Found problems: 310
Given are positive reals $a, b, c$, such that $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=3$. Prove that
$\frac{\sqrt{a+\frac{b}{c}}+\sqrt{b+\frac{c}{a}}+\sqrt{c+\frac{a}{b}}}{3}\leq \frac{a+b+c-1}{\sqrt{2}}$.
[i]Albania[/i]
Let $M$ be a point in the interior of triangle $ABC$. Let $A'$ lie on $BC$ with $MA'$ perpendicular to $BC$. Define $B'$ on $CA$ and $C'$ on $AB$ similarly. Define
\[
p(M) = \frac{MA' \cdot MB' \cdot MC'}{MA \cdot MB \cdot MC}.
\]
Determine, with proof, the location of $M$ such that $p(M)$ is maximal. Let $\mu(ABC)$ denote this maximum value. For which triangles $ABC$ is the value of $\mu(ABC)$ maximal?
Let $a, b, c$ be positive real numbers such that $a + b + c + ab + bc + ca + abc = 7$. Prove
that $\sqrt{a^2 + b^2 + 2 }+\sqrt{b^2 + c^2 + 2 }+\sqrt{c^2 + a^2 + 2 } \ge 6$ .
$a,b,c$ are positive numbers such that $a+b+c=3$, show that: \[\frac{a^{2}+9}{2a^{2}+(b+c)^{2}}+\frac{b^{2}+9}{2b^{2}+(a+c)^{2}}+\frac{c^{2}+9}{2c^{2}+(a+b)^{2}}\leq 5\]
Let $a_0,a_1,\cdots ,a_n$ be numbers from the interval $(0,\pi/2)$ such that \[ \tan (a_0-\frac{\pi}{4})+ \tan (a_1-\frac{\pi}{4})+\cdots +\tan (a_n-\frac{\pi}{4})\geq n-1. \] Prove that \[ \tan a_0\tan a_1 \cdots \tan a_n\geq n^{n+1}. \]
If $a$, $b$, $c \geq 1$; $y \geq x \geq 1$; $p$, $q$, $r > 0$ then$$\left(\frac{1+y\left(a^pb^qc^r\right)^{\frac{1}{p+q+r}}}{1+x\left(a^pb^qc^r\right)^{\frac{1}{p+q+r}}}\right)^{\frac{p+q+r}{\left(a^pb^qc^r\right)^{\frac{1}{p+q+r}}}}\left(\frac{1+xa}{1+ya}\right)^{\frac{p}{a}}\left(\frac{1+xb}{1+yb}\right)^{\frac{q}{b}}\left(\frac{1+xc}{1+yc}\right)^{\frac{r}{c}}$$ $$\geq \prod \limits_{cyc}\left(\frac{1+y\left(a^pb^q\right)^{\frac{1}{p+q}}}{1+x\left(a^pb^q\right)^{\frac{1}{p+q}}}\right)^{\frac{p+q}{\left(a^pb^q\right)^{\frac{1}{p+q}}}}$$
For an integer $m\geq 1$, we consider partitions of a $2^m\times 2^m$ chessboard into rectangles consisting of cells of chessboard, in which each of the $2^m$ cells along one diagonal forms a separate rectangle of side length $1$. Determine the smallest possible sum of rectangle perimeters in such a partition.
[i]Proposed by Gerhard Woeginger, Netherlands[/i]
Given an arbitrary triangle $ ABC$, denote by $ P,Q,R$ the intersections of the incircle with sides $ BC, CA, AB$ respectively. Let the area of triangle $ ABC$ be $ T$, and its perimeter $ L$. Prove that the inequality
\[\left(\frac {AB}{PQ}\right)^3 \plus{}\left(\frac {BC}{QR}\right)^3 \plus{}\left(\frac {CA}{RP}\right)^3 \geq \frac {2}{\sqrt {3}} \cdot \frac {L^2}{T}\]
holds.
Let $r,s,u,v$ be real numbers. Prove that:
$$min\{r-s^2,s-u^2, u-v^2,v-r^2\}\le \frac{1}{4}$$
Let $a, b, c$ be positive reals such that $abc=1$. Show that \[\frac{1}{a^5(b+2c)^2} + \frac{1}{b^5(c+2a)^2} + \frac{1}{c^5(a+2b)^2} \ge \frac{1}{3}.\]