be real numbers . Prove that
![$$-\frac{1}{18}\leq \frac{ab+a+b+3}{(a^2+3)(b^2+3)}\leq \frac{2+\sqrt[3] 2+\sqrt[3] {4}}{12} $$](http://latex.artofproblemsolving.com/9/9/9/9997397eb38cd4bceec2ff8c85951c78e2797d0f.png)
and 
.
for any 
and 
such that 
and exist 
odd numbers 
satisfied all conditions below:![$\sqrt[k]{a_1}+\sqrt[k]{a_2} + ... + \sqrt[k]{a_n} = \sqrt[k]{b_1}+\sqrt[k]{b_2} + ... + \sqrt[k]{b_m} $](http://latex.artofproblemsolving.com/c/9/e/c9e051a562376b2b9e02e82d46a41126874cdf60.png)
for all positive number 
Y by Adventure10, Mango247
Squares 
are constructed from the outside at sides of triangle 
and 
are the centres of these squares. Let 
be the intersection points of the straight lines 
and 
and 
and respectively. Prove that the straight lines 
and 
are concurrent.
are constructed from the outside at sides of triangle 
and 
are the centres of these squares. Let 
be the intersection points of the straight lines 
and 
and 
and respectively. Prove that the straight lines 
and 
are concurrent.
