Questions on proving and manipulating inequalities.

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43 views

How to show that $\frac {q + \frac {1}{2}}{p - \frac {1}{2}} > \sum_{i = p}^q\frac {1}{i}$ if $q\ge p > 0?$

How to show that : $$\frac{2q+1}{2p-1}>\sum_{i=p}^q\frac{1}{i}$$ if $q\ge p>0$
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2answers
55 views

Prove Schwarz inequality in $R^2$

Can someone please show me how you would prove the following in $R^2$ $\int f(x)* g(x) dx \leqslant \int f(x)^2 dx * \int g(x)^2 dx $ starting from $\int [\lambda*f(x) - g(x)]^2 dx \geqslant ...
3
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2answers
54 views

Proving an convexity-looking inequality

If $0 \le \alpha \le 1$ and $0 \le \lambda \le 1$, then $$\lambda^\alpha x^\alpha +(1-\lambda^\alpha) y^\alpha \ge (\lambda x + (1-\lambda)y)^\alpha$$ whenever $0 \le y \le x$. This looks ...
8
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6answers
559 views

How to solve the inequality $x^2>10$ using square roots?

Solve the inequality: $$x^2>10$$ How am I supposed to do this? It doesn't make sense when I take into account that if $x^2=10$ then $x=+\sqrt{10}$ and $x=-\sqrt{10}$ But how am I supposed to ...
5
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1answer
60 views

Prove $\sin^{2m}\alpha\cdot\cos^{2n}\alpha\leq\frac{m^m n^n}{(m+n)^{(m+n)}}$

If $n$ and $m$ are natural numbers, Prove: $$\sin^{2m}\alpha\cos^{2n}\alpha\leq\frac{m^mn^n}{(m+n)^{(m+n)}}$$ Additional info:We should only use AM-GM inequality.We can use Trigonometry ...
2
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4answers
57 views

Prove the inequality $2\sqrt{x}\ge3-\frac1x $

Given that $x\gt0$ prove the following inequality: $2\sqrt{x}\ge3-\frac1x $ I have done it using calculus but how can I do it using elementary methods?Thanks!!
9
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1answer
75 views

If a polynomial has only real zeros then $a_{0}+a_{1}+\cdots+a_{n}\le\frac{(n+1)^n}{\binom{n}{s}(n-s)^{n-s}(s+1)^s}\cdot\max_{k}a_{k}$

Question: For all real polynomials $P(x)=a_{0}+a_{1}x+\cdots+a_{n}x^n$ of degree $n$, with only real zeros,we have ...
1
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1answer
49 views

Regarding 'non-square- free ' numbers.

Call an integer 'n' that is not a square or a prime power or a square-free a 'square-in'.Let n be square-in. Then between n and (2 n) is there another square-in? This is a kind of 'variation' on ...
2
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2answers
55 views

Trigonometric inequation $\sin x \ne \sin y$

How can I solve the following trigonometric inequation? $$\sin\left(x\right)\ne \sin\left(y\right)\>,\>x,y\in \mathbb{R}$$ Why I'm asking this question... I was doing my calculus homework, ...
2
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2answers
52 views

Writing solutions of inequalities: $3<x$ versus $x>3$

My son wrote a solution to a number line graph as 3 < x instead of what his teacher said was the correct answer of x > 3. When he brought his paper back in to bring it up he was told that the ...
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1answer
46 views

Show That $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ And $\frac{x^2}{A^2} - \frac{y^2}{B^2} = 1$ Are Orthogonal Trajectories

Show that the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ and the hyperbola $\frac{x^2}{A^2} - \frac{y^2}{B^2} = 1$ are orthogonal trajectories if $A^2< a^2$ and $a^2-b^2 = A^2+B^2$. What I've ...
2
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1answer
36 views

How prove that $q \geq b+d$ for $ad-bc = 1$ and $\frac{a}{b} > \frac{p}{q} > \frac{c}{d}$?

Let $a,b,c,d,p$, and $q$ be natural numbers such that $ad-bc = 1$ and $\frac{a}{b} > \frac{p}{q} > \frac{c}{d}$. How prove that $q \geq b+d$?
2
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1answer
58 views

Using logical OR to combine inequalities.

