AoPS Academy
with 
, prove that
positive integers 
such that for every index 
, with 
, 
is a multiple of 
. Determine the greatest possible value of the largest of the numbers.
and 
be positive integers satisfying the equation ![$(a, b) + [a, b]=2021^c$](http://latex.artofproblemsolving.com/2/4/6/246ccbb22acecb7adc72219afb8afb34c9b05101.png)
. If 
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 are either all odd or all even.
, prove that there exists a positive integer 
, such that for any integer 
, 
can be expressed by a sum of positive integers 
's as![\[n=a_1+a_2+\dots+a_m,\]](http://latex.artofproblemsolving.com/6/c/2/6c2a979a5ff6c40bf2d7152c8c32088fc36f848b.png)
where 
, 
, 
, 
and 
.
equally spaced points on a circular track 
of circumference . The points are labeled 
in some order, each label used once. Initially, Bunbun the Bunny begins at 
. She hops along from to 
, then from to 
, until she reaches 
, after which she hops back to . When hopping from 
to 
, she always hops along the shorter of the two arcs 
of ; if 
is a diameter of , she moves along either semicircle. arcs which Bunbun traveled, over all possible labellings of the points.
such that: 
. Find the maximum: 
such that: . Find the maximum: 
such that: . Find the maximum: 
so we only need to study the inequality for negative numbers and the function is convex for 
.
and the maximum value is equal to 
.
so we only need to study the inequality for negative numbers and the function is convex for .
, we get:
Therefore, it is enough to study the function for 
. But the function 
is convex for 
with convex constraints, so it is enough to check the borders where we get a maximum at equal to .
, we get:Therefore, it is enough to study the function for . But the function is convex for with convex constraints, so it is enough to check the borders where we get a maximum at equal to .
we obtain a greater value. Why don't we get a greater value with a condition 
?
we obtain a greater value. Why don't we get a greater value with a condition ? as it does not verify the condition !!!! I know you are a very smart guy 
, but not sure what do you mean with your suggestion.
be real numbers such as 
and . Prove that:
we obtain a greater value. Why don't we get a greater value with a condition ? as it does not verify the condition !!!!
is convex on ![$[-2,0]$](http://latex.artofproblemsolving.com/5/5/a/55a2deada78acc1831c348210e10b1e6a6ede5a9.png)
and without condition you can get a maximal value by checking 
.? If for any you'll obtain that the condition indeed occurs, so your reasoning is true. Otherwise, your reasoning is total wrong, I think.
is convex on and without condition you can get a maximal value by checking .? If for any you'll obtain that the condition indeed occurs, so your reasoning is true. Otherwise, your reasoning is total wrong, I think. is essential to finding not just the border conditions. Check my example:
and 
.. But the function is convex for with convex constraints, so it is enough to check the borders where we get a maximum at equal to .
Prove that 
Prove that