A1. Consider a set \(S\) and a binary operation \(\ast,\) that is, for each \(a,b\in S, a\ast b\in S.\) Assume \((a\ast b)\ast a = b\) for all \(a,b\in S.\) Prove that \(a\ast (b\ast a)=b\) for all \(a,b\in S.\)
Proof. We have \(((b\ast a)\ast b)\ast (b\ast a)=b.\) But we also have \((b\ast a)\ast b=a.\) Thus \(a\ast (b\ast a)=b.\)\(\square\)
B2. Find all pairs of real numbers \((x,y)\) satisfying the system of equations
\[\begin{aligned} & \frac 1 x + \frac 1 2y = (x^2+3y^2)(3x^2+y^2)\\ & \frac 1 x - \frac 1 2y = 2(y^4-x^4). \end{aligned}\]