**16.433 inequality questions.**

This question is an offshoot of this earlier MSE question.
Let $\sigma(z)$ denote the sum of divisors of $z \in \mathbb{N}$, the set of positive integers. Denote the abundancy index of $z$ by $I(z) :...

Solve the inequality
$$\frac{\log_ax}{x^2+(a-4)x+4-2a}\le0$$
My work:
If $1>a>0$ then $x \in (0;1] \cup (2; + \infty)$ .
If $3>a>1$ then $x \in (0;2) $ .
If $3 \le a<4$ then $x \...

Calculus tools render the minimization of
$$\frac{(1-p)^2}{p} + \frac{p^2}{1-p}$$
on the interval $(0,1/2]$ to be a trivial task. But given how much symmetry there is in this expression, I was curious ...

Given a triangle $ABC$, let $I$ be the incenter. The internal bisectors of angles $A$, $B$, $C$ meet the opposite sides in $A'$, $B'$, $C'$, respectively. Prove that $$\frac 14 < \frac{AI \cdot BI \...

https://mathhelpboards.com/pre-algebra-algebra-2/range-values-inequalities-19110.html
I came across this thread while looking for some inequality practice problems for the GRE. (I paraphrase the ...

Let $f(x) = (x-1)\ln x$, and give $0 < a < b$.
If $f(a) = f(b)$, how to prove that $\frac{1}{\ln a}+\frac{1}{\ln b} < \frac{1}{2}$?

The problem is to find the values of n such $n^2\le n!$.
I have found this set to be {${n\in Z^+| n\le1 \lor n\ge 4}$}
I've used a proof by cases to prove the first part ($\forall n\le 1$) which was ...

I am wondering if someone can provide some geometric intuition, or some simple way to visualize why
$$
c-d<a-b \implies b<a+d-c
$$
The way I have been trying to do this is to think of $a,b,c,d$ ...

I wish to solve this inequality:
$\sin (2x) \gt \sqrt 2 \sin (x)$
My approach:
I tried to isolate the $x$ on the left side by using the sine sum formula:
$2\sin(x)\cos(x) \gt \sqrt2\sin(x)$
then ...

My proof uses some calculus, and I was wondering and there are any other ways (namely a more elementary way).
Let (for $x\ge1$)$$g(x) = \frac{\ln^2x}{x}$$
Then taking the derivative:
$$g'(x) = \...

My question is so elementary. But, I don't know correct answer.
$f(n)$ and $f(n+\lambda)$ are finite functions and following fraction is convergent:
If $0≤ \lim_{n \to \infty}\frac {f(\lfloor ...

A basic-closed semialgebraic set in $\mathbb{R}^{n}$ is defined as $$ M=\lbrace x \in \mathbb{R}^{n} \mid f_{1}(x) \geq 0, \ldots, f_{m}(x) \geq 0 \rbrace$$ for some polynomials $f_{i}$ in $n$ ...

Sometimes in proofs related to limits I see the use of something like this: $$|a-b| = \mathbf{|(a-c)+(c-b)|} \leq |a-c|+|c-b|.$$
What is the logic of doing such thing apart from the reason that it ...

I'm solving the following trigonometric inequation ( $cos(x)-sin(x)+1 \gt 0$) on the interval $[0,2\pi]$. And I found that
$-\pi +2k \pi \lt x \lt {\pi\over 2} + 2k\pi $, $k\in \mathbb Z$
So, if I'...

A GRE prep question asks, "If $-13 < a < -2$ and $1 < b < 9$, which of the following could be equal to the product of $a$ and $b$?"
Potential answers: $-20$, $-18$, $-15$, $-14$, $-13$, $-...

Let $f: S \to \mathbb{R}$ be a Lebesgue integrable function on a measure space $(S,\Sigma,\mu)$, then we may have
\begin{align}
\|f\|_{L^1} &:= \int_S |f| d\mu \\
\|f\|_{L^2} &:= \left(\...

