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• ### peace*out

1. 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).
2. 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)
3. (f(x+y)-f(xy))/(3x) = f(y/(3x))-11-y Find f(x) where f(x) is a polynomial.
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• 4. 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.
5. 2f(1/x)-f(x)+2f(2/x)-f(x/2) = x, x is defined on the reals except where x =0 find f(x) =
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• 6. ## Solving a system of functional equations

h(f(x)) + g(h(x)) + f(g(x)) = 2x^2 + 11x + 14 f(h(x)) + h(g(x)) + g(f(x)) = 2x^2 - 15x + 66 f(g(x)) = g(f(x)) h(g(x)) - g(h(x)) = -16x + 72 h(f(x)) + f(h(x)) = 2x^2 + 10x + 30 f(x) * g(x) = h(x) - 3x - 40 ----- f(x) = g(x) = h(x) =
7. How many different paths can I make up a flight of 20 stairs if I can take the steps either one at a time or two at a time (in any order)?
8. ## Easy question about a system (or is it)

On the right track but see if you can find what x is approaching.
9. ## Easy question about a system (or is it)

I have a negative value for x as my min and a different x max This is not the answer but as an example of this possibility:
10. 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?
11. I think you are on the right track rocdocmac
12. you have not proven nor disproven this
13. 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?
14. ## Bringing back an old one.... Making Toast

15. ## Bringing back an old one.... Making Toast

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?
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