Questions on proving, manipulating and applying inequalities.

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2
votes
2answers
40 views

For which values of $a$ does $d\ge ac\ln c\implies d\ge c\ln d$?

For which values of $a>0$ is it true that for all $c,d>0$, $\hspace{.2 in}d\ge ac\ln c\implies d\ge c\ln d$? I believe that this is true for $a\ge2$, (see Showing if $n \ge 2c\log(c)$ then $n\...
3
votes
1answer
38 views

Is there an $\alpha\in\mathbb{R}^m$, such that $\alpha_i > 0$ and $A\alpha\in S$?

$A$ is a real $n\times m$ matrix and set $S\subseteq \mathbb{R}^n$ is defined as $$S = \{(x_1,\dots, x_n)\in \mathbb{R}^n\mid \forall(i,j)\in I.\; x_i< x_j\}\text{,}$$ where $I$ is a possibly empty ...
2
votes
0answers
31 views

Partitioning a set of integers (with Alice and Bob)

Let $ d_1,\ldots,d_n \in \mathbb{N}_{\ge 2} $ (not necessarily distinct) be given. Define $ D:=\operatorname{lcm}(d_1,\ldots,d_n) $ and $ d:=\sum_{i=1}^n d_i $. (1) Alice claims that whenever $ \...
5
votes
3answers
83 views

prove this inequality $a^n>b^n+c^n$

We know a,b and c are positive and $a^2=b^2+c^2$ How we can conclude this inequality: $a^n>b^n+c^n$ , $n>2$ I tried Binomial Theorem but I can't prove this. Thanks
0
votes
0answers
21 views

Uniform Boundary for S.D.E with Lipschitz Coefficients

Edit of progress: Since the SDE is linear, I got a solution in the form of $e^\int...$$\cdot e^\int$. By Jensen's inequality I can change the order of the left factor to have the integral on the ...
4
votes
1answer
62 views

How prove this inequality

Let $a,b,c>0$ and such $a+b+c=1$ show that $$\dfrac{1}{1-a}+\dfrac{1}{1-b}+\dfrac{1}{1-c}\ge \dfrac{1}{ab+bc+ac}+\dfrac{1}{2(a^2+b^2+c^2)}$$ Let $p=a+b+c=1,ab+bc+ac=q,abc=r$ $$\Longleftrightarrow -...
2
votes
1answer
35 views

The first step in the proof of the Pólya-Vinogradov Inequality.

The well-known Pólya-Vinogradov Inequality states: $\forall m, n \in \mathbb{N}: \displaystyle \left|{\sum_{k \mathop = m}^{m+n} \left({\frac k p}\right)}\right| < \sqrt p \ \ln p$,. where $(k/p)$...
1
vote
2answers
52 views

What's the mistake on my answer for this inequality $ \frac{\left(x+1\right)}{\sqrt{x^2+1}}>\frac{\left(x+2\right)}{\sqrt{x^2+4}} $

Good evening to everyone! I have the following inequality: $$ \frac{\left(x+1\right)}{\sqrt{x^2+1}}>\frac{\left(x+2\right)}{\sqrt{x^2+4}} $$. I don't know what's wrong with my answer: $$ \frac{\...
2
votes
3answers
52 views

Concept of roots in Quadratic Equation

$a$ , $b$, $c$ are real numbers where a is not equal to zero and the quadratic equation \begin{align} ax^2 + bx +c =0 \end{align} has no real roots then prove that $c(a+ b+ c)>0$ and $a(a+ ...
8
votes
2answers
126 views

Prove inequality $\sqrt{\frac{1}n}-\sqrt{\frac{2}n}+\sqrt{\frac{3}n}-\cdots+\sqrt{\frac{4n-3}n}-\sqrt{\frac{4n-2}n}+\sqrt{\frac{4n-1}n}>1$

