1. 3^{x}+4^{y}=5^{z}find all natural x, y, z numbers

2. 2^{2k+1}+9*2^{k}+5=n^{2}k=? k is an integer

3. If there's a point P inside the triangle ABC where PAB=10 PBA=20 PCA=30 PAC=40 prove that triangle ABC is isosceles

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- Jan 25th 2013, 02:59 AManod1996Mongolia's province math competition 9th grade.
1. 3

^{x}+4^{y}=5^{z}find all natural x, y, z numbers

2. 2^{2k+1}+9*2^{k}+5=n^{2}k=? k is an integer

3. If there's a point P inside the triangle ABC where PAB=10 PBA=20 PCA=30 PAC=40 prove that triangle ABC is isosceles - Jan 25th 2013, 07:11 AMtopsquarkRe: Mongolia's province math competition 9th grade.
- Jan 25th 2013, 10:42 AMearthboyRe: Mongolia's province math competition 9th grade.
I guess the competition is over or the problems are from old exams....

btw, the problems listed are very common olympiad problems.....

1.Common problem, with a little long solution, got posed on Nick's mathematical puzzle number:98. I am posting the link:

Nick's Mathematical Puzzles: Solution 98

2. we have to make the given expression a square number:

[taking ]

=(#now we complete the squares)

for the expression to be a perfect square, , which happens when .as k is integer, we take 1

so , hence k can only be

3.This problem is both tricky and famous(& i'm giving the most well known solution):

consider and we know , hence

now we have to use the trigonometric version of ceva's theorem:

=>

=>

=> [#using product-sum formula]

=>

=>

so as , we can write or

hence ,therefore triangle ABC is isosceles.