267. Dubious Equality. The statement «$342=97$» can be made valid by inserting several algebraic signs between the digits, for example $(-3+4) \cdot 2=9-7$. Can the equality be given meaning without inserting any signs?
267. For such a statement to be true, it is necessary that these two numbers are written in different number systems, that is, $342_{a}=97_{b}$. If $b=10$, then since $3(4)^{2}=48$ and $3(6)^{2}=108, a=5$. Indeed, $3(5)^{2}+4(5)+2=97$.

In the general case, from $3 a^{2}+4 a+2=9 b+7$, it follows that $b=\frac{3 a^{2}+4 a-5}{9}$. But such a value will be an integer only when $a=9 x+5$; in this case, $b=27 x^{2}+34 x+10$. Therefore, there are infinitely many solutions to this problem; for example, $342_{5}=97_{10}$, $342_{14}=97_{71}$, etc.