Found problems: 592
For positive real numbers $a,b,c$ with $abc=1$ prove that $\left(a+\frac{1}{b}\right)^{2}+\left(b+\frac{1}{c}\right)^{2}+\left(c+\frac{1}{a}\right)^{2}\geq 3(a+b+c+1)$
The least number is $m$ and the greatest number is $M$ among $ a_1 ,a_2 ,\ldots,a_n$ satisfying $ a_1 \plus{}a_2 \plus{}...\plus{}a_n \equal{}0$. Prove that
\[ a_1^2 \plus{}\cdots \plus{}a_n^2 \le\minus{}nmM\]
Let $n$ be a positive integer and let $a_1, a_2, \ldots, a_n$ be positive reals. Show that $$\sum_{i=1}^{n} \frac{1}{2^i}(\frac{2}{1+a_i})^{2^i} \geq \frac{2}{1+a_1a_2\ldots a_n}-\frac{1}{2^n}.$$
For two arbitrary reals $x, y$ which are larger than $0$ and less than $1.$ Prove that$$\frac{x^2}{x+y}+\frac{y^2}{1-x}+\frac{(1-x-y)^2}{1-y}\geq\frac{1}{2}.$$
If $a, b$ are positive reals such that $a+b<2$. Prove that $$\frac{1}{1+a^2}+\frac{1}{1+b^2} \le \frac{2}{1+ab}$$ and determine all $a, b$ yielding equality.
[i]Proposed by Gottfried Perz[/i]
Define the numbers $a_0, a_1, \ldots, a_n$ in the following way:
\[ a_0 = \frac{1}{2}, \quad a_{k+1} = a_k + \frac{a^2_k}{n} \quad (n > 1, k = 0,1, \ldots, n-1). \]
Prove that \[ 1 - \frac{1}{n} < a_n < 1.\]
Let $x,y,z$ be nonnegative positive integers.
Prove $\frac{x-y}{xy+2y+1}+\frac{y-z}{zy+2z+1}+\frac{z-x}{xz+2x+1}\ge 0$
Let $n\geqslant 1$ be an integer, and let $x_0,x_1,\ldots,x_{n+1}$ be $n+2$ non-negative real numbers that satisfy $x_ix_{i+1}-x_{i-1}^2\geqslant 1$ for all $i=1,2,\ldots,n.$ Show that \[x_0+x_1+\cdots+x_n+x_{n+1}>\bigg(\frac{2n}{3}\bigg)^{3/2}.\][i]Pakawut Jiradilok and Wijit Yangjit, Thailand[/i]
Determine the minimum value of $\dfrac{m^m}{1\cdot 3\cdot 5\cdot \ldots \cdot(2m-1)}$ for positive integers $m$.
For positive $a,b,c$, \[a^{2}+b^{2}+c^{2}+abc=4\] Prove $a+b+c \leq3$
$\boxed{\text{A2}}$ Prove that for all Positive reals $a,b,c$ $\frac{a^2-bc}{2a^2+bc}+\frac{b^2-ca}{2b^2+ca}+\frac{c^2-ab}{2c^2+ab}\leq 0$
Prove that for positive real numbers $a, b, c$ the following inequality holds:
\begin{align*}
\frac{a}{9bc+1}+\frac{b}{9ca+1}+\frac{c}{9ab+1}\geq \frac{a+b+c}{1+(a+b+c)^2}
\end{align*}
When does equality occur?
Given that $a, b, c, d \in \mathbb{R}$, prove that $(ab+cd)^2 \leq (a^2+c^2)(b^2+d^2)$.
Let $a \geq b \geq 1 \geq c \geq 0$ be real numbers such that $a+b+c=3$. Show that
$$3 \left( \frac{a}{b}+\frac{b}{a} \right ) \geq 4c^2+\frac{a^2}{b}+\frac{b^2}{a}$$
Find the max and minimum without using dervivate:
$\sqrt{x} +4 \cdot \sqrt{\frac{1}{2} - x}$
Let $a$ and $b$ be two fixed positive real numbers. Find all real numbers $x$, such that inequality holds $$\frac{1}{\sqrt{x}}+\frac{1}{\sqrt{a+b-x}} < \frac{1}{\sqrt{a}} + \frac{1}{\sqrt{b}}$$
Prove the following inequality where positive reals $a$, $b$, $c$ satisfies $ab+bc+ca=1$.
\[
\frac{a+b}{\sqrt{ab(1-ab)}} + \frac{b+c}{\sqrt{bc(1-bc)}} + \frac{c+a}{\sqrt{ca(1-ca)}} \le \frac{\sqrt{2}}{abc}
\]
For all positive real numbers $a,b,c$ with $a+b+c=3$, prove the inequality
$$\frac{a^6}{c^2+2b^3} + \frac{b^6}{a^2+2c^3} + \frac{c^6}{b^2+2a^3} \geq 1.$$
Let $x,y,z$ be positive real numbers . Prove:
$\frac{x}{\sqrt{\sqrt[4]{y}+\sqrt[4]{z}}}+\frac{y}{\sqrt{\sqrt[4]{z}+\sqrt[4]{x}}}+\frac{z}{\sqrt{\sqrt[4]{x}+\sqrt[4]{y}}}\geq \frac{\sqrt[4]{(\sqrt{x}+\sqrt{y}+\sqrt{z})^7}}{\sqrt{2\sqrt{27}}}$
For positive numbers $a, b, c$ define $A = \frac{(a + b + c)}{3}$, $G = \sqrt[3]{abc}$, $H = \frac{3}{(a^{-1} + b^{-1} + c^{-1})}.$ Prove that
\[ \left( \frac AG \right)^3 \geq \frac 14 + \frac 34 \cdot \frac AH.\]
Determine all pairs $(a,b)$ of non-negative integers such that
$$ \frac{a+b}{2}-\sqrt{ab}=1.$$
Let $a$, $b$, $c$ be real numbers such that $0 < a \le b \le c$. Prove that
$(a + 3b)(b + 4c)(c + 2a) \ge 60abc$.
When does equality hold?
Let $x,y$ and $z$ be positive real numbers such that $xyz=1$. Prove that
\[
\frac{x^{3}}{(1 + y)(1 + z)}+\frac{y^{3}}{(1 + z)(1 + x)}+\frac{z^{3}}{(1 + x)(1 + y)}
\geq \frac{3}{4}.
\]
For positive integers $x, y, $ $r_x(y)$ to represent the smallest positive integer $ r $ such that $ r \equiv y(\text{mod x})$ .For any positive integers $a, b, n ,$ Prove that $$\sum_{i=1}^{n} r_b(a i)\leq \frac{n(a+b)}{2}$$
Let $x, y, z$ be positive real numbers. Prove that $\frac{y^2 + z^2}{x}+\frac{z^2 + x^2}{y}+\frac{x^2 + y^2}{z}\ge 2(x + y + z)$.