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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.
3. ## Relating sides of a quadrilateral

Suppose we have a quadrilateral with Angles A,B,C,D, corresponding sides a,b,c,d, and the following fact: CosA/a=CosB/b=CosC/c=CosD/d What can be claimed about this quadrilateral?
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.
6. ## Convex Quadrilaterals

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?
16. ## Find the missing value

I think you are on the right track rocdocmac
17. ## Find the missing value

you have not proven nor disproven this
18. ## Find the missing value

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?

20. ## 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?
21. ## Fair division of sausages

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.
22. ## Fair division of sausages

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?
23. ## Impoo

Assume that only four buttons work on your calculator: 5, 7, enter, and plus. What whole numbers can you not use your calculator to make?
24. ## Arc length = area

yes, this is what I mean.
25. ## distance vs time

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?
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