Questions tagged [tangent-line-method]

For proofs inequalities by Tangent Line method.

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Inequality $\sum_{k=1}^{n} \frac{\log(a_k)}{1+a_{k}^{2}} \leqslant 0$

Let $a_1,a_2,...,a_n$ be positive real numbers such that $a_1 \cdot a_2 \cdot ... \cdot a_n=1$. Prove that $\sum_{k=1}^{n} \frac{\log(a_k)}{1+a_{k}^{2}} \leqslant 0$. I tried using Jensen inequality, ...
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Is this Factorization?

I'm doubtful about the some parts of the solution to this question: Suppose that the real numbers $a, b, c > 1$ satisfy the condition $${1\over a^2-1}+{1\over b^2-1}+{1\over c^2-1}=1$$ Prove ...
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Inequality with a High Degree Constraint

This question- Suppose that $x, y, z$ are positive real numbers and $x^5 + y^5 + z^5 = 3$. Prove that $${x^4\over y^3}+{y^4\over z^3}+{z^4\over x^3} \ge 3$$ The inequality has a high degree ...
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A more elementary proof that if $x_i>0$ for $1\leq i\leq n$, and $\sum x_i=1$, then $(x_1+\frac{1}{x_1})\cdots(x_n+\frac{1}{x_n})\geq(n+\frac1n)^n$

For $x_i>0$, $1\leq i\leq n$ and $\sum_i x_i=1$, show that $$\left(x_1+\frac{1}{x_1}\right)\cdots \left(x_n+\frac{1}{x_n}\right)\geq \left(n+\frac{1}{n}\right)^n$$ I think this can be proved easily ...
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If $a$, $b$, $c$, $d$ are positive reals so $(a+c)(b+d) = 1$, prove the following inequality would be greater than or equal to $\frac {1}{3}$.

Let $a$, $b$, $c$, $d$ be real positive reals with $(a+c)(b+d) = 1$. Prove that $\frac {a^3}{b + c + d} + \frac {b^3}{a + c + d} + \frac {c^3}{a + b + d} + \frac {d^3}{a + b + c} \geq \frac {1}{3}$. ...
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Show that $(a^3+a+1)(b^3+b+1)(c^3+c+1)\le 27$

Let $a,b,c\ge 0$ be such that $a^2+b^2+c^2=3$. Show that $$(a^3+a+1)(b^3+b+1)(c^3+c+1)\le 27$$ I want to consider the function $$f(x)=\ln{(x^{3/2}+x^{1/2}+1)}$$ Maybe it isn't the case $f''(x)\le 0$, ...
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Prove that $21(a^2+b^2+c^2)\ge 20 +9(a^3+b^3+c^3)$ [closed]

Let $a,b,c$ be the length of sides of triangle such that $a+b+c=2$. Prove that $$21(a^2+b^2+c^2)\ge 20 +9(a^3+b^3+c^3)$$ It was in my exam. It can be solved easy by BW but it takes alot of time to ...
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If $abc=1$, then how do you prove $\frac{b-1}{bc+1}+\frac{c-1}{ac+1}+\frac{a-1}{ab+1} \geq 0$?

If $abc=1$, then how do you prove $\frac{b-1}{bc+1}+\frac{c-1}{ac+1}+\frac{a-1}{ab+1} \geq 0$? I tried substitution on the bottom (for example $\frac{b-1}{\frac{1}{a}+1}$), but I then a very similar ...
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$\frac{3x+1}{x+1}+\frac{3y+1}{y+1}+\frac{3z+1}{z+1} \le \frac{9}{2}$

I'm having trouble proving that for any $x,y,z>0$ such that $x+y+z=1$ the following inequality is true: $\frac{3x+1}{x+1}+\frac{3y+1}{y+1}+\frac{3z+1}{z+1} \le \frac{9}{2}$ It seems to me that ...
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Maximize $\sum\limits_{k =1}^n x_k (1 - x_k)^2$

Given problem for maximizing \begin{align} &\sum_{k =1}^n x_k (1 - x_k)^2\rightarrow \max\\ &\sum_{k =1}^n x_k = 1,\\ &x_k \ge 0, \; \forall k \in 1:n. \end{align} My attempt: first of ...
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Prove that $a^2+b^2+c^2 \le a^3 +b^3 +c^3$

