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such that for all real numbers 
and 
,![\[f(xf(y))+f(y)=f(x+y)+f(xy).\]](http://latex.artofproblemsolving.com/d/2/f/d2ff6ed448cf39e7ec9bce6b944965d5e89b9878.png)
table filled with natural numbers, we say it is a divisor table if:
-th row are exactly all the divisors of some natural number 
,
-th column are exactly all the divisors of some natural number 
,
for every 
.
is given. Determine the smallest natural number 
, divisible by , such that there exists an divisor table, or prove that such does not exist.
, a line 
to tangent , and another circle 
disjoint from such that and lie on opposite sides of . The tangents to from a variable point 
on meet at 
and 
. Prove that, as varies over , the circumcircle of 
is tangent to two fixed circles.
of natural numbers such that for every pair of natural numbers 
and 
, the following inequality holds:![\[
\frac{1}{2} < \frac{\gcd(a_r, a_s)}{\gcd(r, s)} < 2
\]](http://latex.artofproblemsolving.com/1/6/7/167679c1707b957d87311298ea5b72347a9bdc45.png)
+1 w
, 
, 
is the point on 
such that 
. Let 
be the circle through 
tangent to 
at , and 
the circle through tangent to 
at 
. Let 
be the second intersection of and . Prove that 
lies on 
.
be a point on ray beyond is tangent to circle 
and is tangent to circle 
is cyclic.
, giving the required collinearity.
be the intersection of and . We will prove that 
, which implies![\[
(!) \quad AB \text{ is tangent to } (CF'B).
\]](http://latex.artofproblemsolving.com/0/9/1/0912a5d89c9b96991df593e9811fe2ccb3ad79db.png)
Due to the Power of a Point theorem,![\[
AD^2 = AF \cdot AC.
\]](http://latex.artofproblemsolving.com/c/7/c/c7ce7f6916003527f4d1812c1cfcaf61d3bf0bf6.png)
Since 
, we have![\[
AB^2 = AD^2 = AF \cdot AC.
\]](http://latex.artofproblemsolving.com/e/f/f/eff6902bf20790e7dfdb273eee89ca8dac4c58c8.png)
. Note that 
is the radical axis of 
and 
, and so this implies that 
lies on extended, which then implies lies on .
be 
,
is tangent to 
and 
meet on