Problem 1498

1. Understanding the Problem:
We are given two circles $O_1$ and $O_2$ with different radii, both tangent to a larger circle $O$ at points $S$ and $T$ respectively. The intersections of $O_1$ and $O_2$ are $M$ and $N$. We need to prove that the necessary and sufficient condition for $OM \perp MN$ is that $S, N, T$ are collinear.

2. Restating the Condition:
The condition $OM \perp MN$ implies that the quadrilateral $OMTN$ is cyclic. This is because if $OM$ is perpendicular to $MN$, then $\angle OMN = 90^\circ$, which is a property of a cyclic quadrilateral where one of the angles is $90^\circ$.

3. Cyclic Quadrilateral Property:
For $OMTN$ to be cyclic, the opposite angles must sum to $180^\circ$. Therefore, we need:
$$\angle OMT + \angle ONT = 180^\circ$$

4. Collinearity Condition:
If $S, N, T$ are collinear, then $\angle SNT = 180^\circ$. This implies that $\angle OMT + \angle ONT = 180^\circ$ because $\angle OMT$ and $\angle ONT$ are subtended by the same arc $ST$ in the larger circle $O$.

5. Equivalence of Conditions:
- Sufficient Condition:
If $S, N, T$ are collinear, then $\angle SNT = 180^\circ$. This implies that $\angle OMT + \angle ONT = 180^\circ$, making $OMTN$ a cyclic quadrilateral. Hence, $OM \perp MN$.
- Necessary Condition:
If $OM \perp MN$, then $\angle OMN = 90^\circ$. This implies that $OMTN$ is a cyclic quadrilateral, and thus $\angle OMT + \angle ONT = 180^\circ$. This can only happen if $S, N, T$ are collinear, as $\angle SNT = 180^\circ$.

6. Conclusion:
We have shown that $OM \perp MN$ if and only if $S, N, T$ are collinear.

$\blacksquare$