Whether non-zero integers $a, b, c$ with the property that $$\frac{a}{b} + \frac{b}{c} + \frac{c}{a} = m \in \mathbb{Z}$$ and $$\frac{a}{c} + \frac{c}{b} + \frac{b}{a }= n \in \mathbb{Z}$$ Calculate all possible values for $m + n$.

I have a problem regarding supply distribution. I distribute widgets on a monthly basis; I have many customers and each of them request a different quantity each month. My monthly supply is limited and I cannot fill every order. How do I distribute fairly, across the board? There must be some sort proportional way to […]

Start with the set {3, 4, 12}. You are allowed to perform a sequence of replacements, each time replacing two numbers a and b from your set with the new pair 0.6 a – 0.8b and 0.8 a + 0.6b. Can you transform the set into {4, 6, 12}? Look for an invariant. I am […]

$ \frac{a\sin A+b\sin B+c\sin C}{a\cos A+b\cos B+c\cos C}=R\left(\frac{a^2+b^2+c^2}{abc}\right) $ This is what I have so far: I know that $A + B + C = 180^\circ$, so $C = 180^\circ – (A+B)$. Plugging this in, I get that $\sin C = \sin(A+B)$ and $\cos C = -\cos(A+B)$. When I plug this back into the equation, […]

Please show me how to manipulate $\dfrac{x}{x+1}\;\;$ to get $\;\;1 – \dfrac{1}{x+1}$

Prove that there exist infinitely many Pythagorean integers $a²+b²=c²$ My key idea is to show that there exists infinitely many integers that can be the length of the sides of a right triangle, but I fail at it. Other try is that $\sqrt{a^2+b^2}=c$ and so it is an equation of a circle, so I tried […]

How could we solve $$\sqrt{x} + \ln(x) -1 = 0$$ without using Mathematica? Obviously a solution is $x = 1$, but what are the other exact solutions?

The formula for finding the roots of a polynomial is as follows $$x = \frac {-b \pm \sqrt{ b^2 – 4ac }}{2a} $$ what happens if you want to find the roots of a polynomial like this simplified one $$ 3x^2 + x + 24 = 0 $$ then the square root value becomes $$ […]

I was asked to find the zeros of $y = x^4 + 5x^2 +6$. I tried to turn this into a quadratic to factor it as follows: $y = x^4 + 5x^2 +6 = {(x^2)}^2 + 5{(x^2)}^1 + 6$ Put another way: Let $t = {x^2}$ $y = t^2 + 5t +6$ $y = (t […]

This is a simple problem I am having a bit of trouble with. I am not sure where this leads. Given that $\vec a = \begin{pmatrix}4\\-3\end{pmatrix}$ and $|\vec b|$ = 3, determine the limits between which $|\vec a + \vec b|$ must lie. Let, $\vec b = \begin{pmatrix}\lambda\\\mu\end{pmatrix}$, such that $\lambda^2 + \mu^2 = 9$ […]

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