Found problems: 592
Assuming that real numbers $x$ and $y$ satisfy $y\left(1+x^2\right)=x\left(\sqrt{1-4y^2}-1\right)$, find the maximum value of $xy$.
Find all pair $(x,y)$ of positive integers satisfying $$\left|\frac{x}{y}-\sqrt2\right|<\frac{1}{y^3}.$$
Consider infinite sequences $\{x_n\}$ of positive reals such that $x_0=1$ and $x_0\ge x_1\ge x_2\ge\ldots$.
[b]a)[/b] Prove that for every such sequence there is an $n\ge1$ such that: \[ {x_0^2\over x_1}+{x_1^2\over x_2}+\ldots+{x_{n-1}^2\over x_n}\ge3.999. \]
[b]b)[/b] Find such a sequence such that for all $n$: \[ {x_0^2\over x_1}+{x_1^2\over x_2}+\ldots+{x_{n-1}^2\over x_n}<4. \]
Let $x,y$ be positive real numbers .Find the minimum of $x+y+\frac{|x-1|}{y}+\frac{|y-1|}{x}$.
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]
Let $z_1,\ldots,z_n$ and $w_1,\ldots,w_n$ $(n\in\mathbb N)$ be complex numbers such that
$$|\epsilon_1z_1+\ldots+\epsilon_nz_n|\le|\epsilon_1w_1+\ldots+\epsilon_nw_n|$$holds for every choice of $\epsilon_1,\ldots,\epsilon_n\in\{-1,1\}$. Prove that
$$|z_1|^2+\ldots+|z_n|^2\le|w_1|^2+\ldots+|w_n|^2.$$
Given positive real numbers $a_1, a_2, \dots, a_n$. Let
\[ m = \min \left( a_1 + \frac{1}{a_2}, a_2 + \frac{1}{a_3}, \dots, a_{n - 1} + \frac{1}{a_n} , a_n + \frac{1}{a_1} \right). \]
Prove the inequality
\[ \sqrt[n]{a_1 a_2 \dots a_n} + \frac{1}{\sqrt[n]{a_1 a_2 \dots a_n}} \ge m. \]
If $x,y,z$ satisfies $x^2+y^2+z^2=1$, find the maximum possible value of $$(x^2-yz)(y^2-zx)(z^2-xy)$$
Suppose that $ a$,$ b$,$ c$ be three positive real numbers such that $ a\plus{}b\plus{}c\equal{}3$ . Prove that :
$ \frac{1}{2\plus{}a^{2}\plus{}b^{2}}\plus{}\frac{1}{2\plus{}b^{2}\plus{}c^{2}}\plus{}\frac{1}{2\plus{}c^{2}\plus{}a^{2}} \leq \frac{3}{4}$
For positive real numbers $a,b,c$ with $abc=1$ prove that $\left(a+\frac{1}{b}\right)^{2}+\left(b+\frac{1}{c}\right)^{2}+\left(c+\frac{1}{a}\right)^{2}\geq 3(a+b+c+1)$
Prove that for all real numbers $a, b, c > 1$ the inequality
\[a(b^2 + c) + b(c^2 + a) + c(a^2 + b) \ge a^2 + b^2 + c^2 + 3abc\]
holds. When does equality hold?
Proposed by [i]Ilija Jovcevski[/i]
Prove the trigonometric inequality $\cos x < 1 - \frac{x^2}{2} + \frac{x^4}{16},$ when $x \in \left(0, \frac{\pi}{2} \right).$
Let $a,b,c$ be positive real numbers such that $a+b+c=3$. Determine, with certainty, the largest possible value of the expression $$ \frac{a}{a^3+b^2+c}+\frac{b}{b^3+c^2+a}+\frac{c}{c^3+a^2+b}$$
Let $(a_n)_{n=1}^{\infty}$ be a strictly increasing sequence such that inequality
$$a_n(a_n-2a_{n-1})+a_{n-1}(a_{n-1}-2a_{n-2})\geq 0$$
holds for all $n \geq 3$. Prove that for all $n\geq2$ the inequality
$$a_n \geq a_{n-1}+a_{n-2}+\dots+a_1$$
holds as well.
Let $n \in \mathbb{N}, n \ge 2$ and the positive real numbers $a_1,a_2,…,a_n$ and $b_1,b_2,…,b_n$ such that $a_1+a_2+…+a_n=b_1+b_2+…+b_n=S.$
$\textbf{a)}$ Prove that $\sum\limits_{k=1}^n \frac{a_k^2}{a_k+b_k} \ge \frac{S}{2}.$
$\textbf{b)}$ Prove that $\sum\limits_{k=1}^n \frac{a_k^2}{a_k+b_k}= \sum\limits_{k=1}^n \frac{b_k^2}{a_k+b_k}.$
Prove that for positive reals $a,b,c$,
\[\frac{8a^2+2ab}{(b+\sqrt{6ac}+3c)^2}+\frac{2b^2+3bc}{(3c+\sqrt{2ab}+2a)^2}+\frac{18c^2+6ac}{(2a+\sqrt{3bc}+b})^2\geq 1\]
The set $ \{a_0, a_1, \ldots, a_n\}$ of real numbers satisfies the following conditions:
[b](i)[/b] $ a_0 \equal{} a_n \equal{} 0,$
[b](ii)[/b] for $ 1 \leq k \leq n \minus{} 1,$ \[ a_k \equal{} c \plus{} \sum^{n\minus{}1}_{i\equal{}k} a_{i\minus{}k} \cdot \left(a_i \plus{} a_{i\plus{}1} \right)\]
Prove that $ c \leq \frac{1}{4n}.$
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]
Let $a,b,c$ be positive reals satisfying :
\[ \frac{a}{1+b+c}+\frac{b}{1+c+a}+\frac{c}{1+a+b}\ge \frac{ab}{1+a+b}+\frac{bc}{1+b+c}+\frac{ca}{1+c+a} \]
Then prove that :
\[ \frac{a^2+b^2+c^2}{ab+bc+ca}+a+b+c+2\ge 2(\sqrt{ab}+\sqrt{bc}+\sqrt{ca}) \]
[i]Proposed by Dimitar Trenevski[/i]
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.
Prove the inequality
\[ \frac{a_1+ a_3}{a_1 + a_2} + \frac{a_2 + a_4}{a_2 + a_3} + \frac{a_3 + a_1}{a_3 + a_4} + \frac{a_4 + a_2}{a_4 + a_1} \geq 4, \]
where $a_i > 0, i = 1, 2, 3, 4.$
Let $a$, $b$, $c$ be positive reals such that $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=1$. Show that $$a^abc+b^bca+c^cab\ge 27bc+27ca+27ab.$$
[i]Proposed by Milan Haiman[/i]
Let $a, b$ and $c$ be pairwise different natural numbers. Prove $\frac{a^3 + b^3 + c^3}{3} \ge abc + a + b + c$.
When does equality holds?
(Karl Czakler)
Prove that for all positive real numbers $a,b,c,d$,
$$\frac{2}{(a+b)(c+d)+(b+c)(a+d)} \leq \frac{1}{(a+c)(b+d)+4ac}+\frac{1}{(a+c)(b+d)+4bd}$$
and determine when equality occurs.
Suppose $a$, $b$ and $c$ are nonzero real numberss satisfying $abc=2$. Prove that among the three numbers $2a-\frac{1}{b}$, $2b-\frac{1}{c}$ and $2c-\frac{1}{a}$, at most two of them are greater than $2$.