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1. ## A simple Coffee Problem

Every morning, I brew 3 cups of coffee in my French press. I prepare a large mason jar with ice, sweetener, and cream which fills the container half-way. I add enough coffee to fill to the top of the jar. Throughout the morning I drink the jar down halfway just to top it up once again. I am able to refill it twice fully. For the third refill, I am only able to add 2 FL oz. of coffee. How big is my Mason Jar? Bonus question: what % of coffee is in my last cups mixture after adding the 2 oz? For added clarity, when I top it up, I mean that I am only adding my coffee to the mixture.
2. ## Relating sides of a quadrilateral

That is true. For this problem it doesn't matter whether it is the corresponding side on the left or right. Just be consistent.

4. ## n^5 has the same last digit as n

I am a math professor.
5. ## n^5 has the same last digit as n

Assume that n is a natural number, prove that n and n5 will always have the same one's digit. e.g. 13 and 135=371,293 both end in 3.

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).
7. ## A taylor series problem....

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)
8. ## One last functional equation

(f(x+y)-f(xy))/(3x) = f(y/(3x))-11-y Find f(x) where f(x) is a polynomial.
9. ## Emoji based math puzzle - level hard!

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.
10. ## Another Functional Equation

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) =
11. ## 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) =
12. ## Climbing Stairs

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)?
13. ## Easy question about a system (or is it)

On the right track but see if you can find what x is approaching.
14. ## 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:
15. ## Easy question about a system (or is it)

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?
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