An operations engineer teaches a new work method to some men and women by having them pass on the method. Indeed, every man passes the method to 1 man and as many women as already know it from the previous hand, and every woman passes the method to 1 woman and as many men as already know it from the previous hand. Thus, in the third hand, 195 men and 192 women took over the new method. How many men and how many women did the engineer teach?
I. Solution: Suppose the new method is received first-hand by

$x$ men and

$y$ women

then, according to the problem,

second-hand,

$$
x+x y \text { men and } \quad y+x y \text { women }
$$

third-hand, 

$$
x+x y+(x+x y)(y+x y) \text { men and } y+x y+(y+x y)(x+x y) \text { women }
$$

receive it.

Thus, according to the problem,

$$
\begin{aligned}
& x+x y+(x+x y)(y+x y)=195 \\
& y+x y+(y+x y)(x+x y)=192
\end{aligned}
$$

The difference between (1) and (2) is

$$
x-y=3
$$

Substituting the value of $x$ from (3) into (1), after first factoring out $(x+x y)$

$$
(x+x y)(1+y+x y)=\left(y^{2}+4 x+3\right)\left(y^{2}+4 y+1\right)=195
$$

Let $y^{2}+4 y+1=z$, then (4) becomes

$$
(z+2) z=195=3 \cdot 5 \cdot 13
$$

from which, considering only the positive solution,

$$
z=13 \quad \text { and thus } \quad y^{2}+4 y+1=13
$$

or

$$
y^{2}+4 y-12=(y-2)(y+6)=0
$$

from which (again, only the positive root)

$$
y=2, \quad x=y+3=5
$$

Ferenc Gyapjas (Bp., VIII., Széchenyi g. IV. D. t.)

II. Solution: Let the number of men and women who received the innovation second-hand be $u$ and $v$ respectively. Then, according to the problem, third-hand $u+u v$ men and $v+u v$ women received the new method, and thus

$$
\begin{array}{r}
u+u v=195 \\
v+u v=192
\end{array}
$$

From the difference between (1) and (2), $u=v+3$, substituting this value into (2)

$$
v^{2}+4 v-192=(v-12)(v+16)=0
$$

from which the positive root

$$
v=12 \quad \text { and thus } \quad u=v+3=15
$$

If $x$ men and $y$ women received it first-hand, then according to the problem

$$
\begin{aligned}
& x+x y=15 \\
& y+x y=12
\end{aligned}
$$

from which

$$
x=5 . \quad \text { and } \quad y=2 .
$$

Elemér Lábos (Sátoraljaújhely, Kossuth g. II. o. t.)