Q.Find two numbers whose A.M. exceeds their G.M. by $\dfrac12$ and their H.M. by $\dfrac{25}{26}$.

For two numbers $x$ and $y$, the Arithmetic Mean is $A=\dfrac{x+y}2$ (so $x, A, y$ form an A.P.), the Geometric Mean is $G=\sqrt{xy}$ for same-signed numbers (so $x, G, y$ form a G.P.), and the Harmonic Mean is $H=\dfrac{2xy}{x+y}$ (so $x, H, y$ form an H.P., since $\tfrac1x, \tfrac1H, \tfrac1y$ are in A.P.). These extend to $n$ positive numbers as $A=\dfrac{x_1+\cdots+x_n}n$, $G=(x_1x_2\cdots x_n)^{1/n}$, $H=\dfrac{n}{\frac1{x_1}+\cdots+\frac1{x_n}}$. A key identity links all three: $G^2=AH$; and for two positive, unequal numbers, $A\ge G\ge H$ always holds (equality only when $x= …