Found problems: 592
Given are positive reals $a, b, c$, such that $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=3$. Prove that
$\frac{\sqrt{a+\frac{b}{c}}+\sqrt{b+\frac{c}{a}}+\sqrt{c+\frac{a}{b}}}{3}\leq \frac{a+b+c-1}{\sqrt{2}}$.
[i]Albania[/i]
Prove that if $a,b,c \ge 1$ and $a+b+c=9$, then $\sqrt{ab+bc+ca} \le \sqrt{a} +\sqrt{b} + \sqrt{c}$
Let $x,y$ and $z$ be positive real numbers such that $xy+yz+xz=3xyz$. Prove that \[ x^2y+y^2z+z^2x \ge 2(x+y+z)-3 \] and determine when equality holds.
[i]UK - David Monk[/i]
Let $a, b, c$ be positive real numbers such that $a + b + c + ab + bc + ca + abc = 7$. Prove
that $\sqrt{a^2 + b^2 + 2 }+\sqrt{b^2 + c^2 + 2 }+\sqrt{c^2 + a^2 + 2 } \ge 6$ .
Suppose that $a, b,$ and $c$ are positive real numbers such that
$$a + b + c \ge \frac{1}{a} + \frac{1}{b} + \frac{1}{c}.$$
Find the largest possible value of the expression
$$\frac{a + b - c}{a^3 + b^3 + abc} + \frac{b + c - a}{b^3 + c^3 + abc} + \frac{c + a - b}{c^3 + a^3 + abc}.$$
Let $f,g,h : \mathbb R \to \mathbb R$ be continuous functions and \(X\) be a random variable such that $E(g(X)h(X))=0$ and $E(g(X)^2) \neq 0 \neq E(h(X)^2)$. Prove that $$E(f(X)^2) \geq \frac{E(f(X)g(X))^2}{E(g(X)^2)} + \frac{E(f(X)h(X))^2}{E(h(X)^2)}.$$ You may assume that all expected values exist.
[i]Proposed by Cristi Calin[/i]
Let $a < b < c < d < e$ be positive integers. Prove that
$$\frac{1}{[a, b]} + \frac{1}{[b, c]} + \frac{1}{[c, d]} + \frac{2}{[d, e]} \le 1$$
where $[x, y]$ is the least common multiple of $x$ and $y$ (e.g., $[6, 10] = 30$). When does equality hold?
For every positive $a_1, a_2, \dots, a_6$, prove the inequality
\[ \sqrt[4]{\frac{a_1}{a_2 + a_3 + a_4}} + \sqrt[4]{\frac{a_2}{a_3 + a_4 + a_5}} + \dots + \sqrt[4]{\frac{a_6}{a_1 + a_2 + a_3}} \ge 2 \]
For all non-negative real numbers $x,y,z$ with $x \geq y$, prove the inequality
$$\frac{x^3-y^3+z^3+1}{6}\geq (x-y)\sqrt{xyz}.$$
1) Two nonnegative real numbers $x, y$ have constant sum $a$. Find the minimum value of $x^m + y^m$, where m is a given positive integer.
2) Let $m, n$ be positive integers and $k$ a positive real number. Consider nonnegative real numbers $x_1, x_2, . . . , x_n$ having constant sum $k$. Prove that the minimum value of the quantity $x^m_1+ ... + x^m_n$ occurs when $x_1 = x_2 = ... = x_n$.
The matrix
\[A=\begin{pmatrix} a_{11} & \ldots & a_{1n} \\ \vdots & \ldots & \vdots \\ a_{n1} & \ldots & a_{nn} \end{pmatrix}\]
satisfies the inequality $\sum_{j=1}^n |a_{j1}x_1 + \cdots+ a_{jn}x_n| \leq M$ for each choice of numbers $x_i$ equal to $\pm 1$. Show that
\[|a_{11} + a_{22} + \cdots+ a_{nn}| \leq M.\]
Let $a_1,a_2,\cdots ,a_{100}$ be non-negative integers such that
$(1)$ There are positive integers$ k\leq 100$ such that $a_1\leq a_2\leq \cdots\leq a_{k}$
and $a_i=0$ $(i>k);$
$(2)$ $ a_1+a_2+a_3+\cdots +a_{100}=100;$
$(3)$ $ a_1+2a_2+3a_3+\cdots +100a_{100}=2022.$
Find the minimum of $ a_1+2^2a_2+3^2a_3+\cdots +100^2a_{100}.$
$x, y, z$ are positive reals with sum $3$. Show that $$6 < \sqrt{2x+3} + \sqrt{2y+3} + \sqrt{2z+3}\le 3\sqrt5$$
Let $a,b$ and $c$ be positive real numbers satisfying $abc=1$. Prove that
\[\dfrac{1}{a^3+2b^2+2b+4}+\dfrac{1}{b^3+2c^2+2c+4}+\dfrac{1}{c^3+2a^2+2a+4}\leq \dfrac13.\]
Given real numbers $x_1,x_2,y_1,y_2,z_1,z_2$ satisfying $x_1>0,x_2>0,x_1y_1>z_1^2$, and $x_2y_2>z_2^2$, prove that: \[ {8\over(x_1+x_2)(y_1+y_2)-(z_1+z_2)^2}\le{1\over x_1y_1-z_1^2}+{1\over x_2y_2-z_2^2}. \] Give necessary and sufficient conditions for equality.
Let $a, b, c $ be positive real numbers such that $abc = \frac {2} {3}. $ Prove that:
$$\frac {ab}{a + b} + \frac {bc} {b + c} + \frac {ca} {c + a} \geqslant \frac {a+b+c} {a^3+b ^ 3 + c ^ 3}.$$
Let $a, b, c$ be real numbers with the property as $ab + bc + ca = 1$. Show that:
$$\frac {(a + b) ^ 2 + 1} {c ^ 2 + 2} + \frac {(b + c) ^ 2 + 1} {a ^ 2 + 2} + \frac {(c + a) ^ 2 + 1} {b ^ 2 + 2} \ge 3 $$.