For any Convex Quadrilateral, show that the ratio of the Area to its Perimeter^2 is always <1/16, bonus points if you can show that it holds for concave quadrilaterals (not squares).

Let F(t)=f(t)/g(t) be a rational function with integer coefficients, assume g(0)=1, then the Taylor expansion of F(t) at 0 has integer coefficients, and more over, these coefficients satisfy a recursion relation of the form c_n+k=a_{k-1}c_{n+k-1}+ ... + a_0c_n (k and all a_i are all fixed integers) for all but finitely many n?
(for example try computing a MacLauren series for (1+2x)/(1-x^3)

I get two possible solutions: 1100 or 76461. Though if we want only positive values for each emoji then my answer of 1100 is the correct one.
Though I am treating the fact that like how one row has two emojis of alligators and is different than the other rows, then
the eagles being doubled is significant.

Suppose we have the following system
x^2+y^2=r; x+y=r, such that the line crosses the circle at exactly two places.
Obviously with two equations and three variables, we have a solution set of answers that can satisfy the given conditions. What I want to know is of the given solutions that satisfies this problem, what is the smallest and largest values x can possibly be?

I have in mind a number which, when you remove the units digit and place it at the front, gives the same result as multiplying the original number by 2. Am I telling the truth?

Three slices of bread are to be toasted under a grill. The grill can hold two slices at once but only one side is toasted at a time. It takes 30 seconds to toast one side of a piece of bread, 5 seconds to put a piece in or take a piece out and 3 seconds to turn a piece over. What is the shortest time in which the three slices can be toasted?

You have the right idea with your comment. I an designing a cut as a swipe of the knife, so lining them up and then cutting with one swipe would count as one cut.

I wish to share 30 identical individual sausages equally amongst 18 people. What is the minimum number of cuts I need to make? What is the minimum number of pieces I need to create?

James and Mike are running in a race. They both walked and ran for part of the rate. They each walked and ran at the same speed. James ran for half the distance and walked for half the distance. Mike ran for half of his time and walked for half of his time. Who finished first?

Say we have the function:
y=x^x^x^x^x.....
Find an x value for which the derivative of this function converges.
If you are really clever you'll find the interval that converges.