Found problems: 1671
Find the smallest constant $ k$ such that
$ \frac {x}{\sqrt {x \plus{} y}} \plus{} \frac {y}{\sqrt {y \plus{} z}} \plus{} \frac {z}{\sqrt {z \plus{} x}}\leq k\sqrt {x \plus{} y \plus{} z}$
for all positive $ x$, $ y$, $ z$.
Let $a$, $b$, and $c$ be positive numbers satisfying $abc=1$. Show that
\[\frac{a^3}{a^3+2}+\frac{b^3}{b^3+2}+\frac{c^3}{c^3+2}\ge1\text{ and }\frac1{a^3+2}+\frac1{b^3+2}+\frac1{c^3+2}\le1\]
Given that $x^2 + y^2 = 14x + 6y + 6$, what is the largest possible value that $3x + 4y$ can have?
$\text{(A)}\ 72 \qquad \text{(B)}\ 73 \qquad \text{(C)}\ 74 \qquad \text{(D)}\ 75\qquad \text{(E)}\ 76$
Find all non-zero real numbers $a, b, c$ such that the following polynomial has four (not necessarily distinct) positive real roots.
$$P(x) = ax^4 - 8ax^3 + bx^2 - 32cx + 16c$$
Find all positive integers $m$ such that there exists integers $x$ and $y$ that satisfies $$m \mid x^2+11y^2+2022.$$
Prove that \[ \frac{a}{b+2c+3d} +\frac{b}{c+2d+3a} +\frac{c}{d+2a+3b}+ \frac{d}{a+2b+3c} \geq \frac{2}{3} \] for all positive real numbers $a,b,c,d$.
Let be given positive reals $a$, $b$, $c$. Prove that: $\frac{a^{3}}{\left(a+b\right)^{3}}+\frac{b^{3}}{\left(b+c\right)^{3}}+\frac{c^{3}}{\left(c+a\right)^{3}}\geq \frac{3}{8}$.
Let $\omega$ be a $n$ -th primitive root of unity. Given complex numbers $a_1,a_2,\cdots,a_n$, and $p$ of them are non-zero. Let
$$b_k=\sum_{i=1}^n a_i \omega^{ki}$$
for $k=1,2,\cdots, n$. Prove that if $p>0$, then at least $\tfrac{n}{p}$ numbers in $b_1,b_2,\cdots,b_n$ are non-zero.
Show that
\[ \sum_{cyc} \sqrt[4]{\frac{(a^2+b^2)(a^2-ab+b^2)}{2}} \leq \frac{2}{3}(a^2+b^2+c^2)\left(\frac{1}{a+b}+\frac{1}{b+c}+\frac{1}{c+a}\right) \]
for all positive real numbers $a, \: b, \: c.$
Let $ A\subset \mathbb C$ be a closed and countable set. Prove that if the analytic function $ f: \mathbb C\backslash A\longrightarrow \mathbb C$ is bounded, then $ f$ is constant.
Let $x_1,x_2,\cdots,x_n$ be real numbers with $x_1^2+x_2^2+\cdots+x_n^2=1$. Prove that
\[\sum_{k=1}^{n}\left(1-\dfrac{k}{{\displaystyle \sum_{i=1}^{n} ix_i^2}}\right)^2 \cdot \dfrac{x_k^2}{k} \leq \left(\dfrac{n-1}{n+1}\right)^2 \sum_{k=1}^{n} \dfrac{x_k^2}{k}\]
Determine when does the equality hold?
Show that the inequality \[3(1 + a^2 + a^4)\geq (1 + a + a^2)^2\]
holds for all real numbers $a.$
Let $a\in\mathbb{R}-\{0\}$. Find all functions $f: \mathbb{R}\to\mathbb{R}$ such that $f(a+x) = f(x) - x$ for all $x\in\mathbb{R}$.
[i]Dan Schwartz[/i]
Given is a triangle $ABC$, the inscribed circle $G$ of which has radius $r$. Let $r_a$ be the radius of the circle touching $AB$, $AC$ and $G$. [This circle lies inside triangle $ABC$.] Define $r_b$ and $r_c$ similarly. Prove that $r_a + r_b + r_c \geq r$ and find all cases in which equality occurs.
[i]Bosnia - Herzegovina Mathematical Olympiad 2002[/i]
Let us divide by straight lines a quadrangle of unit area into $ n$ subpolygons and draw a circle into each subpolygon. Show that the sum of the perimeters of the circles is at most $ \pi \sqrt{n}$ (the lines are not allowed to cut the interior of a subpolygon).
[i]G. and L. Fejes-Toth[/i]
Prove that any positive real numbers a,b,c satisfy the inequlaity $$\frac{1}{(a+b)b}+\frac{1}{(b+c)c}+\frac{1}{(c+a)a}\ge \frac{9}{2(ab+bc+ca)}$$
I.Voronovich
Prove that for all positive real numbers $a,b,c$, \[ \frac{a}{\sqrt{a^2 + 8bc}} + \frac{b}{\sqrt{b^2 + 8ca}} + \frac{c}{\sqrt{c^2 + 8ab}} \geq 1. \]
The tetrahedron $OABC$ has all angles at vertex $O$ equal to $60^o$. Prove that $$AB \cdot BC + BC \cdot CA + CA \cdot AB \ge OA^2 + OB^2 + OC^2$$
For every set $S$ with $n(\ge3)$ distinct integers, show that there exists a function $f:\{1,2,\dots,n\}\rightarrow S$ satisfying the following two conditions.
(i) $\{ f(1),f(2),\dots,f(n)\} = S$
(ii) $2f(j)\neq f(i)+f(k)$ for all $1\le i<j<k\le n$.
For all positive real numbers $a, b, c$ satisfying $a+b+c=1$, prove that
\[ \frac{a^4+5b^4}{a(a+2b)} + \frac{b^4+5c^4}{b(b+2c)} + \frac{c^4+5a^4}{c(c+2a)} \geq 1- ab-bc-ca \]
$a,b,c \geq-3/4$ and $a+b+c=1$. Show that: $\frac{a}{1+a^{2}}+\frac{b}{1+b^{2}}+\frac{c}{1+c^{2}}\leq \frac{9}{10}$
Prove the inequality
\[
{x\over y^2-z}+{y\over z^2-x}+{z\over x^2-y} > 1,
\]
where $2 < x, y, z < 4.$
[i]Proposed by A. Golovanov[/i]
$ u,v,w > 0$,such that $ u \plus{} v \plus{} w \plus{} \sqrt {uvw} \equal{} 4$
prove that $ \sqrt {\frac {uv}{w}} \plus{} \sqrt {\frac {vw}{u}} \plus{} \sqrt {\frac {wu}{v}}\geq u \plus{} v \plus{} w$
Let $a_1>a_2>.....>a_r$ be positive real numbers .
Compute $\lim_{n\to \infty} (a_1^n+a_2^n+.....+a_r^n)^{\frac{1}{n}}$
Let $a, b, c > 0$ satisfy $a + b + c \ge \frac{1}{a} +\frac{1}{b} +\frac{1}{c}$. Prove that $a^3 + b^3 + c^3 \ge a + b + c$