1. 3x+4y=5z find all natural x, y, z numbers
2. 22k+1+9*2k+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
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:
=(#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.