Problem 1081

The task only makes sense if neither $A$ nor $B$ falls inside the circle. If neither $A$ nor $B$ is on the circumference of the circle, then the statement is true; otherwise, it is not. If, for instance, $A$ is on the circumference and $B$ is outside the circle, then $a=0, b>AB$, so $a+b>AB$, but the segment $AB$ has a common point with $k$, so in this case, the statement is not true (Figure 1).

If both $A$ and $B$ are outside the circle, we need to examine the following three cases:

(i) $AB$ is outside $k$,

(ii) $AB$ touches $k$ at a point $E$,

(iii) $AB$ intersects $k$.

The statement of the problem is that in case (i), $a+b>AB$, while in the other two cases, $a+b \leq AB$. In all three cases, it is true that the two tangent segments drawn from $A$ and $B$ to $k$ are of equal length, so it is always sufficient to examine the tangent segment that is more favorably positioned for us.

In case (i), we can assume that one of the two tangent lines separates $AB$ and $k$ (i.e., the circle and the segment are on different sides of the tangent line), while the other does not. In our diagram, the tangent from $A$ is the separating one. Then the intersection point $C$ of the two tangent lines will be an internal point of the tangent segment from $B$, but not of the tangent segment from $A$. The points will be arranged as shown in Figure 2. Using the notation from the diagram:

$$a+b=AE_A + BE_B = AE_A + CE_B + CB = AE_A + CE_A + CB = AC + CB$$

However, by the triangle inequality, $AC + CB > AB$, so the statement of the problem is true in this case.

In case (ii) (Figure 3), $a+b = AE + EB = AB$, so the statement is clearly true in this case.

In case (iii), we draw the tangents whose points of tangency are separated from the center $O$ of the circle $k$ by $AB$ (Figure 4). Let the intersection points of the radii $E_AO$ and $E_BO$ with $AB$ be $M_A$ and $M_B$, respectively. Then the points $A, M_A, M_B, B$ are arranged in this order, and $M_A$ and $M_B$ may coincide if $O$ is on $AB$, so $a+b < AM_A + M_BB \leq AB$.

This proves that the statement of the problem is true precisely when both $A$ and $B$ are outside the circle.

Based on the work of Gábor Nyul (Debrecen, Fazekas M. Gymnasium, 1st year)

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