[/quote]
,
positive integers 
such that for every index 
,
,
is a multiple of 
.
and 
be positive integers satisfying the equation ![$(a, b) + [a, b]=2021^c$](http://latex.artofproblemsolving.com/2/4/6/246ccbb22acecb7adc72219afb8afb34c9b05101.png)
.
is a prime number, prove that the number 
is composite.
be the set of real numbers. Determine all functions 
such that ![\[ f(x^2 - y^2) = x f(x) - y f(y) \]](http://latex.artofproblemsolving.com/6/8/f/68f78e0df627f6ac04052433c01627bf16f0af8e.png)
for all pairs of real numbers 
and 
.
into 
or 
can sometimes be quite complex. That's why I’ve written a program to assist with this process.
be real numbers such that 
Prove that 
Where 
with odd integer side lengths is divided into small rectangles with integer side lengths. Prove that there is at least one among the small rectangles whose distances from the four sides of 
Prove that 
,
,
,
can be expressed by a sum of positive integers 
'![\[n=a_1+a_2+\dots+a_m,\]](http://latex.artofproblemsolving.com/6/c/2/6c2a979a5ff6c40bf2d7152c8c32088fc36f848b.png)
where 
,
,
,
and 
.
equally spaced points on a circular track 
of circumference 
in some order, each label used once. Initially, Bunbun the Bunny begins at 
.
,
,
,
to 
,
of 
is a diameter of 
with the property that ![\[f(x-f(y))=f(f(x))-f(y)-1\]](http://latex.artofproblemsolving.com/f/2/5/f25cc1e8ae0be1fd02b347fd94be4fab88af1d46.png)
holds for all 
.
be the set of positive integers. Find all functions 
such that![\[ m^2 + f(n) \mid mf(m) +n \]](http://latex.artofproblemsolving.com/2/f/4/2f409d1de993f1af8fd839bb8e9f87a57e1b8608.png)
and 
Y by Adventure10, Mango247, and 1 other user
Given are three pairwise externally tangent circles 
, 
and 
. denote by 
tangent point of and and by 
tangent point of and .
Let
(
and 
are different from tangency points) be a diameter of circle . Line 
intersects circle (for second time) at point 
and line 
intersects circle (for second time) at 
.
If
is intersection point of lines 
and 
prove that points , and are collinear.
, 
and 
. denote by 
tangent point of and and by 
tangent point of and .Let
(
and 
are different from tangency points) be a diameter of circle . Line 
intersects circle (for second time) at point 
and line 
intersects circle (for second time) at 
.If
is intersection point of lines 
and 
prove that points , and are collinear.
cut 
![$ \frac {\overline{XP_1}}{\overline{XA}} = \frac {S_[BXP_1]}{S_[BAX]} = - \frac {BP_1}{AP_2}.\frac {P_1P_2}{AB}$](http://latex.artofproblemsolving.com/d/d/3/dd34b9e08ae290baa92a5e71d04e870f59d6ca50.png)
so if 
cut 
so 
is collinear .
the tangent point of 
and 
,
the intersection of 
,
.
is cyclic, denote 
the intersection of its circumcircle and 
,
is the second intersection of 
.
.
,
.
is cyclic. If we assume this is true, then the task becomes trivial. Therefore I think, that the major problem is to prove that 
be 
,
,
respectively. Let 
and 
,
.
,
,
.
.
.
.
and since 
.
,
is cyclic.
are collinear and 
,
.
and 
are collinear. Since 
are collinear.
and 
are diameters of 
.
,
and 
.
is cyclic (which was proved by The QuattoMaster 6000). First we show that 
.
(where the last is true, since 
)
are collinear
are collinear.
and 
.
are collinear 
are collinear.