I have a physical system that must satisfy one of two inequalities: $x\leq y$ OR $p\leq q$ But not necessarily both simultaneously. Is there a way to combine this into a single inequality? ...
6
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5answers
199 views

Alternate Proof for $e^x \ge x+1$

This is just a standard problem from my high school's calculus text, but my proof seems sort of off. This is it: Let $f(x) = e^x$. The tangent line of $f(x)$ at $x=0$ is $g(x)=x+1$. Since $f''(x_0) ...
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2answers
38 views

Radical Inequality

$\sqrt{2x-1}$ + $\sqrt{3x-2}$ > $\sqrt{4x-3}$ + $\sqrt{5x-4}$ I have attempted to solve this by squaring each side, resulting in $5x + 2\sqrt{2x-1}\sqrt{3x-2} - 3 > 9x + 2\sqrt{(4x-3)(5x-4)} - 7 ...
5
votes
1answer
52 views

How prove that $ x+y+z>4$ for $ a+b+c=4$ and $ ax+by+cz=xyz$?

Given positive reals $ a,b,c,x,y,z$ such that $ a+b+c=4$ and $ ax+by+cz=xyz$. How prove that $ x+y+z>4$?
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3answers
361 views

How to solve inequalities with absolute values on both sides?

If you have an inequality that has two absolute value bars like $|4x+1|<|3x|$, how do you go about doing this? I know that if $4x+1<3x$, then those $x$'s will work but what else do I do? I think ...
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2answers
31 views

Regarding square-free numbers and their doubles.

Is it true that between any non-prime square-free number and it's double is another non-prime square-free number?
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5answers
147 views

How to solve this inequality? From MSU entrance exam '66

$\frac{\log _{10}\left(2\right)}{\log _{10}\left(\sin \left(x\right)\right)}\le \frac{\log _{10}\left(4\sin ^2\left(x\right)\right)}{\log _{10}\left(\sin \left(x\right)\right)}$ From the title. Not ...
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1answer
34 views

How prove $ \frac{\cos x\cos y-4}{\cos x+\cos y-4}\le1+\frac{1}{2}\cos(\frac{x+y}{\cos x+\cos y-4}) $?

For any $x,y\in[0,\frac{\pi}{2}]$ , how prove the inequality $\frac{\cos x\cos y-4}{\cos x+\cos y-4}\le1+\frac{1}{2}\cos(\frac{x+y}{\cos x+\cos y-4})$?
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1answer
23 views

$S$, $I$, $O$ are circumcenter, incenter and orthocenter then $SO\ge IO \sqrt2$

Let $S$, $I$ and $O$ be the circumcenter, incenter and orthocenter of $\triangle ABC$ then prove that $SO\ge IO \sqrt2$, or equivalently $SO^2\ge 2IO^2$. I was able to derive an expression for $SO^2$ ...
6
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2answers
139 views

Prove two of $\frac{2}{a}+\frac{3}{b}+\frac{6}{c}\geq 6,\frac{2}{b}+\frac{3}{c}+\frac{6}{a}\geq 6,\frac{2}{c}+\frac{3}{a}+\frac{6}{b}\geq 6$ are True

if $a,b,c$ are positive real numbers that $a+b+c\geq abc$, Prove that at least $2$ of following inequalities are true. $\frac{2}{a}+\frac{3}{b}+\frac{6}{c}\geq 6, ...
1
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5answers
61 views

Inequality involving a finite sum

this is my first post here so pardon me if I make any mistakes. I am required to prove the following, through mathematical induction or otherwise: $$\frac{1}{\sqrt1} + \frac{1}{\sqrt2} + ...
3
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2answers
74 views

Prove the inequality $n!\lt n^{n+\frac12} e^{-n+1}$ [closed]

Prove the following inequality: $$n!\lt n^{n+\frac12} e^{-n+1}.$$ Try to avoid induction if possible. Thanks!!
2
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3answers
48 views

if $ax-2by+cz=0$ and $ac-b^2>0$ , Prove $zx-y^2\leq0$

for real numbers like $a,b,c,x,y,z$ that $ax-2by+cz=0$ and $ac-b^2>0$ Prove:$$zx-y^2\leq0$$ Additional info: The Proof should be by contradiction.we can use Cauchy , AM-GM and other simple ...
2
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1answer
34 views

How find all $n\in \mathbb N$ such that $\cot \left(\frac{x}{2^{n+1}}\right)-\cot(x)>2^n$?

