Found problems: 2529
Let $S$ be a set of positive integers such that $\lfloor \sqrt{x}\rfloor =\lfloor \sqrt{y}\rfloor $ for all $x, y \in S$. Show that the products $xy$, where $x, y \in S$, are pairwise distinct.
Prove that for all positive real numbers $a,b,c$, \[ \frac{a}{\sqrt{a^2 + 8bc}} + \frac{b}{\sqrt{b^2 + 8ca}} + \frac{c}{\sqrt{c^2 + 8ab}} \geq 1. \]
Solve in the set $R$ the equation $$\frac{3x+3}{\sqrt{x}}-\frac{x+1}{\sqrt{x^2-x+1}}=4$$
Two points $A$ and $B$ are given in the plane. A point $X$ is called their [i]preposterous midpoint[/i] if there is a Cartesian coordinate system in the plane such that the coordinates of $A$ and $B$ in this system are non-negative, the abscissa of $X$ is the geometric mean of the abscissae of $A$ and $B$, and the ordinate of $X$ is the geometric mean of the ordinates of $A$ and $B$. Find the locus of all the [i]preposterous midpoints[/i] of $A$ and $B$.
(K. Tyschu)
Let $a,b,c$ be nonnegative real numbers.Prove that $3(a^2+b^2+c^2)\ge (a+b+c)(\sqrt{ab}+\sqrt{bc}+\sqrt{ca})+(a-b)^2+(b-c)^2+(c-a)^2\ge (a+b+c)^2$.
For all positive real numbers $a, b, c$ satisfying $a+b+c=1$, prove that
\[ \frac{a^4+5b^4}{a(a+2b)} + \frac{b^4+5c^4}{b(b+2c)} + \frac{c^4+5a^4}{c(c+2a)} \geq 1- ab-bc-ca \]
Determine all functions $f:(0,\infty)\to\mathbb{R}$ satisfying $$\left(x+\frac{1}{x}\right)f(y)=f(xy)+f\left(\frac{y}{x}\right)$$ for all $x,y>0$.
Given $n$ positive real numbers $x_1,x_2,x_3,...,x_n$ such that
$$\left (1+\frac{1}{x_1}\right )\left(1+\frac{1}{x_2}\right)...\left(1+\frac{1}{x_n}\right)=(n+1)^n$$
Determine the minimum value of $x_1+x_2+x_3+...+x_n$.
[i]Proposed by Loh Kwong Weng[/i]
Let $P(x)$ be a polynomial with real non-negative coefficients. Let $k$ be a positive integer and $x_1, x_2, \dots, x_k$ positive real numbers such that $x_1x_2\cdots x_k=1$. Prove that
$$P(x_1)+P(x_2)+\cdots+P(x_k)\geq kP(1).$$
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]
For any positive real numbers $x,y,z$, prove that
$\frac{x}{y}+\frac{y}{z}+\frac{z}{x} \geq \frac{z(x+y)}{y(y+z)} + \frac{x(z+y)}{z(x+z)} + \frac{y(x+z)}{x(x+y)}$
Given a triangle $ \,ABC,\,$ let $ \,I\,$ be the center of its inscribed circle. The internal bisectors of the angles $ \,A,B,C\,$ meet the opposite sides in $ \,A^{\prime },B^{\prime },C^{\prime }\,$ respectively. Prove that
\[ \frac {1}{4} < \frac {AI\cdot BI\cdot CI}{AA^{\prime }\cdot BB^{\prime }\cdot CC^{\prime }} \leq \frac {8}{27}.
\]
Let be $f:\left[0,1\right]\rightarrow\left[0,1\right]$ a continuous and bijective function,such that :
$f\left(0\right)=0$.Then the following inequality holds:
$\left(\alpha+2\right)\cdotp\int_{0}^{1}x^{\alpha}\left(f\left(x\right)+f^{-1}\left(x\right)\right)\leq2,\forall\alpha\geq0 $
$ 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 $x$ and $y$ be two positive real numbers, such that $x + y = 1$. Prove that $$\left(1 +\frac{1}{x}\right)\left(1 +\frac{1}{y}\right) \ge 9$$
Let $a\geq b\geq c\geq 0$ are real numbers such that $a+b+c=3$. Prove that $ab^{2}+bc^{2}+ca^{2}\leq\frac{27}{8}$ and find cases of equality.
Prove that for integer $n$ we have:
\[n! \le \left( \frac{n+1}{2} \right)^n\]
[size=75][i](please note that the pupils in the competition never heard of AM-GM or alikes, it is intended to be solved without any knowledge on inequalities)[/i][/size]
Let $ABC$ be a triangle whose side lengths are opposite the angle $A,B,C$ are $a,b,c$ respectively. Prove that $$\frac{ab\sin{2C}+bc\sin{ 2A}+ca\sin{2B}}{ab+bc+ca}\leq\frac{\sqrt{3}}{2}$$.
[i](nooonuii)[/i]
For all reals $a, b, c,d $ prove the following inequality:
$$\frac{a + b + c + d}{(1 + a^2)(1 + b^2)(1 + c^2)(1 + d^2)}< 1$$
Show that for any $a,b,c$ positive reals,
\[ (a+b)(a+c) \geq 2 \sqrt{abc(a+b+c)} \]
Let $0\le a\le b\le c$ be real numbers. Prove that
\[(a+3b)(b+4c)(c+2a)\ge 60abc \]
If $a,b,c$ are triangle sides then prove that $(\sum_{cyc}\sqrt{\frac{a}{-a+b+c}} \geq 3$
A right triangle is inscribed in parabola $y=x^2$. Prove that it's hypotenuse is not less than $2$.
Let $\alpha$ be an arbitrary positive real number. Determine for this number $\alpha$ the greatest real number $C$ such that the inequality$$\left(1+\frac{\alpha}{x^2}\right)\left(1+\frac{\alpha}{y^2}\right)\left(1+\frac{\alpha}{z^2}\right)\geq C\left(\frac{x}{z}+\frac{z}{x}+2\right)$$
is valid for all positive real numbers $x, y$ and $z$ satisfying $xy + yz + zx =\alpha.$ When does equality occur?
[i](Proposed by Walther Janous)[/i]
Let $0\leq p,r\leq 1$ and consider the identities
$$a)\; (px+(1-p)y)^{2}=a x^2 +bxy +c y^2, \;\;\;\, b)\; (px+(1-p)y)(rx+(1-r)y) =\alpha x^2 + \beta xy + \gamma y^2.$$
Show that
$$ a)\; \max(a,b,c) \geq \frac{4}{9}, \;\;\;\; b)\; \max( \alpha, \beta , \gamma) \geq \frac{4}{9}.$$