Problems

Observe that \(14\) isn’t a square number but \(144=12^2\) and \(1444=38^2\) are both square numbers. Let \(k_1^2=\overline{a_n...a_1a_0}\) the decimal representation of a square number.
Is it possible that \(\overline{a_n...a_1a_0a_0}\) and \(\overline{a_n...a_1a_0a_0a_0}\) are also both square numbers?

Let \(ABCDEF\) be a regular hexagon. Points \(G\) and \(H\) lie on \(EF\) and \(DE\) respectively such that \(|EG|=|EH|\). Furthermore, the area of quadrilateral \(ABGF\) is equal to the area of quadrilateral \(BGEH\), which are both equal to the area of \(BCDH\). What’s the ratio \(\frac{|EG|}{|EF|}\)?