Inequality

Let $x, y,z>0$ such that $x+y+z+2=xyz$ . Prove that:

$$x+y+z+6 \geq 2\left( \sqrt{xy}+ \sqrt{yz}+\sqrt{zx}\right)$$

Solution

Inequality

For all $x_1, x_2, \dots, x_n \geq 0$ , let $x_{n+1}=x_1$. Prove that:

$$\sum_{k=1}^{n}\sqrt{\frac{1}{\left ( x_k+1 \right )^2} + \frac{x_{k+1}^2}{\left ( x_{k+1}+1 \right )^2}} \geq \frac{n}{\sqrt{2}}$$

Solution

Inequality with products

Let $a_i, \;\; i =1, \dots, n$  be positive real numbers. Prove that:

$$\sqrt[n]{\prod_{i=1}^{n}\prod_{j=1}^{n}\left ( 1+\frac{a_i}{a_j} \right )}\geq 2^n, \;\; n \in \mathbb{N}$$

Solution

Inequality from Memo 2012

Let $a, b, c$ be positive real numbers such that $abc=1$. Prove that:

$$\sqrt{9+16a^2}+\sqrt{9+16b^2}+\sqrt{9+16c^2} \geq 3 + 4(a+b+c)$$

 Solution

Inequality with roots

Let $x, y, z$ be positive real number such that $x+y+z=1$. Prove that:

$$\sqrt{\frac{xy}{xy+z}}+ \sqrt{\frac{yz}{yz+x}}+ \sqrt{\frac{zx}{zx+y}} \leq \frac{3}{2}$$

Solution

Archimedes's inequality

Let $m, n \in \mathbb{N}$. Prove that:

$$1^m +2^m +3^m +\cdots+ (n-1)^m < \frac{n^{m+1}}{m+1}< 1^m +2^m +3^m +\cdots +n^m$$

Solution

Minimum of expression

Let $x_1, x_2, \dots, x_n$ be positive real numbers which sum to $1$. Find the minimum of:

$$\mathcal{A}=\max\left \{ \frac{x_1}{1+x_1}, \frac{x_2}{1+x_1+x_2}, \cdots, \frac{x_n}{1+x_1+x_2+\cdots+x_n} \right \}$$

Solution (G. Basdekis)

Inequality

Let $x, y, z >0$ . Prove that:

$$\frac{3xy}{xy+x+y}+ \frac{3yz}{yz+y+z}+ \frac{3zx}{zx+z+x}\leq 2+ \frac{x^2+y^2+z^2}{3}$$

(Câtâlin Cristea)

Solution

An inequality involving complex numbers

Let $z_1, z_2, \dots, z_n$ be complex numbers . Prove that:

$$\left ( \sum_{k=1}^{n}\left | z_k \right | \right )^2 -\left | \sum_{k=1}^{n}z_k \right |^2 \geq \left ( \sum_{k=1}^{n}\left | \mathfrak{Re}\left ( z_k \right ) \right | - \left | \sum_{k=1}^{n}\mathfrak{Re}\left ( z_k \right ) \right | \right )^2$$

(G. Stoika, Canada)

Solution  (by Roberto Tauraso/ University of Rome/Italy)

Gautschi's Inequality for Gamma function

Prove that:

$$x^{1-s} < \frac{\Gamma(x+1)}{\Gamma(x+s)} < (x+1)^{1-s},\qquad x > 0,\; 0 < s < 1$$

which is better known as Gautschi's Inequality , due to Walter Gautschi.

Solution

Inequality within a triangle

Let $ABC$ be a triangle. Prove that:

$$\prod \frac{\sin^2 A + \sin B \sin C}{\sin B + \sin C} \geq \frac{E}{2R^2}$$

where $E$ is the area of the triangle.

Solution

Gronwall Inequality

Let $f, g$ be two continuous functions , non negative in $[a, b]$ and let $c>0$. If $\displaystyle f(x) \leq c + \int_a^b f(t) g(t) \, {\rm d}t$ then prove that:

$$f(x) \leq ce^{\displaystyle \int_a^x g(t)\, {\rm d}t}$$

Solution

Inequality

Let $a, b>0$. Prove that:

$$\left ( 1+ \frac{a}{b} \right )^{2014}+ \left ( 1+ \frac{b}{a} \right )^{2014}\geq 2^{2015}$$

Solution

Inequality with radicals

Let $a, b, c$ be positive real numbers such that $a^2+b^2 +c^2=48$. Prove that:

$$a^2\sqrt{2b^3+16}+ b^2 \sqrt{2c^3+16}+ c^2 \sqrt{2a^3+16}\leq 24^2$$

Solution (G. Bas)

Inequality

An inequality proposed by one of our readers.

Let $a, b, c >0$. Prove that:

$$ \frac{ab}{a+b+2c}+ \frac{bc}{b+c+2a}+ \frac{ca}{c+a+2b}\leq \frac{a+b+c}{4}$$

Solution

Existence of constant

Let $f$ be a non constant function defined on $[a, b]$ and  $f(a)=f(b)=0$. Prove that there exists a $\xi \in (a, b)$ such that:

$$\bigl|{f'(\xi)}\bigr|>\frac{4}{(b-a)^2}\int_{a}^{b}{|f(x)|\,{\rm d}x}$$

 (Qualifying Exams, Wisconsin-Madison, 2015)

Solution

Inequality

Let $a, \; b , \; c $ be real positive numbers that their product is equal to $1$. Prove that:

$$\frac{1}{a^3 (b+c)} +\frac{1}{b^3(c+a)}+\frac{1}{c^3(a+b)} \geq \frac{3}{2}$$

Solution

Inequality

Let $x, y, z$ be positive real numbers such that $x+y+z=1$. Prove that:

$$\sum_{cyc}\frac{x^3}{\left ( x+y \right )^2}\geq \frac{1}{4}$$

Solution

Inequality with sines and cosines

Let $0<\theta< \frac{\pi}{2}$. Prove that:

$$\sqrt{\sin^2 \theta +\frac{1}{\sin^2 \theta}}+ \sqrt{\cos^2 \theta + \frac{1}{\cos^2 \theta}}\geq \sqrt{10}$$

Solution (George Basdekis)

Series and inequality

Prove that:

$$\sum_{n=1}^{\infty} \frac{1}{\sqrt{n} (n+1)}<2$$

IMC 2015 / Round 2 Problem 1

Solution