Thread: Mongolia'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=n² 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

Originally Posted by anod1996

1. 3^x+4^y=5^z find all natural x, y, z numbers
2. 2^2k+1+9*2^k+5=n² 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

Surely it can't be legal for you to ask for help on this.

-Dan

Originally Posted by anod1996

1. 3^x+4^y=5^z find all natural x, y, z numbers
2. 2^2k+1+9*2^k+5=n² 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

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.