Math Is Fun Forum

The graph of the equation

is an ellipse with center [ma­th]h,k)[/ma­th], horizontal axis length [ma­th]2a[/ma­th], and vertical axis length [ma­th]2b[/ma­th]. Find parametric equations whose graph is an ellipse with center [ma­th](h,k)[/ma­th], horizontal axis length[ma­th]2a[/ma­th], and vertical axis length [ma­th]2b[/ma­th], and explain why your answer is correct.

Let [ma­th]\mathcal{G}[/ma­th] be the graph of the parametric equations
[ma­th]\begin{align*}
x &= \cos(4t),\\
y &= \sin(6t).
\end{align*}[/ma­th]
What is the length of the smallest interval [ma­th]I[/ma­th] such that the graph of these equations for all [ma­th]t\in I[/ma­th] produces the entire graph [ma­th]\mathcal{G}[/ma­th]?

1. In $\triangle ABC$, we have $AB = 16$, $BC = 14$, and $\angle A = 60^\circ$. Find the sum of all possible values of $AC$.
2. In $\triangle ABC$, we have $AB = 14$, $BC = 16$, and $\angle A = 60^\circ$. Find the sum of all possible values of $AC$.
3. Let

be positive real numbers. Prove that \[\sqrt{a^2 - ab + b^2} + \sqrt{a^2 - ac + c^2} \ge \sqrt{b^2 + bc + c^2}.\]
Under what conditions does equality occur? That is, for what values of $a$, $b$, and $c$ are the two sides equal?

2. a) Which positive integers can be represented as the sum of two or more consecutive positive integers? For instance, 1 and 2 cannot be represented this way, because the smallest thesum of two or more consecutive positive integers can be is 1 + 2 = 3. Thus 3 can be represented this way. So can 5050, because as Gauss showed as a boy, 1 + 2 + 3 + · · · +99 + 100 = 5050.
You must prove that every positive integer you include in your list can be represented as a sum of consecutive positive integers and every other positive integer cannot be so represented.
b) Which integers can be represented as the sum of two or more consecutive integers? Again, proof required.

I found that every single integer can be represented except numbers that are 2^n. I have used patterns to find this. I need help proving that either every integer that is 2^n cannot be represented as the sum of two or more consecutive positive integers.

Start with any positive integer. If it is 1, stop. Otherwise, if it is odd, subtract 1; if it is even, divide by 2. Either way call this one step. Keep repeating steps, stopping only when you reach 1. For instance, if you start with 6, you must go 6 → 3 → 2 → 1; thus you reach 1 in 3 steps. Consider the set S10 of positive integers from which you reach 1 in exactly 10 steps. For each of the following parts, justify your answer.
a) What is the largest number in S10? The smallest?
b) How many numbers are in S10? How many even numbers?
c) What is the sum of the numbers in S10?
d) Obviously there is nothing special about 10 steps. Can you generalize your results for
Parts a–c to Sn?
e) Can you think of other questions to ask about Sn? Can you answer them, with proof? If
so do so.

I have figure out a and b, but I don't know how to prove b very well, the best I can say is that I noticed it using patterns.