How find all $n\in \mathbb N$ such that $\cot \left(\frac{x}{2^{n+1}}\right)-\cot(x)>2^n$ for $x \in (0,\pi)$?
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0answers
25 views

Proof of inequality related to complex function

Let $f(z)$ be a nonsingular complex function whose domain is $D=\{z\in C\ ;\ |z|<R\}$. $f(z)$ satisfies $f(0)=1$, and Re$f(z)$ is positive everywhere on $D$. Let $g(z)$ be $$g(z) = ...
0
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1answer
42 views

Prove $\sum\limits_{cyc}\frac{(b+c-a)^4}{a(a+b-c)}\geq ab+bc+ca$

if $a,b,c$ are positive real numbers,Prove:$$\sum\limits_{cyc}\frac{(b+c-a)^4}{a(a+b-c)}\geq ab+bc+ca$$ Additional info: Problem should be solved with AM-GM inequality only. Things i have tried ...
8
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2answers
119 views

Prove $(1-a)(1-b)(1-c)(1-d)\geq abcd$ if $a^2+b^2+c^2+d^2=1$

Let $a,b,c,d\geq0$, $a^2+b^2+c^2+d^2=1$ Prove $\displaystyle (1-a)(1-b)(1-c)(1-d)\geq abcd$ I mutiplied both with $\displaystyle (1+a)(1+b)(1+c)(1+d)$ to use $1-a^2=b^2+c^2+d^2$ and try using the ...
5
votes
1answer
62 views

Prove $\frac{(xy)^7}{x^8+(xy)^7+y^8}+\frac{(yz)^7}{y^8+(yz)^7+z^8}+\frac{(zx)^7}{z^8+(zx)^7+x^8}\leq1$

If $x,y,z$ are positive real numbers that $xyz=1$ , Prove a) $\frac{xy}{x^8+xy+y^8}+\frac{yz}{y^8+yz+z^8}+\frac{zx}{z^8+zx+x^8}\leq1$ ...
0
votes
0answers
24 views

Minkowski inequality for $0<p<1$

I'm trying to prove this, $$\left ( \sum_{i=1}^{n}(x_i+y_i)^p \right) \geq \left ( \sum_{i=1}^{n}(x_i)^p \right)^\frac{1}{p} + \left ( \sum_{i=1}^{n}(y_i)^p \right)^\frac{1}{p} $$ for $0<p<1$. ...
3
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5answers
59 views

Determine variables that fit this criterion…

There is a unique triplet of positive integers $(a, b, c)$ such that $a ≤ b ≤ c$. $$ \frac{25}{84} = \frac{1}{a} + \frac{1}{ab} + \frac{1}{abc} $$ Just having trouble with this Canadian Math ...
6
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1answer
66 views

How to show $ \left(\frac{1-x}{2}\right)^p+\left(\frac{1+x}{2}\right)^p \leq \frac{1+x^p}{2}$ [duplicate]

When $p\geq 2$ and $0\leq x\leq1$, how does one show the inequalities $$ \left(\frac{1-x}{2}\right)^p+\left(\frac{1+x}{2}\right)^p \leq \frac{1+x^p}{2}$$ and $$ 2(1+x^p)\leq (1+x)^p + (1-x)^p \ ?$$ ...
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2answers
42 views

Upper bound on the rational function of $z$ in terms of $|z|$

Show that: $$\frac{|2z^2-5|}{|z^2+1||z^2+4|} \le \frac{2|z|^2+5}{(|z|^2-1)(|z|^2-4)}$$ I started by considering that for the above to hold $|2z^2-5|\le (2|z|^2+5)$, and $|z^2+1|\ge (|z|^2-1)$, ...
2
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1answer
150 views

How prove this ineuality$\sqrt{x+1}+\sqrt{y+1}+\sqrt{z+1}\le\sqrt{xy+yz+zx+9}$

let $$x,y,z\in(-1,1), x+y+z=-xyz$$ show that $$\sqrt{x+1}+\sqrt{y+1}+\sqrt{z+1}\le\sqrt{xy+yz+zx+9}$$ This problem is my frends ask me,I remenber this is old inequality,But Now I can't it ...
6
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2answers
101 views

Prove $\frac{a^2+b^2+c^2}{ab+bc+ca} + 8\frac{abc}{(a+b)(b+c)(c+a)} \ge 2$

Let $a,b,c>0$, prove that $$\frac{a^2+b^2+c^2}{ab+bc+ca}+\frac{8abc}{(a+b)(b+c)(c+a)}\ge 2.$$ I tried using the equality $(a+b)(b+c)(c+a)=(a+b+c)(ab+bc+ca)-abc$ and the Schur inequality but it's ...
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2answers
47 views

Given $(x+3)(y−4)=0$, what is the relationship between $xy$ and $-12$?

Given $(x+3)(y−4)=0 $ Quantity $A = xy $ Quantity $B = -12 $ A Quantity $A$ is greater. B Quantity $B$ is greater. C The two quantities are equal. D The relationship cannot be determined from ...
3
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2answers
61 views

Prove that $\frac{a^3}{x} + \frac{b^3}{y} + \frac{c^3}{z} \ge \frac{(a+b+c)^3}{3(x+y+z)}$ a,b,c,x,y,z are positive real numbers.