For any $n\ge2, n \in \mathbb N$ prove that $$\sqrt{\frac{1}n}-\sqrt{\frac{2}n}+\sqrt{\frac{3}n}-\cdots+\sqrt{\frac{4n-3}n}-\sqrt{\frac{4n-2}n}+\sqrt{\frac{4n-1}n}>1$$ My work so far: 1) $$\...
8
votes
1answer
141 views

range of a fractional function

I encountered the following problem and was hinted to consider the edge case. Determine all possible values of $$S = \frac{a}{a+b+d}+\frac{b}{a+b+c}+\frac{c}{b+c+d}+\frac{d}{a+c+d}$$ where $a$, $b$, ...
0
votes
2answers
1k views

Combining inequalities into one inequality

Let's say we are given $a$, $b$, $d$ with $1 \leq a, b, d \leq 1000$ and inequalities $x \geq a$, $y \geq b$, and $a+b < x + y \leq a+b+d$. I need to combine all this and the following into one ...
-4
votes
0answers
30 views

proving: Let x, y, p, q be positive numbers with 1/p + 1/q = 1. Prove that xy ≤ x^p/ p + y^q/q [on hold]

4) Let $x, y, p, q$ be positive numbers with $ 1/p + 1/q = 1$. Prove that $$xy ≤ x^p/ p + y^q/q $$
1
vote
4answers
44 views

$\cos2\theta +\cos\theta +k = 0 $ - set of all values of $k$ for which there is a solution

The set of all values of $k$ (real), such that the equation $\cos2\theta +\cos\theta +k = 0 $ admits a solution for $\theta$ is? MY ATTEMPT: I substituted $\cos2\theta$ with $2\cos^2\theta - 1 $. On ...
0
votes
0answers
15 views

How to find the maximal length of a system?

Let P be the set of $(a,b,c)^t \in \mathbb{R}$ which satisfies the following inequalities: $-2a+b+c \leq 4$ $a-2b + c \leq 1$ $2a + 2b-c \leq 5$ where $a \geq 1 $, $b \geq 2$, and $c \geq 3 $. ...
-1
votes
2answers
22 views

Solve an easy inequality

How can I find the solution of the following inequality analytically in terms of $x$? $$i_1x^3+i_2x^2+i_3x \ge 0$$ where $i_k$ is a constant value.
6
votes
2answers
116 views

The number of positive integer solutions to the equation $x_1+2x_2+…+nx_n=n^2.$

Let $n \ge 2, n \in \mathbb N$. $A_n$ denotes the number of positive integer solutions to the equation $$x_1+2x_2+...+nx_n=n^2.$$ Prove inequality $$\frac{n^n(n-1)^{n-1}}{2^{n-1}\left(n!\right)^...
1
vote
3answers
57 views

Inequality with square root $x+\sqrt{x^2-10x+9}\ge \sqrt{x+\sqrt{x^2-10x+9}}$

Good morning to everyone! The inequality is the following:$$ x+\sqrt{x^2-10x+9}\ge \sqrt{x+\sqrt{x^2-10x+9}} $$. I don't know how to solve it. Here's what I tried: $$x+\sqrt{x^2-10x+9}\ge \sqrt{x+\...
3
votes
2answers
73 views

The number of positive integer solutions to the equation $x_1+x_2+…+x_n=n^2.$

I'm working on this problem. To solve it I need this lemma: Let $n\ge2, n\in \mathbb N$. Let $X$ be the number of solutions in positive integers to the equation $x_1+x_2+...+x_n=n^2$. Let $Y$ be ...
0
votes
2answers
48 views

Where am I going wrong with this inequality?

