughts: I have no idea of where to begin. A hint would be nice on this one.

Let a1, a2, a3... be the numbers which can be written as a sum of one of more different powers of 4 with a1<a2<a3... e.g. 4^0=a1, 4^1=a2, 4^0+4^1=a3. find a70

Thoughts: It sounds to me that its going by the binomial theorem for example, 210. 3!/(1!)(2!)= 3 ways of having only one term. 3!/(2!)(1!)= 3 ways of having two terms. 1 way of having all. I hope I am on the right track. So do i set up the sigma:
_x_
\ x!/{(x-n)!n!} = 70, or around there.
/__
n=1
Im so lost!

The sides of triangle ABC have length AB=7, BC=8, and AC=9. The distance between the incenter and the orthocenter of triangle ABC is

Thoughts: So the incenter is where the medians intersect?, and the orthocenter is?


Thanks in advance, -Matt

 

[StatusX]

Unless I'm missing something about the second one, it seems a lot easier than the other two. It just comes down to finding 70 in binary and then converting back to base ten using 4 in the place of 2. For the third one, www.mathworld.com will have the definitions you need.

 

[tongos]

okay, i think i got number two, it should be 4116. note that the 64th term is 4^6. And its easy to figure out from there.

Hints would still be helpful on the prime question.

 

[Muzza]

how many primes, p, make 13p^4-888 a positive prime?

Hint: look at 13p^4 - 888 modulo some small integer...

 

[Gokul43201]

tongos, where are these questions from ? Are they part of a test/competition ?

 

[tongos]

artofproblemsolving.com. I'm not at school right now, i stayed home today so I am doing study in order to make the USAMO.

Another question that bugs me is this one...
5. Compute, to the nearest integer, the area of the region enclosed by the graph of 13x^2-20xy+52y^2-10x+52y=563.

I figured that i should solve for the center of the ellipse and then begin to maximize x^2+y^2...