I stumbled upon it on some olympiad papers. Tried to AM>GM but didn't get any idea to move forward.
2
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1answer
67 views

How prove $(\ln{\frac{1-\sin{xy}}{1+\sin{xy}}})^2 \geq \ln{\frac{1-\sin{x^2}}{1+\sin{x^2}}}\ln{\frac{1-\sin{y^2}}{1+\sin{y^2}}}$

How prove that if $x, y \in (0,\sqrt{\frac{\pi}{2}})$ and $x \neq y$, then $(\ln{\frac{1-\sin{xy}}{1+\sin{xy}}})^2 \geq \ln{\frac{1-\sin{x^2}}{1+\sin{x^2}}}\ln{\frac{1-\sin{y^2}}{1+\sin{y^2}}}$?
3
votes
3answers
76 views

Proof by induction: $n$th Fibonacci number is at most $ 2^n$

I'm trying to find the proof by induction of the following claim: For all $n\in\mathbb N$, $\operatorname{fibonacci}(n) \le 2^n$ My Proof: Base case: $n = 1$ $\operatorname{fibonacci}(1) \le 2^ 1$ ...
4
votes
1answer
99 views

How find this maximum and minimum of the value $\sum_{i=1}^{n-1}[x_{i+1}-x_{i}]$

Question: let $x_{1},x_{2},\cdots,x_{n}\in \mathbb{R}$,and Assume that the following two sets are equivalent; $$\{[x_{1}],[x_{2}],[x_{3}],\cdots,[x_{n}],\}=\{1,2,3,\cdots,n\},n\ge 2 $$ ...
2
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0answers
75 views

Diophantine inequality that comes up after Vieta Jumping Hurwitz technique

I am blaming this on Prove the equality EDITTTTT: allowing $x_1 \geq x_2$ and $x_2 \geq x_n,$ I would rather not explain what that was about and the only changes are in $n=3,4,$ already settled. ...
1
vote
3answers
50 views

$\exists x \in N, \forall y \in N, x \ge y$

$\exists x \in N, \forall y \in N, x \ge y$ Why is this a false statement? Intuitively, it seems that if you let x always equal y, then the statement always holds true.
0
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2answers
41 views

How to solve the system $ax>y+z$, $by>x+z$, $cz>x+y$ in positive numbers?

Let $a>b>c>1$. How to find solutions in positive numbers of the following system? \begin{cases} ax>y+z \\ by>x+z \\ cz>x+y \end{cases}
3
votes
3answers
230 views

How to prove this inequality $\sum_{i=1}^{n}\left(x^k_{i}\ln{x_{i}}\ln{\frac{x_{i}}{n}}\right)\le 0$

Let $x_{i}\ge 0$ for $i\in\{1,2,\cdots,n\}$ and $x_{1}+x_{2}+\cdots+x_{n}=n$ for $n\ge 3$ Show that for all strictly positive integers $k\ge2$ the following inequality holds : ...
0
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0answers
30 views

Comparing Fractional Numbers

Does a formula exist for comparing two fractional numbers, without resolving to using anything other than integers and fractions? (Thus not real numbers). In other words: given $\dfrac{a}{b}$ and ...
0
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2answers
51 views

Mathematical Induction - Inequality

Does anyone have any idea on how to complete the inductive step? Thm: For all $n >= 0~~~~ 6^n + 4 > n^3$ Pf: by Induction     Let $P(n)$ be proposition that $~6^n + 4 ...
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1answer
30 views

How prove this inequality?

Question: let sequence $x_{i}\neq 0,i=1,2,\cdots,n$ are real numbers, show that $$\left|x_{l}-\dfrac{\displaystyle\max_{1\le k\le n}|x_{k}|\min_{1\le k\le n}|x_{k}|}{x_{l}}\right| \le \max_{1\le ...
0
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3answers
73 views

$a,b,c > 0$ such that $\sqrt{a^2+b^2+c^2}=\sqrt[3]{ab+bc+ca} $ prove that $a^2b+b^2c+c^2a+abc \le \frac{4}{27}$

I have a series of problems in inequalities that I can not solve,please help me if you can. problem 1 :$a,b,c \geq 0$ such that $\sqrt{a^2+b^2+c^2}=\sqrt[3]{ab+bc+ca} $ prove that ...
2
votes
1answer
56 views

How prove $(x+\sqrt{x^{2}-1})^{n}+(x-\sqrt{x^{2}-1})^{n}\leq 2(1+n(x-1))^{n}$ for $n\in\mathbb{N}$?

Let $x\ge 1$. How prove that $(x+\sqrt{x^{2}-1})^{n}+(x-\sqrt{x^{2}-1})^{n}\leq 2(1+n(x-1))^{n}$ for $n\in\mathbb{N}$?