Good evening to everyone! I got this inequality: $$\frac{x-1}{x-2}<\frac{x-1}{x}.$$ If I try to solve this, it gives me $$ \frac{x-1}{x-2}<\frac{x-1}{x} \Rightarrow \frac{x-1}{x-2}-\frac{x-1}{x}...
0
votes
0answers
27 views

Solutions to Diophantine Moving Window Inequations

I am interested in finding the number of non-negative integer solutions, $N(m,h,u)$, to this system of inequations $$ \left\{ \matrix{ 0 \le x_{\,0} + x_{\,1} + \cdots + x_{\,m} \le u \hfill \...
3
votes
5answers
234 views

Sum of real powers: $\sum_{i=1}^{N}{x_i^{\beta}} \leq \left(\sum_{i=1}^{N}{x_i}\right)^{\beta}$

Let $\{x_i\}_{i=1}^{N}$ be positive real numbers and $\beta \in \mathbb{R}$. Can we say that: $$ \sum_{i=1}^{N}{x_i^{\beta}} \leq \left(\sum_{i=1}^{N}{x_i}\right)^{\beta}$$ I know that this holds if $...
2
votes
1answer
47 views

Prove inequality $\sqrt[3]{\frac{a^3+b^3}{2}}+\sqrt[3]{\frac{b^3+c^3}{2}}+\sqrt[3]{\frac{c^3+d^3}{2}}+\sqrt[3]{\frac{d^3+a^3}{2}} \le 2(a+b+c+d)-4$

Let $a,b,c,d$ positive real numbers, such that $$\frac1a+\frac1b+\frac1c+\frac1d=4.$$ Prove inequality $$\sqrt[3]{\frac{a^3+b^3}{2}}+\sqrt[3]{\frac{b^3+c^3}{2}}+\sqrt[3]{\frac{c^3+d^3}{2}}+\sqrt[3]...
3
votes
2answers
80 views

Find $\lim\sqrt{\frac{1}n}-\sqrt{\frac{2}n}+\sqrt{\frac{3}n}-\cdots+\sqrt{\frac{4n-3}n}-\sqrt{\frac{4n-2}n}+\sqrt{\frac{4n-1}n}$ [on hold]

The question arise in connection with this problem Prove that $$\lim_{n\rightarrow \infty}\sqrt{\frac{1}n}-\sqrt{\frac{2}n}+\sqrt{\frac{3}n}-\cdots+\sqrt{\frac{4n-3}n}-\sqrt{\frac{4n-2}n}+\sqrt{\...
2
votes
2answers
39 views

Inequality with $\arctan$

I try to show that $x \cdot \arctan\left( \frac{1}{x^2} \right)$ is monotonically decreasing, but I can't solve this inequality with $\arctan$. Can somebody show me how to do this? $$ x \in [1, \...
11
votes
6answers
5k views

Proving $(1 + 1/n)^{n+1} \gt e$

I'm trying to prove that $$ \left(1 + \frac{1}{n}\right)^{n+1} > e $$ It seems that the definition of $e$ is going to be important here but I can't work out what to do with the limit in the ...
0
votes
0answers
43 views

Probabilistic Modeling Parameters Request

Before posing the question itself, it is indispensable to give the definition from which it arises. First of all, let us restrict our attention to the vectors $\overrightarrow{x} = (x_{1},x_{2},\ldots,...
3
votes
0answers
29 views

Can you please comment on and check these couple of induction proofs?

So the following statements need to be proved: 1) $(1+a_1)(1+a_2)\cdots(1+a_n)>1+a_1+a_2+\cdots+a_n$ for $a_i>0,(i=1,2,\ldots,n)$ and $n\ge2$ 2) $(1-a_1)(1-a_2)\cdots(1-a_n)<1-(a_1+a_2+\...
2
votes
1answer
36 views

Greatest integer function inequality solution

If $\lfloor x+\lfloor x\rfloor\rfloor \le 2$ then what are all the possible values of $x$? Please tell how to proceed and tell the solution by graph method of possible. Explain how to sketch the graph ...
3
votes
0answers
35 views

On the conjectured nonexistence of even almost perfect numbers (other than powers of two) and odd perfect numbers