Generally,the length of the sum of two vectors is not equal to the sum of their lengths. To see this consider the vectors $u$ and $v$ as shown below.
By considering $u$ and $v$ as two sides of a ...

Prove that, given $a,b,c > 0$ and $n$ a positive integer,
$$\frac{a^n}{b+c}+\frac{b^n}{a+c}+\frac{c^n}{a+b} \geq \frac{a^{n-1}+b^{n-1}+c^{n-1}}{2}\ .$$
I've tried every rathole for hours on this ...

$x,y,z$ are positive real numbers such that $$xyz=1$$ Prove that $\dfrac{1}{y(x+y)}+\dfrac{1}{z(y+z)}+\dfrac{1}{x(z+x)} \geqslant \dfrac{3}{2}$. I have no idea how to solve this problem.
I've tried ...

If $abc=1$ for positive $a,b,c$, then $\sum\limits_{cyc}^{}{\dfrac{1}{b(a+b)}}\ge \dfrac{3}{2}$
I have tried the following,in decreasing order of success:
1)AM-GM:$a+b+c\ge 3$ and $ab+bc+ca\ge 3$
2)...

enter image description here
Solved 2nd part but how to solve first part?

A large, hollow sphere of internal radius R contains a smaller solid
sphere of radius r. Write an inequality which would indicate that the
small sphere has impacted the interior surface of the ...

I was just wondering if someone can help me with a real basic proof.
Prove that $-\frac{2n+1}{n+1} \leq 0 \forall n \in \mathbb N$.
Is it just enough to show that $-\frac{2n+1}{n+1} > 0$ cannot ...

I have a polynomial $P(A,B,C)$ where $A,B,C \in \mathbb{R}$ and $A>0,B>0$.
$$P(A,B,C)=C^2+A^2-2CA+4AB-2B+C-A$$
When will
$P(A,B,C)>0$
$P(A,B,C)=0$
$P(A,B,C)<0$
I have tried factorizing ...

Given two integers $n$ and $k$ such that $n\geq k+1$.
Can we find any relation between $\left\lfloor\dfrac{n}{k}\right\rfloor$ and $\left\lfloor \dfrac{n}{k+1}\right\rfloor$?
At first, I thought ...

Let $f(x)=ax^3+bx^2+cx+d$, where $a,b,c,d$ are real is an increasing function. Given that $3b^2<c^2$. Define $g(x)= af’(x) + bf’’(x) + c^2$. Also , define $p(x) = \int_x^m g(t)dt$. Classify $p(x)$ ...

The inequality for x is:
$(\log_3 x)^2 \lt \log_9( x^4)$
I plotted the graphs to find as answer: $1\lt x \lt 9$
My approach:
In my attempt to solve it, I used the logarithm properties to equal the ...

Can someone please explain the last three inequality expression to me, as I do not understand how these are obtained, in particular, why is the square root term not included in the inequality for $ \...

I have the following inequality on my class notes that I haven't been able to prove, I was even wondering if it is actually true:
$$
\forall a,b \in \mathbb{R}^{\ge0} \left( \left| \sqrt{a}-\sqrt{b} \...

Let $a$, $b$ and $c$ be positive numbers such that
$$ \dfrac{b+c}{a}+ \dfrac{c+a}{b}+ \dfrac{a+b}{c}
= 2\left(\dfrac{1}{ab}+\dfrac{1}{bc}+\dfrac{1}{ac} \right).$$
Prove that
$$a^2+b^2+c^2+3\ge 2(ab+...

I am a bit confused about the following step in a proof:
for all $\epsilon>0$ we have that $b<c+2 \epsilon$, we can conclude that $b\leq c$.
The proof goes as follows:
We wish to prove that if ...

This might be a simple and even stupid question. Why does the following equation hold true?
\begin{align}
\|a+b\|^2 = \|a\|^2 + \|b\|^2 + 2\langle a,b \rangle
\end{align}
where the norm can be ...

