.
,
,
.
be the set of permutations of n-degree and 
.
,
(where 
is the identical permutation).
be subgroups of the group 
and 
.
such as 
.
,
for any integer 
.
and 
be positive integers such that 
is coprime with 
.
for 
.
for some 
where 
.
is said to be good if there exists a partition of 
into disjoint proper subsets such that the sum of the numbers in each subset of the partition is 
the product of whose digits equals 
.
,
in the last term in the sum. How many times does the digit 
occur in the number 
wants the 5th derivative, raise the function to the 5th power, or plug the thing back in the function five times?
Given that 
is a prime, find the number of nonnegative integers 
such that 
where 
are positive integers, 
and 
is as small as possible with 
.
Y by
Suppose a function 
satisfies 
for all real 
. Then, we have 
We also have ![\[f(\tfrac{x+y}{2}) + f(\tfrac{x-y}{2}) \ge 2f(x).\]](//latex.artofproblemsolving.com/5/3/d/53d5e6a96b03a5c4a05a9da1945cbaa81829271e.png)
Adding the first two inequalities and using the third, we have ![\[2f(x) + f(y)+f(-y) \ge 2(f(\tfrac{x+y}{2}) + f(\tfrac{x-y}{2})) \ge 4f(x),\]](//latex.artofproblemsolving.com/a/9/c/a9c9255deeb88e31603a55d6cf32fbf864ca0758.png)
which implies 
. But this means is bounded above which is absurd -- take the identity function. I feel I am making some blithely stupid mistake, but I can't see what. Any help is appreciated 
satisfies 
for all real 
. Then, we have 
We also have ![\[f(\tfrac{x+y}{2}) + f(\tfrac{x-y}{2}) \ge 2f(x).\]](http://latex.artofproblemsolving.com/5/3/d/53d5e6a96b03a5c4a05a9da1945cbaa81829271e.png)
Adding the first two inequalities and using the third, we have ![\[2f(x) + f(y)+f(-y) \ge 2(f(\tfrac{x+y}{2}) + f(\tfrac{x-y}{2})) \ge 4f(x),\]](http://latex.artofproblemsolving.com/a/9/c/a9c9255deeb88e31603a55d6cf32fbf864ca0758.png)
which implies 
. But this means is bounded above which is absurd -- take the identity function. I feel I am making some blithely stupid mistake, but I can't see what. Any help is appreciated This post has been edited 3 times. Last edited by Sedro, Jun 17, 2024, 3:57 PM
on the RHS.
Thanks.
