1) Let \(\alpha\) and \(\beta\) be positive integers such that \(\alpha \beta +1\) divides \(\alpha ^{2} + \beta ^{2}\). Show that  \(\frac{\alpha ^{2} + \beta ^{2}}{ \alpha \beta +1}\) is a square of some integer?
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2) Following the criteria of Euler infinite sum evaluation; can you give some value for this sum:
     $$\sum_{n=0}^{\infty }{n} = 1+1+1+1+1+... = ?? $$
Hint: read this article http://www.aac-m.com/?app=article.show.5
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3) Can you solve the following calculus problem without using PC software? 
$$3(x^2+y^2+z^2)=10(xy+yz+zx)$$
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