(Note: This question has been cross-posted from MO.) Let $\sigma(a) = \sigma_{1}(a)$ be the sum of the divisors of the positive integer $a$. A number $M$ is called almost perfect if $\sigma(M) = 2M ...
1
vote
3answers
91 views

Proof of AM-GM inequality for $n=3$: $\frac{a+b+c}{3}\geq\sqrt[3]{abc}$ [duplicate]

Sorry for bad formatting, I couldn't mark the 3rd root on the right hand side... I've figured this out into the point where (and yeah, the problem is to prove that this applies to all non-negative ...
9
votes
0answers
189 views

Bounds on derivative of real positive coefficient polynomial satisfying certain properties

While thinking about this question of Clin, I wanted to consider the polynomial: $P(z) = 1+x_1z+x_2z^2+\cdots+x_nz^n$, satisfying: (I) $1\geq x_{1}\geq x_2\geq\cdots\geq x_{n}\geq0$ and $\...
5
votes
1answer
37 views

Is it true that $\left(\frac{a^2+b^2+c^2}{a+b+c}\right)^{a+b+c}≥a^ab^bc^c$?

Let $a,b,c\in\mathbb{R}_{>0}$. Is it true that: $$ \left(\frac{a^2+b^2+c^2}{a+b+c}\right)^{a+b+c}≥a^ab^bc^c $$ I remarked that the inequality is (a bit weirdly) homogeneous, but couldn't use it. ...
-1
votes
1answer
35 views

Solution of the inequality $2ec{\sqrt{ad}}\lt dc^2+ae^2$

I just want to find a way to prove that the inequality $2ec{\sqrt{ad}}\lt dc^2+ae^2$ is true because I need it for a prove. Thanks for your help!
3
votes
3answers
36 views

inequality proof

I have come across a problem: Let $a,b$ and $c$ be real numbers where $a > b$. Prove that if $ac \leq bc$, then $c \leq 0$. I tried using the Indirect proof If $a > b$, and $c > 0$, ...
2
votes
1answer
32 views

Upper bound for $\sum_{i=1}^{N}{x_i^{\beta_i}}$

$\{x_i\}_{i=1}^{N}$ a sequence of positive real numbers and $\{\beta_i\}_{i=1}^{N} $ are real numbers such that $\underset{1 \leq i \leq N}\min{\beta_i}$ > 1. Is it possible to find an upper bound of ...
2
votes
2answers
129 views

Let $a, b, c>0$, such that $a+b+c=1$, prove that $\frac{a}{(b+c)^2}+\frac{b}{(a+c)^2}+\frac{c}{(a+b)^2}\ge\frac{9}{4}$ [on hold]

Let $a, b, c>0$, such that $a+b+c=1$, prove that: $$\frac{a}{(b+c)^2}+\frac{b}{(a+c)^2}+\frac{c}{(a+b)^2}\ge\frac{9}{4}$$
0
votes
3answers
56 views

Solve $3^{2x}-3^x\geq2$

How to solve: $$3^{2x}-3^x\geq2$$ I tried with $y=3^x$ and solved as equation: $y^2-y-2 \geq 0$ and I get: $y<2$ $y>-1$ How should I proceed?
1
vote
3answers
62 views

How to solve $x<\frac{1}{x+2}$

Need some help with: $$x<\frac{1}{x+2}$$ This is what I have done: $$Domain: x\neq-2$$ $$x(x+2)<1$$ $$x^2+2x-1<0$$ $$x_{1,2} = \frac{-2\pm\sqrt{4+4}}{2} = \frac{2 \pm \sqrt{8}}{2} = \frac{...
4
votes
2answers
65 views

If $x,y,z>0\;,$ and $xyz=1$ Then minimum value of $\frac{x^2}{y+z}+\frac{y^2}{z+x}+\frac{z^2}{x+y}$

