Let be a positive integer. Prove that is a permutation polynomial mod iff is a power of 3.
Note: we call a permutation polynomial mod iff the induced map is bijective.

Are there four natural numbers such that ?

Two circles and with equal radii intersect at P and Q. Points B and C are located on the circles and so that they are inside the circles and , respectively. Also, points X and Y distinct from P are located on and , respectively, so that:
The intersection point of the circumcircles of triangles XPC and YPB is called S. Prove that BC, XY and QS are concurrent.
Thanks.

1)Let what is range of values of

2) Let what is range of values of

Let . Define as the average of all for . Determine the value of

I want to learn how to integrate, I'm a ms student with knowledge about math counts ,amc 10 even tho that want help mebut I don't want to dwell in calc, I just want to learn how to integrate and nothing else like I don't want to study it deep, how can I learn how to integrate its for an integration bee hosted near me its a state uni and I want to join so in the span of 2 months how can I learn to integrate without learning calc like fully

Let be a sequence of nonnegative real numbers whose sum is . Find the maximum possible value of