Let $a,b,c$ be positive real numbers such that $abc\geq 1$. Prove that \[ \frac{1}{1+b+c}+\frac{1}{1+c+a}+\frac{1}{1+a+b}\leq 1. \] [hide="Remark"]This problem derives from the well known inequality given in [url=http://www.mathlinks.ro/Forum/viewtopic.php?p=185470#p185470]USAMO 1997, Problem 5[/url]. [/hide]

Find the max and minimum without using dervivate: $\sqrt{x} +4 \cdot \sqrt{\frac{1}{2} - x}$

Let $ n \geq 2, n \in \mathbb{N}$ and let $ p, a_1, a_2, \ldots, a_n, b_1, b_2, \ldots, b_n \in \mathbb{R}$ satisfying $ \frac{1}{2} \leq p \leq 1,$ $ 0 \leq a_i,$ $ 0 \leq b_i \leq p,$ $ i \equal{} 1, \ldots, n,$ and \[ \sum^n_{i\equal{}1} a_i \equal{} \sum^n_{i\equal{}1} b_i.\] Prove the inequality: \[ \sum^n_{i\equal{}1} b_i \prod^n_{j \equal{} 1, j \neq i} a_j \leq \frac{p}{(n\minus{}1)^{n\minus{}1}}.\]

We consider two sequences of real numbers $x_{1} \geq x_{2} \geq \ldots \geq x_{n}$ and $\ y_{1} \geq y_{2} \geq \ldots \geq y_{n}.$ Let $z_{1}, z_{2}, .\ldots, z_{n}$ be a permutation of the numbers $y_{1}, y_{2}, \ldots, y_{n}.$ Prove that $\sum \limits_{i=1}^{n} ( x_{i} -\ y_{i} )^{2} \leq \sum \limits_{i=1}^{n}$ $( x_{i} - z_{i})^{2}.$

The sum of the squares of five real numbers $a_1, a_2, a_3, a_4, a_5$ equals $1$. Prove that the least of the numbers $(a_i - a_j)^2$, where $i, j = 1, 2, 3, 4,5$ and $i \neq j$, does not exceed $\frac{1}{10}.$

We say that a $2023$-tuple of nonnegative integers $(a_1,\hdots,a_{2023})$ is [i]sweet[/i] if the following conditions hold: [list] [*] $a_1+\hdots+a_{2023}=2023$ [*] $\frac{a_1}{2}+\frac{a_2}{2^2}+\hdots+\frac{a_{2023}}{2^{2023}}\le 1$ [/list] Determine the greatest positive integer $L$ so that \[a_1+2a_2+\hdots+2023a_{2023}\ge L\] holds for every sweet $2023$-tuple $(a_1,\hdots,a_{2023})$ [i]Ivan Novak[/i]

Let $a$, $b$ and $c$ be positive real numbers such that $abc=1$. Prove the inequality: $\frac{ab}{a^5+b^5+ab} +\frac{bc}{b^5+c^5+bc}+\frac{ac}{c^5+a^5+ac}\leq 1$

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]

$\boxed{\text{A1}}$Let $a,b,c$ be positive reals numbers such that $a+b+c=1$.Prove that $2(a^2+b^2+c^2)\ge \frac{1}{9}+15abc$

Prove that for positive reals $a,b,c$, \[\frac{8a^2+2ab}{(b+\sqrt{6ac}+3c)^2}+\frac{2b^2+3bc}{(3c+\sqrt{2ab}+2a)^2}+\frac{18c^2+6ac}{(2a+\sqrt{3bc}+b})^2\geq 1\]

Let $n$ be a positive integer and let $a_1, a_2, \ldots, a_n$ be positive reals. Show that $$\sum_{i=1}^{n} \frac{1}{2^i}(\frac{2}{1+a_i})^{2^i} \geq \frac{2}{1+a_1a_2\ldots a_n}-\frac{1}{2^n}.$$

Let $x_0 = 5$ and $x_{n+1} = x_n + \frac{1}{x_n} \ (n = 0, 1, 2, \ldots )$. Prove that \[45 < x_{1000} < 45. 1.\]

$A = \{a_1, a_2, . . . , a_k\}$ is a set of positive integers for which the sum of some (we can have only one number too) different numbers from the set is equal to a different number i.e. there $2^k - 1$ different sums of different numbers from $A$. Prove that the following inequality holds: $\frac{1}{a_1}+\frac{1}{a_2}+...+\frac{1}{a_k}<2$

Let $a$, $b$, $c$ be positive real numbers such that $\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2} = \frac{27}{4}.$ Show that: $$\frac{a^3+b^2}{a^2+b^2} + \frac{b^3+c^2}{b^2+c^2} + \frac{c^3+a^2}{c^2+a^2} \geq \frac{5}{2}.$$ [i]Authored by Nikola Velov[/i]

Prove for positive $a,b,c$ that $$ (a^2+\frac{b^2}{c^2})(b^2+\frac{c^2}{a^2})(c^2+\frac{a^2}{b^2}) \geq abc (a+\frac{1}{a})(b+\frac{1}{b})(c+\frac{1}{c})$$

Prove that for every pair of positive real numbers $a, b$ and for every positive integer $n$, $$(a+b)^n-a^n-b^n \ge \frac{2^n-2}{2^{n-2}} \cdot ab(a+b)^{n-2}.$$

Let $x$ and $y$ are real numbers such that $x^2y^2+2yx^2+1=0.$ If $S=\frac{2}{x^2}+1+\frac{1}{x}+y(y+2+\frac{1}{x})$, find (a)max$S$ and (b) min$S$.

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.

Show that $\frac{20}{60} <\sin 20^{\circ} < \frac{21}{60}.$

Let $x$ and $y$ be real numbers where at least one of them is bigger than $2$ and $xy+4 > 2(x+y)$ holds. Show that $xy>x+y$.

For each positive integer $k$ greater than $1$, find the largest real number $t$ such that the following hold: Given $n$ distinct points $a^{(1)}=(a^{(1)}_1,\ldots, a^{(1)}_k)$, $\ldots$, $a^{(n)}=(a^{(n)}_1,\ldots, a^{(n)}_k)$ in $\mathbb{R}^k$, we define the score of the tuple $a^{(i)}$ as \[\prod_{j=1}^{k}\#\{1\leq i'\leq n\textup{ such that }\pi_j(a^{(i')})=\pi_j(a^{(i)})\}\] where $\#S$ is the number of elements in set $S$, and $\pi_j$ is the projection $\mathbb{R}^k\to \mathbb{R}^{k-1}$ omitting the $j$-th coordinate. Then the $t$-th power mean of the scores of all $a^{(i)}$'s is at most $n$. Note: The $t$-th power mean of positive real numbers $x_1,\ldots,x_n$ is defined as \[\left(\frac{x_1^t+\cdots+x_n^t}{n}\right)^{1/t}\] when $t\neq 0$, and it is $\sqrt[n]{x_1\cdots x_n}$ when $t=0$. [i]Proposed by Cheng-Ying Chang and usjl[/i]

If a b c positive reals smaller than 1, prove: a+b+c+2abc>ab+bc+ca+2(abc)^(1/2)

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?

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]

Let $x$, $y$, and $z$ be positive real numbers such that $x + y + z = 1$. Prove that $\frac{(x+y)^3}{z} + \frac{(y+z)^3}{x} + \frac{(z+x)^3}{y} + 9xyz \ge 9(xy + yz + zx)$. When does equality hold?