The two-variable function $f(x, y)$ defined on natural numbers satisfies the following conditions:

$$
f(x, x)=x
$$

$$
f(x, y)=f(y, x)
$$

$$
(x+y) \cdot f(x, y)=f(x, x+y) \cdot y
$$

What is $f(980,1980)$?
(1) gives the function on the line $y=x$ of the coordinate system, and because of (3), the value of $f$ on the $x$-axis can only be 0. Therefore, according to (2), it is sufficient to determine the other values of $f$ in the positive quadrant with the condition $y<x$. If we write the third condition in the form

$$
\frac{f(x, x+y)}{x+y}=\frac{f(x, y)}{y}
$$

we can see that the function

$$
g(x, y)=\frac{f(x, y)}{y}
$$

is periodic in $y$ for a fixed $x$, with period $x$. From this, the following procedure for determining the values of the functions $f, g$ follows.

If we seek the value of the function $g$ at some point $(x, y)$ satisfying $0<y<x$, based on (2) and (5) we get the relationship

$$
g(x, y)=\frac{x}{y} g(y, x)
$$

which allows us to reduce this question to determining $g(y, x)$, and then to $g(y, x-y)$ according to (4),

$$
g(y, x)=g(y, x-y)
$$

where $z$ is the remainder of the division of $x$ by $y$. If $x$ happens to be divisible by $y$, we will eventually find that $g(y, x)=g(y, y)=1$, so $g(x, y)=\frac{x}{y}$. If not, we continue.

We already know the value of the function $g$ at points where the first coordinate is divisible by the second. For the other points, we know that

$$
g(x, y)=\frac{x}{y} g(y, z)
$$

where $z$ is the remainder of the division of $x$ by $y$. Notice that if we continue, and denote the remainder of the division of $y$ by $z$ as $v$, then $g(x, y)=\frac{x}{z} g(z, v)$, so we can "forget" the intermediate numbers. Since our numbers are decreasing, the procedure will certainly end, and we will eventually reach a point $(z, v)$ where $v$ divides $z$, possibly $v=1$. In any case, we get that $g(x, y)=\frac{x}{v}$, where $v$ is the last non-zero remainder. It is well known that our procedure is the so-called Euclidean algorithm for determining the greatest common divisor, so the sought function is $f(x, y)=x y / v$, where $v$ is the greatest common divisor of $(x, y)$. The number $x y / v$ is nothing other than the least common multiple of $x, y$, denoted by $[x, y]$. By convention, if either $x$ or $y$ is 0, we define $[x, y]$ to be 0.

If there is a function satisfying the conditions, it can only be $f(x, y)=[x, y]$. (1) and (2) are clearly satisfied by this, and we will show that (3) is also true. Let the greatest common divisor of $x, y$ be denoted by $v$: this will also be the greatest common divisor of $x$ and $x+y$, so (3) represents the identity

$$
\frac{(x+y) x y}{v}=\frac{y x(x+y)}{v}
$$

Therefore, only one function satisfies the given conditions, which assigns the least common multiple of its arguments to them. Specifically, $f(980,1980)=49 \cdot 1980=97020$.