If $x,y,z>0\;,$ and $xyz=1$ Then find the minimum value of $\displaystyle \frac{x^2}{y+z}+\frac{y^2}{z+x}+\frac{z^2}{x+y}$ $\bf{My\; Try::}$Using Titu's Lemma $$\frac{x^2}{y+z}+\frac{y^2}{z+x}+\...
-1
votes
0answers
12 views

Convex subset and linear equalities

Let S denote the set of $(a,b,c)$ $\in$ ${\mathbb{R^3}}$ which satisfies the following equalities: $-2a+b+c \leq 4 $ $a-2b+c \leq 1 $ $2a+2b-c \leq 5 $ $ a \geq 1 $ $ b \geq 2 $ $ c \geq 3 $ ...
-3
votes
0answers
42 views

Lets a, b, c>0 such that a+b+c=6, prove that: [on hold]

Let a, b, c>0 such that a+b+c=6, prove that: $$\sum_{cyc} \frac{a^7+b^7}{a^5+b^5}\ge12$$
3
votes
0answers
40 views

Where does the premise of this idea come from?

Let $x$ , $y$ be positive real numbers. Prove the inequality $$x^ y + y^x \ge 1$$ This is the solution provided by my textbook: Where does this first idea (proving that $a^b \ge \frac{a}{a+ b - ...
4
votes
1answer
185 views
+50

$f \in C^2(\mathbb R)$ , $(f(x))^2 \le 1$ ; $(f'(x))^2+(f''(x))^2 \le 1 $ ; then is $(f(x))^2+(f'(x))^2 \le 1 $?

Let $f \in C^2(\mathbb R)$ be such that $$(f(x))^2 \le 1 ; (f'(x))^2+(f''(x))^2 \le 1 , \forall x \in \mathbb R$$ Then is it true that $(f(x))^2+(f'(x))^2 \le 1 , \forall x \in \mathbb R$ ? I ...
0
votes
3answers
42 views

Values of x that satisfy this inequality

$||x-2|-3|>1$. I have made some cases but still the complete values don't come,plus I don't have any idea of how to sketch the graph for the lefy hand side of the inequality.
1
vote
1answer
39 views

Lagrange multipliers: when is local extremum a global extremum?

Consider the following Olympiad problem from the IMO shortlist: Let the real numbers $a,b,c,d$ satisfy the relations $a+b+c+d=6$ and $a^2+b^2+c^2+d^2=12.$ Prove that: $36 \leq 4 \left(a^3+b^3+c^3+d^...
13
votes
3answers
463 views

Prove $\frac{xy}{5y^3+4}+\frac{yz}{5z^3+4}+\frac{zx}{5x^3+4} \leqslant \frac13$

$x,y,z >0$ and $x+y+z=3$, prove $$\tag{1}\frac{xy}{5y^3+4}+\frac{yz}{5z^3+4}+\frac{zx}{5x^3+4} \leqslant \frac13$$ My first attempt is to use Jensen's inequality. Hence I consider the function $...
0
votes
1answer
14 views

The validity of normalization in homogeneous inequalities?

I'm going through a book on inequalities right now, and the author describes normalization with the following example. Prove that $a^2 + b^2 + c^2 \ge ab + bc + ca$ Of course the fundamental ...
1
vote
3answers
112 views

Show the roots of the quadratic equation $z^2 +bz+ c = 0$ lie in or on the unit circle

So I need a little help with the following: Considering separately the cases of real and complex roots show that the roots of the quadratic equation $z^2 +bz+ c = 0$ lie in or on the unit circle (i.e....
5
votes
1answer
250 views
+50

Prove $\sqrt[2]{x+y}+\sqrt[3]{y+z}+\sqrt[4]{z+x} <4$

$x,y,z \geqslant 0$ and $x^2+y^2+z^2+xyz=4$, prove $$\sqrt[2]{x+y}+\sqrt[3]{y+z}+\sqrt[4]{z+x} <4$$ A natural though is that from the condition $x^2+y^2+z^2+xyz=4$, I tried a trig substitutions ...