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 
.
,
,
,
of 
is a diameter of 
Y by
Given a triangle 
. A circle 
passing through points 
and 
, intersects line segments 
and 
(not including endpoints) at points 
and 
, respectively, and 
holds. Let 
be the incenter of the triangle . Let 
and 
be the points, other than 
, where the straight lines 
and 
intersect the circle . Show that 
and lie on the same circle.
. A circle 
passing through points 
and 
, intersects line segments 
and 
(not including endpoints) at points 
and 
, respectively, and 
holds. Let 
be the incenter of the triangle . Let 
and 
be the points, other than 
, where the straight lines 
and 
intersect the circle . Show that 
and lie on the same circle.This post has been edited 2 times. Last edited by parmenides51, Nov 28, 2022, 1:38 PM
