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:
-
,
-
,
for every 
.
is given. Determine the smallest natural number 
,
,
to tangent 
disjoint from 
on 
and 
.
is tangent to two fixed circles.
of natural numbers such that for every pair of natural numbers 
and 
,![\[
\frac{1}{2} < \frac{\gcd(a_r, a_s)}{\gcd(r, s)} < 2
\]](http://latex.artofproblemsolving.com/1/6/7/167679c1707b957d87311298ea5b72347a9bdc45.png)
Y by chessgocube
Let 
be an isosceles triangle with 
and circumcircle 
. The point 
lies on the shorter arc of over the chord 
and is different from 
and 
. Let 
denote the intersection of 
and 
. Prove that the line through and is a tangent of the circumcircle of the triangle 
.
(Karl Czakler)
be an isosceles triangle with 
and circumcircle 
. The point 
lies on the shorter arc of over the chord 
and is different from 
and 
. Let 
denote the intersection of 
and 
. Prove that the line through and is a tangent of the circumcircle of the triangle 
.(Karl Czakler)
This post has been edited 2 times. Last edited by parmenides51, May 31, 2021, 10:58 PM
gives us 
.
,
.
is a cyclic quadrilateral, so 
is isosceles then
~
so we are done by Alternate Segment Theorem. 
flips 
,
on the unit circle with:
,
.
as:![\[ e = \frac{ab(c+d) - cd(a+b)}{ab-cd} = \frac{1+d - da - \frac{d}{a}}{1-d} = \frac{a + ad - da^2 - d}{a(1-d)} \]](http://latex.artofproblemsolving.com/6/8/2/682a79dc5ebc85dd21b075f050137d4c90fa7208.png)
![\[ \vert c-d \vert \cdot \vert c - e \vert = \vert c - b \vert ^2 \]](http://latex.artofproblemsolving.com/c/c/7/cc7ac5581c483da865307b0ee576ba7a07aa7ce1.png)
Indeed:![\[ \vert (1-d)(1-e) \vert^2 = \left\vert (1-d)\frac{a - ad - a - ad + da^2 + d}{a(1-d)} \right\vert^2 = \left\vert \frac{d(a-1)^2}{a} \right\vert^2 = \frac{d(a-1)^2}{a}\cdot \overline{\left(\frac{d(a-1)^2}{a}\right)} = \frac{d(a-1)^2 \cdot \frac{1}{d}(1-\frac{1}{a})^2}{a \cdot \frac{1}{a}} = \left(\left(1-\frac{1}{a} \right)(1-a)\right)^2\]](http://latex.artofproblemsolving.com/2/d/5/2d5cf482dd6bb934a791eb90bd8a354601cf5347.png)
![\[ \vert c-b \vert ^4 = \left(\left(1-\frac{1}{a}\right)(1-a)\right)^2\]](http://latex.artofproblemsolving.com/3/5/7/35709d6c2b304939a8bb2a124991ca9e0811216a.png)
,
.
which is equal to its conjugate
