Found problems: 2529
Show for all integers $ n \ge 2005$ the following chaine of inequalities:
$ (n\plus{}830)^{2005}<n(n\plus{}1)\dots(n\plus{}2004)<(n\plus{}1002)^{2005}$
Prove that $0\le yz+zx+xy-2xyz\le{7\over27}$, where $x,y$ and $z$ are non-negative real numbers satisfying $x+y+z=1$.
If $x$ and $y$ are positive reals, prove that
$$x^2\sqrt{\frac{x}{y}}+y^2\sqrt{\frac{y}{x}} \geq x^2+y^2$$
Let $a$, $b$, $c$ be positive real numbers such that $\min(ab,bc,ca) \ge 1$. Prove that $$\sqrt[3]{(a^2+1)(b^2+1)(c^2+1)} \le \left(\frac{a+b+c}{3}\right)^2 + 1.$$
[i]Proposed by Tigran Margaryan, Armenia[/i]
Prove that if $a_1 , a_2 ,... , a_n$ are positive real numbers, then
$$(a_1 + a_2 + ... + a_n) \left( \frac{1}{a_1}+ \frac{1}{a_1}+...+\frac{1}{a_n}\right)\ge n^2$$.
When is equality valid?
Let $x,y,z$ be positive real numbers such that $x^2+y^2+z^2 = 25.$
Find the minimum of $\frac {xy} z + \frac {yz} x + \frac {zx} y .$
Pierre.
Let $x,y$ and $z$ be positive reals such that $x + y + z = 3$. Prove that
\[ 2 \sqrt{x + \sqrt{y}} + 2 \sqrt{y + \sqrt{z}} + 2 \sqrt{z + \sqrt{x}} \le \sqrt{8 + x - y} + \sqrt{8 + y - z} + \sqrt{8 + z - x} \]
Let $a$ and $b$ be two positive real numbers with $a \le 2b \le 4a$.
Prove that $4ab \le2 (a^2+ b^2) \le 5 ab$.
Denote an acute-angle $\vartriangle ABC $ with sides $a, b, c $ respectively by ${{H}_{a}}, {{H}_{b}}, {{H}_{c}} $ the feet of altitudes ${{h}_{a}}, {{h}_{b}}, {{h}_{c}} $. Prove the inequality:
$$\frac {h_ {a} ^{2}} {{{a} ^{2}} - CH_ {a} ^{2}} + \frac{h_{b} ^{2}} {{{ b}^{2}} - AH_{b} ^{2}} + \frac{h_{c}^{2}}{{{c}^{2}} - BH_{c}^{2}} \ge 3 $$
(Dmitry Petrovsky)
Cells of $11\times 11$ table are colored with $n$ colors (each cell is colored with exactly one color). For each color, the total amount of the cells of this color is not less than $7$ and not greater than $13$.
Prove that there exists at least one row or column which contains cells of at least four different colors.
[i](N. Sedrakyan)[/i]
Let $a,b,c $ be real positive numbers such that $abc(a+b+c)=3$
Prove that $(a+b)(b+c)(c+a) \geq 8$
Let $a, b, c, d > 0$ satisfy $36a + 4b + 4c + 3d = 25$. What is the maximum possible value of $ab^{1/2}c^{1/3}d^{1/4}$ ?
For nonnegative real numbers $a,b,c,x,y,z$, if$a+b+c=x+y+z=1$, find the maximum value of $(a-x^2)(b-y^2)(c-z^2)$.
Given the equation $3x^2 - 4x + k = 0$ with real roots. The value of $k$ for which the product of the roots of the equation is a maximum is:
$\textbf{(A)}\ \dfrac{16}{9} \qquad
\textbf{(B)}\ \dfrac{16}{3}\qquad
\textbf{(C)}\ \dfrac{4}{9} \qquad
\textbf{(D)}\ \dfrac{4}{3} \qquad
\textbf{(E)}\ -\dfrac{4}{3}$
Triangle $ABC$ has $BC=1$ and $AC=2$. What is the maximum possible value of $\hat{A}$.
