Found problems: 592
If for real numbers $x$ and $y$ holds $\left(x+\sqrt{1+y^2}\right)\left(y+\sqrt{1+x^2}\right)=1$ prove that $$\left(x+\sqrt{1+x^2}\right)\left(y+\sqrt{1+y^2}\right)=1$$
Positive numbers $x, y, z$ satisfy $x^2+y^2+z^2+xy+yz+zy \le 1$.
Prove that $\big( \frac{1}{x}-1\big) \big( \frac{1}{y}-1\big)\big( \frac{1}{z}-1\big) \ge 9 \sqrt6 -19$.
Prove that for any positive number $a,b,c$ and any nonnegative integer $p$
$$a^{p+2}+b^{p+2}+c^{p+2}\ge a^pbc+b^pca+c^pab.$$
Let $a,b,c$ be positive real numbers such that $a + b + c = 1$. Show that
$$\left(\frac{1}{a} + \frac{1}{bc}\right)\left(\frac{1}{b} + \frac{1}{ca}\right)\left(\frac{1}{c} + \frac{1}{ab}\right) \geq 1728$$
a) Let $n$ be a positive integer. Prove that $ n\sqrt {x-n^2}\leq \frac {x}{2}$ , for $x\geq n^2$.
b) Find real $x,y,z$ such that: $ 2\sqrt {x-1} +4\sqrt {y-4} + 6\sqrt {z-9} = x+y+z$
Prove that for all positive real $a$, $b$ and $c$ holds: $$ \frac{bc}{a^2+2bc}+\frac{ac}{b^2+2ac}+\frac{ab}{c^2+2ab} \leq 1 \leq \frac{a^2}{a^2+2bc}+\frac{b^2}{b^2+2ac}+\frac{c^2}{c^2+2ab}$$
Let $a,b,c$ be positive real numbers such that $a^2+b^2+c^2=3. $ Prove that $$\frac{a^4+3ab^3}{a^3+2b^3}+\frac{b^4+3bc^3}{b^3+2c^3}+\frac{c^4+3ca^3}{c^3+2a^3}\leq4.$$
The two polynomials $(x) =x^4+ax^3+bx^2+cx+d$ and $Q(x) = x^2+px+q$ take negative values on an interval $I$ of length greater than $2$, and nonnegative values outside of $I$. Prove that there exists $x_0 \in \mathbb R$ such that $P(x_0) < Q(x_0)$.
Let $a_1, \ldots, a_n$ be $n$ positive numbers and $0 < q < 1.$ Determine $n$ positive numbers $b_1, \ldots, b_n$ so that:
[i]a.)[/i] $ a_{k} < b_{k}$ for all $k = 1, \ldots, n,$
[i]b.)[/i] $q < \frac{b_{k+1}}{b_{k}} < \frac{1}{q}$ for all $k = 1, \ldots, n-1,$
[i]c.)[/i] $\sum \limits^n_{k=1} b_k < \frac{1+q}{1-q} \cdot \sum \limits^n_{k=1} a_k.$
Let $\theta_i \in \left ( 0,\frac{\pi}{4} \right ]$ for $i=1,2,3,4$. Prove that:
$\tan \theta _1 \tan \theta _2 \tan \theta _3 \tan \theta _4 \le (\frac{\sin^8 \theta _1+\sin^8 \theta _2+\sin^8 \theta _3+\sin^8 \theta _4}{\cos^8 \theta _1+\cos^8 \theta _2+\cos^8 \theta _3+\cos^8 \theta _4})^\frac{1}{2}$
[hide=edit]@below, fixed now. There were some problems (weird characters) so aops couldn't send it.[/hide]
$a,b,c $ are positive real numbers . Prove that
$\sqrt[7]{\frac{a}{b+c}+\frac{b}{c+a}} +\sqrt[7]{\frac{b}{c+a}+\frac{c}{b+a}}+\sqrt[7]{\frac{c}{a+b}+\frac{a}{b+c}}\geq 3$
Positive numbers $a$, $b$ and $c$ are such that $a^2+b^2+c^2+abc=4$. Prove that \[\sqrt{2-a}+\sqrt{2-b}+\sqrt{2-c}\geqslant 2+\sqrt{(2-a)(2-b)(2-c)}.\]
Prove that for $\forall$ $a,b,c\in [\frac{1}{3},3]$ the following inequality is true:
$\frac{a}{a+b}+\frac{b}{b+c}+\frac{c}{c+a}\geq \frac{7}{5}$.
