BMAD

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.

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

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?

I am a math professor.

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).

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)

(f(x+y)-f(xy))/(3x) = f(y/(3x))-11-y Find f(x) where f(x) is a polynomial.

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.

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) =

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) =

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)?

On the right track but see if you can find what x is approaching.

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:

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?