Prove that every $k$-chromatic graph has size $m\geq \binom k2$.
Here is what I know:
Let $G$ be a $k$-chromatic graph, that mean $\chi(G)=k$. Thus $G$ must have a subgraph of a complete $k$-partite ...

Can someone explain me why this implication is true:
$$\sqrt3 \gt {3\over 2} \implies {\pi\over 6} \lt \operatorname{arccot}({3 \over 2}) $$
where $\operatorname{arccot}$ is defined as $\...

The problem:
Show that, as $n\rightarrow + \infty$, $\sum_{k=1}^{n/2} \dbinom{n}{k}\alpha^{k(n-k)} \rightarrow 0$ for $0<\alpha<1$.
What I tried :
I tried to use $\dbinom{n}{k} \leq \dbinom{...

Find the area of the region bounded by $x^2+y^2 \le 144$ and $\sin(2x+3y) \le 0$
My try:
we have $$\sin(2x+3y) \le 0$$ when
$$-12 \le 2x+3y \le -3 \pi$$
$$ -2 \pi \le 2x+3y \le -\pi$$
$$-\pi \...

Problem:
A vertex of one square is pegged to the centre of an identical square, and the overlapping area is blue. One of the squares is then rotated about the vertex and the resulting overlap is ...

I'm familiar with extraneous roots.
For example $\sqrt{x} = x - 2$
We solve it by squaring both sides
\begin{align*}
& \implies x = x^2 - 4x + 4\\
& \implies x^2 - 5x + 4 = 0\\
& \...

Let $N$ variables $x_i \ge 0$ but not all of them be zero. One may fix $\sum_{i=1}^N x_i = 1$.
There are $P$ equations which need to be solved, with coefficients $a^k_i$ indexed with superscripts $k =...

It doesn't look like it can be solved by the normal substitution method:
It is given in a paper and the solution is given as $m \geq 2 + \sqrt(3)$.

I'm having trouble proving the following:
Let $f$ be analytic on $D(0,R)$ and suppose there is a real constant A for which $\operatorname{Re}f(z) < A$ for all $z \in D(0,R)$. If $f(0) = 0$, show ...

For $a \ge0$ find maximum of $P=$$3x\sqrt{a-y^2}-3y\sqrt{a-x^2}+4xy+4\sqrt{a^2-ax^2-ay^2+x^2y^2}$
I think maximum of P when x=-y but i don’t know how to make it reasonable

If $a,b,c$ positives and $ \frac{1}{a} + \frac{1}{b} + \frac{1}{c} =3,$ I have to prove $\frac {1}{2 a^2+b^2} +\frac {1}{2 b^2+c^2} +\frac {1}{2 c^2+a^2} \le 1.$
Since we have $ \frac{1}{a} + \frac{1}{...

Prove that
$$\frac{1}{x+\sqrt{x^2+2}}<e^{x^2}\int\limits_x^{\infty}e^{-t^2} \, \text dt \le\frac{1}{x+\sqrt{x^2+\displaystyle\tfrac{4}{\pi}}}, \space (x\ge 0)$$

Let $x,y\in \Bbb R$.\begin{align}
|x|-1\le|y| & \iff|x|-1\le y \text{ or }|x|-1\le -y \\
& \iff |x|-1\le y \text{ or } -|x|+1\ge y \\
& \implies |x|-1\le 1-|x| \\
& \implies |x|\...

If $x^2+xy+y^2=\frac{27}{7}$. Then what is the maximum value of $3x+2y$ ?

Let $\{\omega_i\}_{i\ge 1}$ be a bounded sequence of positive numbers and consider the following series of nested integrals
$$
S=\sum_{n=1}^\infty \int_0^{t} \cos(\omega_1t_1)\left(\int_0^{t_1} \cos(\...

Solve the inequality for $x$:
$$e^x-\ln x\leq\frac{e}{x}$$
I know that answer of these inequality is $0<x\leq1$ but I need the proof.

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- complex-analysis
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- elementary-number-theory
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