If $a,b,c$ are three positive real numbers and $abc=1$ then prove that $a^2+b^2+c^2 \le a^3 +b^3 +c^3$ I got $a^2+b^2+c^2\ge 3$ which can be proved $a^2 +b^2+c^2\ge a+b+c$. From here how can I ...
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Given positives $a, b, c$, prove that $$\large \frac{a}{(b + c)^2} + \frac{b}{(c + a)^2} + \frac{c}{(a + b)^2} \ge \frac{9}{4(a + b + c)}$$ Let $x = \dfrac{b + c}{2}, y = \dfrac{c + a}{2}, z = \dfrac{... • 4,228 1 vote 1 answer 102 views Prove that$\sum_{i=1}^n\frac{\sum_{i'=1}^na_{i'}^p - a_i^p}{\sum_{i'=1}^na_{i'}^q - a_i^q}\le n\cdot\frac{\sum_{i=1}^na_i^p}{\sum_{i=1}^na_i^q}$. Given positives $$\large a_1, a_2, \cdots, a_{n - 1}, a_n$$$(n \in \mathbb Z^+, n \ge 3)$. Prove that for all naturals$p$and$q$such that$p \ge q$, $$\large \sum_{i = 1}^n\frac{\displaystyle \... • 4,228 4 votes 3 answers 169 views Inequality \frac{x^3}{x^2+y^2}+\frac{y^3}{y^2+z^2}+\frac{z^3}{z^2+x^2} \geqslant \frac{x+y+z}{2} Help to prove this Inequality: If x,y,z are postive real numbers then: \dfrac{x^3}{x^2+y^2}+\dfrac{y^3}{y^2+z^2}+\dfrac{z^3}{z^2+x^2} \geqslant \dfrac{x+y+z}{2} I tied to use analytic method ... • 49 0 votes 1 answer 73 views Inequality \sum_{cyc}a^3\frac{(16a^2-10ab+12b^2)}{(13a^2+5b^2)^2}\geq \frac{a+b+c}{18} It's a variant of Inequality \sum\limits_{cyc}\frac{a^3}{13a^2+5b^2}\geq\frac{a+b+c}{18} : Let a,b,c>0 then we have :$$\sum_{cyc}a^3\frac{(16a^2-10ab+12b^2)}{(13a^2+5b^2)^2}\geq \frac{a+b+... 0 votes 1 answer 93 views Find all positive real solutions of the system Find all positive real solutions of the system of equations $$\begin{cases} x_1+x_2+...+x_{1994}=1994 \\ x_1^4+x_2^4+...+x_{1994}^4=x_1^3+x_2^3+....+x_{1994}^3 \end{cases}$$ ''By Hölder, we have in ... • 4,266 7 votes 8 answers 855 views Prove that the minimum values of$x^2+y^2+z^2$is$27$with given condition$\dfrac{1}{x}+\dfrac{1}{y}+\dfrac{1}{z}=1$. Question: Prove that the minimum values of$x^2+y^2+z^2$is$27$, where$x,y,z$are positive real variables satisfying the condition$\dfrac{1}{x}+\dfrac{1}{y}+\dfrac{1}{z}=1$. From AM$\ge$GM, we ... • 111 4 votes 4 answers 175 views Proving$\sum_{cyc} \sqrt{a^2+ab+b^2}\geq\sqrt 3$when$a+b+c=3$Good evening everyone, I want to prove the following: Let$a,b,c>0$be real numbers such that$a+b+c=3$. Then $$\sqrt{a^2+ab+b^2}+\sqrt{b^2+bc+c^2}+\sqrt{c^2+ca+a^2}\geq 3\sqrt 3.$$ My attempt: I ... 2 votes 1 answer 104 views Inequality with 5 cyclic variables For postive real numbers$a$,$b$,$c$,$d$and$e$, prove that $$\frac{4a^3}{a^2+2b^2+\frac{2b^3}{a}} + \frac{4b^3}{b^2+2c^2+\frac{2c^3}{b}} + \frac{4c^3}{c^2+2d^2+\frac{2d^3}{c}}+ \frac{4d^3}{d^2+2e^2+... • 1,591 2 votes 1 answer 80 views inequality under condition x+y+z=3 x, y and z being three positive real numbers such that x+y+z=3 It is asked to prove that$$ \dfrac{\sqrt x}{y + z}+ \dfrac{\sqrt y}{x + z} + \dfrac{\sqrt z}{y + x} \ge \dfrac 3 2$$I tried ... • 1,273 4 votes 2 answers 202 views Prove that \sum \frac{x}{x^2+7}\le \frac{3}{8} Let x,y,z>0 such that xy+yz+xz=3. Show that$$\frac{x}{x^2+7}+\frac{y}{y^2+7}+\frac{z}{z^2+7}\le \frac{3}{8}$$We have:$$x+y+z\ge \sqrt{3\left(xy+yz+xz\right)}=3\rightarrow \frac{3}{8\left(x+... • 413 0 votes 1 answer 115 views Chebyshev's sum inequality Given$a,b,c,d>0$satisfying$a+b+c+d=4$. Prove that $$\dfrac{1}{8+a^2}+\dfrac{1}{8+b^2}+\dfrac{1}{8+c^2}+\dfrac{1}{8+d^2}\leq \dfrac{4}{9}.$$ I've tried solving by assuming that$0<a\leq b\...
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Problem Statement: For $a,b,c>0$ and $a+b+c=3$, I'd want to prove that $$\frac{1}{a^2+b+c}+\frac{1}{b^2+a+c} +\frac {1}{c^2+a+b} \leq1 .$$ I am a beginner when it comes to inequalities. This ...