Quadrature on segment Is there a quadrature formula on the segment [0,1], such that on [0,1/2] the points and weights are symetric with respect to 1/4, on [1/2,1] they are symetric with respect to 3/4, and such that the formula exactly integrates polynomials of degree 5. If not, what maximal order can be expected ?

A simple example of such quadrature is
$$
x = \frac{1}{2} + \frac{1}{4}\begin{pmatrix}
-1 - \sqrt\frac{3}{5}&
-1&
-1 + \sqrt\frac{3}{5}&
1 - \sqrt\frac{3}{5}&
1&
1 + \sqrt\frac{3}{5}
\end{pmatrix}\\
w = \frac{1}{36}\begin{pmatrix}5 & 8 & 5 & 5 & 8 & 5\end{pmatrix}
$$
which is just a combination of two three-point Gaussian rules (which are exact for polynomials of degree 5).
It is easy to check that five-point rule of the form you've requested
$$
x = \begin{pmatrix}\frac{1}{4}-a & \frac{1}{4} + a & \frac{3}{4} - b & \frac{3}{4} & \frac{3}{4} + b\end{pmatrix}\\
w = \begin{pmatrix}
w_1 & w_1 & w_2 & w_3 & w_2
\end{pmatrix}
$$
gives an inconsistent system of moment equations, solvable only if requested degree is 4.
$$
w_1 = \frac{1}{4}\quad w_2 = \frac{5}{52}\quad w_3 = \frac{4}{13}\\
a = \frac{1}{4\sqrt{3}}\qquad b = \frac{\sqrt{13}}{4\sqrt{15}}
$$