Let $ ABC$ be a triangle. Points $ D$ and $ E$ are on sides $ AB$ and $ AC,$ respectively, and point $ F$ is on line segment $ DE.$ Let $ \frac {AD}{AB} \equal{} x,$ $ \frac {AE}{AC} \equal{} y,$ $ \frac {DF}{DE} \equal{} z.$ Prove that
(1) $ S_{\triangle BDF} \equal{} (1 \minus{} x)y S_{\triangle ABC}$ and $ S_{\triangle CEF} \equal{} x(1 \minus{} y) (1 \minus{} z)S_{\triangle ABC};$
(2) $ \sqrt [3]{S_{\triangle BDF}} \plus{} \sqrt [3]{S_{\triangle CEF}} \leq \sqrt [3]{S_{\triangle ABC}}.$
Let $a,b,c$ be positive real numbers such that $abc=1$. Determine the minimum value of
$E(a,b,c) = \sum \dfrac{a^3+5}{a^3(b+c)}$ .
Let $P$ be a convex polyhedron. Jaroslav writes a non-negative real number to every vertex of $P$ in such a way that the sum of these numbers is $1$. Afterwards, to every edge he writes the product of the numbers at the two endpoints of that edge. Prove that the sum of the numbers at the edges is at most $\frac{3}{8}$.
Find the maximal value of
\[S = \sqrt[3]{\frac{a}{b+7}} + \sqrt[3]{\frac{b}{c+7}} + \sqrt[3]{\frac{c}{d+7}} + \sqrt[3]{\frac{d}{a+7}},\]
where $a$, $b$, $c$, $d$ are nonnegative real numbers which satisfy $a+b+c+d = 100$.
[i]Proposed by Evan Chen, Taiwan[/i]
Let $a,b,c$ be nonnegative real numbers. Prove that \[ a(a - b)(a - 2b) + b(b - c)(b - 2c) + c(c - a)(c - 2a) \geq 0.\][i]Wenyu Cao[/i]
[i]Version 1[/i]. Let $n$ be a positive integer, and set $N=2^{n}$. Determine the smallest real number $a_{n}$ such that, for all real $x$,
\[
\sqrt[N]{\frac{x^{2 N}+1}{2}} \leqslant a_{n}(x-1)^{2}+x .
\]
[i]Version 2[/i]. For every positive integer $N$, determine the smallest real number $b_{N}$ such that, for all real $x$,
\[
\sqrt[N]{\frac{x^{2 N}+1}{2}} \leqslant b_{N}(x-1)^{2}+x .
\]
Let $\, a_1, a_2, a_3, \ldots \,$ be a sequence of positive real numbers satisfying $\, \sum_{j=1}^n a_j \geq \sqrt{n} \,$ for all $\, n \geq 1$. Prove that, for all $\, n \geq 1, \,$ \[ \sum_{j=1}^n a_j^2 > \frac{1}{4} \left( 1 + \frac{1}{2} + \cdots + \frac{1}{n} \right). \]
Prove that there are only two real numbers $x$ such that \[(x-1)(x-2)(x-3)(x-4)(x-5)(x-6) = 720\]
Suppose that the sequence $\{a_n\}$ of positive reals satisfies the following conditions:
[list]
[*]$a_i \leq a_j$ for every positive integers $i <j$.
[*]For any positive integer $k \geq 3$, the following inequality holds:
$$(a_1+a_2)(a_2+a_3)\cdots(a_{k-1}+a_k)(a_k+a_1)\leq (2^k+2022)a_1a_2\cdots a_k$$
[/list]
Prove that $\{a_n\}$ is constant.
For all pairwise distinct positive real numbers $a, b, c$ such that $abc = 1$, prove that $$\frac{\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+1}{(a+b+c+1)^2}+\frac{3}{8}\sqrt[3]{\frac{(a^2-b^2)^3+(b^2-c^2)^3+(c^2-a^2)^3}{(a-b)^3+(b-c)^3+(c-a)^3}}\geq 1.$$