Articles of algebra precalculus

AGM Inequality Proof

I have been stuck on this one for hours. Let $x$, $y$, $z$ be non-negative real numbers. Also we know $x + z \leq 2$. Prove the following: $(x – 2y + z)^2 \geq 4xz – 8y$. Apparently this can be proven with or without AGM, which is $xy \leq \left(\frac{x + y}{2}\right)^2$. This is […]

The number of ones in a binary representation of an integer

Is there any relation that tells whether the number of ones in a binary representation of an integer is an even or an odd number?

Domain of a function

I am confused about this problem: Find the domain of the function, $$f(x)=\frac{x^3-1}{2x^2+5}.$$ I’m guessing it’s all real numbers but the book gives a different answer. The book gave $$(-\infty,-1)\cup (-1,0)\cup (0,\infty)$$ as the answer.

An elegant way to solve $\frac {\sqrt3 – 1}{\sin x} + \frac {\sqrt3 + 1}{\cos x} = 4\sqrt2 $

The question is to find $x\in\left(0,\frac{\pi}{2}\right)$: $$\frac {\sqrt3 – 1}{\sin x} + \frac {\sqrt3 + 1}{\cos x} = 4\sqrt2 $$ What I did was to take the $\cos x$ fraction to the right and try to simplify ; But it looked very messy and trying to write $\sin x$ in terms of $\cos x$ didn’t […]

Is there a difference between $(x)^{\frac{1}{n}} $ and $\sqrt{x}$?

Is there a difference between $(x)^{\frac{1}{n}}$ and $ \sqrt[n]{x}$ ? I’m confused with this topic. Any ideas or examples ? If $(x)^{\frac{1}{n}} = \sqrt[n]{x}$ Consider $x=\frac{-b \pm \sqrt{b^2-4ac}}{2a}$ . Is it the same if I write $ x=\frac{-b \pm(b^2-4ac)^{\frac{1}{2}}}{2a}$ ?

Solution of a Lambert W function

The question was : (find x) $6x=e^{2x}$ I knew Lambert W function and hence: $\Rightarrow 1=\dfrac{6x}{e^{2x}}$ $\Rightarrow \dfrac{1}{6}=xe^{-2x}$ $\Rightarrow \dfrac{-1}{3}=-2xe^{-2x}$ $\Rightarrow -2x=W(\dfrac{-1}{3})$ $\therefore x=\dfrac{-1}{2} W(\dfrac{-1}{3})$ But when i went to WolframAlpha, it showed the same result but in the graph: WolframAlpha Graph The curves intersect at a point… And hence there is a second […]

Probability that no two consecutive heads occur?

A fair coin is tossed $10$ times. What is the probability that no two consecutive tosses are heads? Possibilities are (dont mind the number of terms): $H TTTTTTH$, $HTHTHTHTHTHTHT$. But except for those, let $y(n)$ be the number of sequences that start with $T$ $T _$, there are two options, $T$ and $H$ so, $y(n) […]

$p$ prime, $1 \le k \le p-2$ there exists $x \in \mathbb{Z} \ : \ x^k \neq 0,1 $ (mod p)

I found this problem in my algebra book, but unfortunately, there is no solution included. Here it is: Let $p$ be a prime, $1 \le k \le p-2$. Show that there exists $x \in \mathbb{Z} \ $ such that $\ x^k \neq 0,1$ (mod $p$). It looks like a very nice problem, but I have […]

Conditions for a unique root of a fifth degree polynomial

Fifth degree polynomials cannot generally be solved analytically, but at least one solution always exists. Given the normal form $$ax^5+bx^4+cx^3+dx^2+ex+f=0,$$ is it possible to find sufficient conditions on $a,b,c,d,e,f$ that guarantee a unique solution? One sufficient condition is obviously $a,b,c,d,e>0$, so that the LHS of the above equation is strictly increasing in $x$. In my […]

How to solve $\mathrm{diag}(x) \; A \; x = \mathbf{1}$ for $x\in\mathbb{R}^n$ with $A\in\mathbb{R}^{n \times n}$?

I would like to solve the following equation for $x\in\mathbb{R}^{n}$ $$\mathrm{diag}(x) \; A \; x = \mathbf{1}, \quad \text{with $A\in\mathbb{R}^{n\times n}$},$$ where $\mathrm{diag}(x)$ is a diagonal matrix whose diagonal elements are the elements of $x$ and $\mathbf{1}$ is a vector whose elements are equal to 1. I will already be very happy to find a […]