Found problems: 93
Let $ a, b, c$ be positive integers for which $ abc \equal{} 1$. Prove that
$ \sum \frac{1}{b(a\plus{}b)} \ge \frac{3}{2}$.
Let $ a, b, c$ be positive real numbers so that $ abc \equal{} 1$. Prove that
\[ \left( a \minus{} 1 \plus{} \frac 1b \right) \left( b \minus{} 1 \plus{} \frac 1c \right) \left( c \minus{} 1 \plus{} \frac 1a \right) \leq 1.
\]
Prove that for all real numbers $x,y,z \in [1,2]$ the following inequality always holds:
\[ (x+y+z)(\frac{1}{x}+\frac{1}{y}+\frac{1}{z})\geq 6(\frac{x}{y+z}+\frac{y}{z+x}+\frac{z}{x+y}). \]
When does the equality occur?
Prove that for every real positive number $a, b, c$ with $1 \le a, b, c \le 8$ the inequality
$$\frac{a+b+c}{5}\le \sqrt[3]{abc}$$
$(YUG 1)$ Suppose that positive real numbers $x_1, x_2, x_3$ satisfy
$x_1x_2x_3 > 1, x_1 + x_2 + x_3 <\frac{1}{x_1}+\frac{1}{x_2}+\frac{1}{x_3}$
Prove that:
$(a)$ None of $x_1, x_2, x_3$ equals $1$.
$(b)$ Exactly one of these numbers is less than $1.$
Find the least real number $k$ with the following property: if the real numbers $x$, $y$, and $z$ are not all positive, then \[k(x^{2}-x+1)(y^{2}-y+1)(z^{2}-z+1)\geq (xyz)^{2}-xyz+1.\]
Let $\displaystyle {x, y, z}$ be positive real numbers such that $\displaystyle {xyz = 1}$. Prove the inequality:$$\displaystyle{\dfrac{1}{x\left(ay+b\right)}+\dfrac{1}{y\left(az+b\right)}+\dfrac{1}{z\left(ax+b\right)}\geq 3}$$
if:
(A) $\displaystyle {a = 0, b = 1}$
(B) $\displaystyle {a = 1, b = 0}$
(C) $\displaystyle {a + b = 1, \; a, b> 0}$
When the equality holds?
Let $a$, $b$ and $c$ be real numbers such that $abc=1$. Prove that the most two of numbers
$$2a-\frac{1}{b},\ 2b-\frac{1}{c},\ 2c-\frac{1}{a}$$
are greater than $1$.
Let $ a, b, c$ be positive real numbers so that $ abc \equal{} 1$. Prove that
\[ \left( a \minus{} 1 \plus{} \frac 1b \right) \left( b \minus{} 1 \plus{} \frac 1c \right) \left( c \minus{} 1 \plus{} \frac 1a \right) \leq 1.
\]
Given real numbers $x,y,z$ such that $x+y+z=0$, show that
\[\dfrac{x(x+2)}{2x^2+1}+\dfrac{y(y+2)}{2y^2+1}+\dfrac{z(z+2)}{2z^2+1}\ge 0\]
When does equality hold?
Let $ a,b,c$ be real numbers such that $ a\plus{}b\plus{}c\equal{}3$. Prove that
\[\frac{1}{5a^2\minus{}4a\plus{}11}\plus{}\frac{1}{5b^2\minus{}4b\plus{}11}\plus{}\frac{1}{5c^2\minus{}4c\plus{}11}\leq\frac{1}{4}\]
Show that the inequality $(x^2 + y^2z^2) (y^2 + x^2z^2) (z^2 + x^2y^2) \ge 8xy^2z^3$ is valid for all integers $x, y$ and $z$.When does equality apply?
Prove that if $x,y,z >1$ and $\frac 1x +\frac 1y +\frac 1z = 2$, then
\[\sqrt{x+y+z} \geq \sqrt{x-1}+\sqrt{y-1}+\sqrt{z-1}.\]
In a non-obtuse triangle $ABC$, prove that
\[ \frac{\sin A \sin B}{\sin C} + \frac{\sin B \sin C}{\sin A} + \frac{\sin C \sin A}{ \sin B} \ge \frac 52. \][i]Proposed by Ryan Alweiss[/i]
Let $ x$, $ y$, $ z \in\mathbb{R}^+$ satisfying $ xyz = 1$. Prove that
\[ \frac {(x + y - 1)^2}{z} + \frac {(y + z - 1)^2}{x} + \frac {(z + x - 1)^2}{y}\geqslant x + y + z\mbox{.}\]
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
\[ \left( \frac AG \right)^3 \geq \frac 14 + \frac 34 \cdot \frac AH.\]
(a) Determine the maximum $M$ of $x+y +z$ where $x, y$ and $z$ are positive real numbers with $16xyz = (x + y)^2(x + z)^2$.
(b) Prove the existence of infinitely many triples $(x, y, z)$ of positive rational numbers that satisfy $16xyz = (x + y)^2(x + z)^2$ and $x + y + z = M$.
Proposed by Karl Czakler
In a non-obtuse triangle $ABC$, prove that
\[ \frac{\sin A \sin B}{\sin C} + \frac{\sin B \sin C}{\sin A} + \frac{\sin C \sin A}{ \sin B} \ge \frac 52. \][i]Proposed by Ryan Alweiss[/i]
Let $a,b,c$ be positive real numbers. Prove that
$\sqrt{\frac{a}{b+c}+\frac{b}{c+a}}+\sqrt{\frac{b}{c+a}+\frac{c}{a+b}}+\sqrt{\frac{c}{a+b}+\frac{a}{b+c}}\ge 3$
Let $a,b,c$ be positive real numbers such that
\[ \cfrac{1}{1+a}+\cfrac{1}{1+b}+\cfrac{1}{1+c}\le 1. \]
Prove that $(1+a^2)(1+b^2)(1+c^2)\ge 125$. When does equality hold?
Let $a,b,c$ be positive real numbers such that $a+b+c=3$. Prove that: \[ \frac 1{a^2}+\frac 1{b^2}+\frac 1{c^2} \geq a^2+b^2+c^2. \]
$a,b,c\in\mathbb{R^+}$ and $a^2+b^2+c^2=48$. Prove that \[a^2\sqrt{2b^3+16}+b^2\sqrt{2c^3+16}+c^2\sqrt{2a^3+16}\le24^2\]
Let $a,b,c$ be positive reals such that $abc=1$, $a+b+c=5$ and $$(ab+2a+2b-9)(bc+2b+2c-9)(ca+2c+2a-9)\geq 0$$.
Find the minimum value of $$\frac {1}{a}+ \frac {1}{b}+ \frac{1}{c}$$
Given real numbers $x,y,z$ such that $x+y+z=0$, show that
\[\dfrac{x(x+2)}{2x^2+1}+\dfrac{y(y+2)}{2y^2+1}+\dfrac{z(z+2)}{2z^2+1}\ge 0\]
When does equality hold?
Prove that $0\le yz+zx+xy-2xyz\le{7\over27}$, where $x,y$ and $z$ are non-negative real numbers satisfying $x+y+z=1$.