![$ a,b \in [0 ,1] . $](http://latex.artofproblemsolving.com/9/1/9/91980d5ca5d97fe4b61797e6fbb378211674d9cd.png)
Prove that
be a polynomial with integer coefficient, It is known that 
for 
distinct integers , Then the number of integer values of 
for which 
of a triangle 
are positive integers. Let
for any positive integer 
.
and 
,
![\[\sum_{0 \le i < j \le n} \sum (i+j) \left(\binom{n}{i}^2 + \binom{n}{j}^2 + \binom ni \binom nj\right)\]](http://latex.artofproblemsolving.com/a/f/8/af8a267e1c11970e807214eebd04d37725aaf323.png)
Y by
Can u solve "The Blue Waterfall" System of equations?
x+y+z+w=7
y*z+w=7
x*y+z=5
z*w-x=2
For real and complex!
The ones who will make it will win my love!
Thanks a million,
George
x+y+z+w=7
y*z+w=7
x*y+z=5
z*w-x=2
For real and complex!
The ones who will make it will win my love!
Thanks a million,
George
.
![\begin{align*}
A_\pm&=\frac{7}{2}-\frac{1}{12}\sqrt{372+6\sqrt[3]{341612-372i\sqrt{25743}}+6\sqrt[3]{341612+372i\sqrt{25743}}}\pm\frac{i}{12}\sqrt{-744+6\sqrt[3]{341612-372i\sqrt{25743}}+6\sqrt[3]{341612+372i\sqrt{25743}}-24\sqrt{-273+3\sqrt[3]{-547003-744i\sqrt{25743}}+3\sqrt[3]{-547003+744i\sqrt{25743}}}}\\
B_\pm&=\frac{3}{4}+\frac{1}{12}\sqrt{-63+6\sqrt[3]{-1836-12i\sqrt{25743}}+6\sqrt[3]{-1836+12i\sqrt{25743}}}\pm\frac{i}{12}\sqrt{126+6\sqrt[3]{-1836-12i\sqrt{25743}}+6\sqrt[3]{-1836+12i\sqrt{25743}}-6\sqrt{-327+6\sqrt[3]{-406772-4332i\sqrt{25743}}+6\sqrt[3]{-406772+4332i\sqrt{25743}}}}\\
C_\pm&=\frac{3}{4}+\frac{1}{12}\sqrt{-183+6\sqrt[3]{-1268-84i\sqrt{25743}}+6\sqrt[3]{-1268+84i\sqrt{25743}}}\pm\frac{i}{12}\sqrt{366+6\sqrt[3]{-1268-84i\sqrt{25743}}+6\sqrt[3]{-1268+84i\sqrt{25743}}-6\sqrt{1449+6\sqrt[3]{183244788-524244i\sqrt{25743}}+6\sqrt[3]{183244788+524244i\sqrt{25743}}}}\\
D_\pm&=2+\frac{1}{6}\sqrt{6+3\sqrt[3]{3329-6i\sqrt{25743}}+3\sqrt[3]{3329+6i\sqrt{25743}}}\pm\frac{i}{6}\sqrt{-12+3\sqrt[3]{3329-6i\sqrt{25743}}+3\sqrt[3]{3329+6i\sqrt{25743}}+6\sqrt{-225+3\sqrt[3]{229041-1176i\sqrt{25743}}+3\sqrt[3]{229041+1176i\sqrt{25743}}}}\\
E_\pm&=\frac{7}{2}+\frac{1}{12}\sqrt{372+6\sqrt[3]{341612-372i\sqrt{25743}}+6\sqrt[3]{341612+372i\sqrt{25743}}}\pm\frac{i}{12}\sqrt{-744+6\sqrt[3]{341612-372i\sqrt{25743}}+6\sqrt[3]{341612+372i\sqrt{25743}}+24\sqrt{-273+3\sqrt[3]{-547003-744i\sqrt{25743}}+3\sqrt[3]{-547003+744i\sqrt{25743}}}}\\
F_\pm&=\frac{3}{4}-\frac{1}{12}\sqrt{-63+6\sqrt[3]{-1836-12i\sqrt{25743}}+6\sqrt[3]{-1836+12i\sqrt{25743}}}\pm\frac{i}{12}\sqrt{126+6\sqrt[3]{-1836-12i\sqrt{25743}}+6\sqrt[3]{-1836+12i\sqrt{25743}}+6\sqrt{-327+6\sqrt[3]{-406772-4332i\sqrt{25743}}+6\sqrt[3]{-406772+4332i\sqrt{25743}}}}\\
G_\pm&=\frac{3}{4}-\frac{1}{12}\sqrt{-183+6\sqrt[3]{-1268-84i\sqrt{25743}}+6\sqrt[3]{-1268+84i\sqrt{25743}}}\pm\frac{i}{12}\sqrt{366+6\sqrt[3]{-1268-84i\sqrt{25743}}+6\sqrt[3]{-1268+84i\sqrt{25743}}+6\sqrt{1449+6\sqrt[3]{183244788-524244i\sqrt{25743}}+6\sqrt[3]{183244788+524244i\sqrt{25743}}}}\\
H_\pm&=2-\frac{1}{6}\sqrt{6+3\sqrt[3]{3329-6i\sqrt{25743}}+3\sqrt[3]{3329+6i\sqrt{25743}}}\pm\frac{i}{6}\sqrt{-12+3\sqrt[3]{3329-6i\sqrt{25743}}+3\sqrt[3]{3329+6i\sqrt{25743}}-6\sqrt{-225+3\sqrt[3]{229041-1176i\sqrt{25743}}+3\sqrt[3]{229041+1176i\sqrt{25743}}}}
\end{align*}](http://latex.artofproblemsolving.com/c/7/a/c7aca4db95efe1aa1442397cb781a97e4f7996f7.png)
