Two rods - one made of steel, the other of zinc - need to be sized such that at $0^{\circ}$ C, $a=1 \mathrm{~m}$, and at $100^{\circ}$ C, their combined length is $b=1.0024 \mathrm{~m}$. What is the length of each rod at $0^{\circ}$ C, given that the linear expansion coefficient of steel is $\lambda^{\prime}=0.000011$, and that of zinc is $\lambda^{\prime \prime}=0.000031$?
Let the length of the steel rod be $x$, and that of the zinc rod be $y$, then

$$
\begin{gathered}
x+y=a \\
x\left(1+100 \lambda^{\prime}\right)+y\left(1+100 \lambda^{\prime \prime}\right)=b,
\end{gathered}
$$

from which

$$
\begin{aligned}
& y=\frac{b-a\left(1+100 \lambda^{\prime}\right)}{100\left(\lambda^{\prime \prime}-\lambda^{\prime}\right)}=0.65 \text{ m} \\
& x=\frac{b-a\left(1+100 \lambda^{\prime \prime}\right)}{100\left(\lambda^{\prime}-\lambda^{\prime \prime}\right)}=0.35 \text{ m}
\end{aligned}
$$

(Pál Sárközy, Pannonhalma.)