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cherman cherman
 
M$5 December 20, 2008 12:05 AM

Can someone resolve these math problems? (Brazilian National Math Olympics - Age 13/14)

Here are the questions:

1 - In each square of a n x n board, we put the numbers 1, 2, 3, 4, in a way that each square has exactly one neighbor with the same number. Is it possible to do this when:

a) n = 2007 ?
b) n = 2008 ?

2 - P is a convex pentagon with all sides equal. Prove that if two of the angles of P egual 180 degrees, it is possible to cover P's plane, without overlapping.

3 - Prove that there are infinite positives "n" that

[ (5^n-2)-1 ] / n

is a whole number.

4 - Show that if p,q are primes, then p^2 + q^2 / p + q, if a whole number, is a prime number as well.

5 - ABC is an acute triangle and O, H its circumcenter, ortocenter (meetup of heights), respectively. Knowing that AB/(square root of 2) = BH = OB, calculate the angles of triangle ABC.

6- Being A a set of whole numbers, we define S (A) as the set formed by the sums of two
elements, not necessarily different elements, and D (A) as the set formed by the differences of two
elements, not necessarily different elements. For example, if A = (1, 2, 3, 10) then S (A) = (2, 3, 4, 5, 6, 11,
12, 13, 20) and D (A) = (-9, -8, -7, -2, -1, 0, 1, 2, 7, 8, 9).

Show that there is a finite set so that S (A) has no more than 10^97 elements and D (A) has at least
10^100 elements.
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bmlhailsto... bmlhailsto...
 
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edd edd
 
December 20, 2008 12:16 AM
I certainly can't without maybe another year of math, but hey, neither could the Yahoo Answers folks :)

However, if those questions were indeed in a math competition, I would suggest that the judges were aware of methods to solve them. One of my math teachers from high school could, no doubt, solve them all in a single classroom setting -- he was a very dedicated mathematician, and studied advanced proofs constantly.
Source(s):
http://answers.yahoo.com/question/index?qid=20081108040937AAec41U


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toddgilmor... toddgilmor...
 
December 20, 2008 12:18 AM
I'm going to save you and us a lot of time if you really don't know...check out http://whyslopes.com/
Source(s):
http://whyslopes.com/


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duane duane
 
December 20, 2008 12:30 AM
Yes.

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mjeezy mjeezy
 
December 20, 2008 02:27 AM
1 is a

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deadringer deadringer
 
December 20, 2008 03:05 AM
I can most definitely resolve those problems, but if theyre for a competition, i dont feel right about posting the answers online. so in short, the answer is yes.

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edd edd
 
December 20, 2008 05:07 AM
Go ahead and post 'em, deadringer -- they were used in a past competition, and they won't reuse the same questions in future competitions. I await your proofs.

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cherman cherman
 
December 20, 2008 07:42 AM
He is lying.

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elly2222 elly2222
 
December 20, 2008 09:18 AM
For #1

a) No. For each square to have EXACTLY one next to it of the same number, each square basically needs to be paired up with exactly one other square of the same number. Therefore, there need to be an even number of squares.

b) yes.

For the bottom row, put 112233441122334411223344.....
For the second row, put 334411223344112233441122....
For the third row, same as the bottom row.
For the fourth row, same as the second row. Keep alternating back and forth.
Source(s):
I took a lot of math; but this is HARD for that age group!


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cherman cherman
 
December 21, 2008 12:42 AM
Yeah, that one is the easy one (only one I got right). Thanks for contribution anyway.

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