Suppose $x_1,x_2,...,x_{60}\in [-1,1]$ , find the maximum of
$$ \sum_{i=1}^{60}x_i^2(x_{i+1}-x_{i-1}),$$
where $x_{i+60}=x_i$.
For all $x \geq 2$, $y \geq 2$ real numbers, prove that
$$x(\frac{4x}{y-1}+\frac{1}{2y+x})+y(\frac{y}{6x-9}+\frac{1}{2x+y}) > \frac{26}{3}$$
Let $p \geq 5$ be a prime and let
\begin{align*} (p-1)^p +1 = \prod _{i=1}^n q_i^{\beta_i} \end{align*}
where $q_i$ are primes. Prove,
\begin{align*} \sum_{i=1}^n q_i \beta_i >p^2 \end{align*}
Find all twice continuously differentiable functions $f: \mathbb{R} \to (0, \infty)$ satisfying $f''(x)f(x) \ge 2f'(x)^2.$
$(HUN 2)$ Prove that $1+\frac{1}{2^3}+\frac{1}{3^3}+\cdots+\frac{1}{n^3}<\frac{5}{4}$
Prove that for all non-negative real numbers $x,y,z$, not all equal to $0$, the following inequality holds
$\displaystyle \dfrac{2x^2-x+y+z}{x+y^2+z^2}+\dfrac{2y^2+x-y+z}{x^2+y+z^2}+\dfrac{2z^2+x+y-z}{x^2+y^2+z}\geq 3.$
Determine all the triples $(x,y,z)$ for which the equality holds.
[i]Milan Mitreski, Serbia[/i]
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 $1<k\leq n$ be positive integers and $x_1 , x_2 , \ldots , x_k$ be positive real numbers such that $x_1 \cdot x_2 \cdot \ldots \cdot x_k = x_1 + x_2 + \ldots +x_k.$
a) Show that $x_{1}^{n-1} +x_{2}^{n-1} + \ldots +x_{k}^{n-1} \geq kn.$
b) Find all numbers $k,n$ and $x_1, x_2 ,\ldots , x_k$ for which equality holds.
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]
Given an integer $n\geq2$, let $x_1<x_2<\cdots<x_n$ and $y_1<y_2<\cdots<y_n$ be positive reals. Prove that for every value $C\in (-2,2)$ (by taking $y_{n+1}=y_1$) it holds that
$\hspace{122px}\sum_{i=1}^{n}\sqrt{x_i^2+Cx_iy_i+y_i^2}<\sum_{i=1}^{n}\sqrt{x_i^2+Cx_iy_{i+1}+y_{i+1}^2}$.
[i]Proposed by Mirko Petrusevski[/i]
A linear binomial $l(z) = Az + B$ with complex coefficients $A$ and $B$ is given. It is known that the maximal value of $|l(z)|$ on the segment $-1 \leq x \leq 1$ $(y = 0)$ of the real line in the complex plane $z = x + iy$ is equal to $M.$ Prove that for every $z$
\[|l(z)| \leq M \rho,\]
where $\rho$ is the sum of distances from the point $P=z$ to the points $Q_1: z = 1$ and $Q_3: z = -1.$
[b]a)[/b] Show that $ \frac{n}{2}\ge \frac{2\sqrt{x} +3\sqrt[3]{x}+\cdots +n\sqrt[n]{x}}{n-1} -x, $ for all non-negative reals $ x $ and integers $ n\ge 2. $
[b]b)[/b] If $ x,y,z\in (0,\infty ) , $ then prove the inequality
$$ \sum_{\text{cyc}} \frac{x}{(2x+y+z)^2+4} \le 3/16 $$