Found problems: 29
Let $a, b, c$ be the side-lengths of a triangle, $r$ be the inradius and $r_a, r_b, r_c$ be the corresponding exradius. Show that
\[ \frac{a+b+c}{\sqrt{a^2+b^2+c^2}} \leq 2 \cdot \frac{\sqrt{{r_a}^2+{r_b}^2+{r_c}^2}}{r_a+r_b+r_c-3r} \]
Let $ x,y,z$ be positive real numbers,satisfying equality $ x^{2}\plus{}y^{2}\plus{}z^{2}\equal{}25$. Find the minimal possible value of the expression $ \frac{xy}{z} \plus{} \frac{yz}{x} \plus{} \frac{zx}{y}$.
Let $a, b, c$ be positive real numbers such that
\[\frac{a}{1+a}+\frac{b}{1+b}+\frac{c}{1+c}=2.\]
Prove that
\[\frac{\sqrt a + \sqrt b+\sqrt c}{2} \geq \frac{1}{\sqrt a}+\frac{1}{\sqrt b}+\frac{1}{\sqrt c}.\]
Let $ n > 3$ be a fixed integer and $ x_1,x_2,\ldots, x_n$ be positive real numbers. Find, in terms of $ n$, all possible real values of
\[ {x_1\over x_n\plus{}x_1\plus{}x_2} \plus{} {x_2\over x_1\plus{}x_2\plus{}x_3} \plus{} {x_3\over x_2\plus{}x_3\plus{}x_4} \plus{} \cdots \plus{} {x_{n\minus{}1}\over x_{n\minus{}2}\plus{}x_{n\minus{}1}\plus{}x_n} \plus{} {x_n\over x_{n\minus{}1}\plus{}x_n\plus{}x_1}\]
Let $ABC$ be a triangle with $\angle BAC=60^\circ$. Consider a point $P$ inside the triangle having $PA=1$, $PB=2$ and $PC=3$. Find the maximum possible area of the triangle $ABC$.
For each positive integer $n$, find the largest real number $C_n$ with the following property. Given any $n$ real-valued functions $f_1(x), f_2(x), \cdots, f_n(x)$ defined on the closed interval $0 \le x \le 1$, one can find numbers $x_1, x_2, \cdots x_n$, such that $0 \le x_i \le 1$ satisfying
\[|f_1(x_1)+f_2(x_2)+\cdots f_n(x_n)-x_1x_2\cdots x_n| \ge C_n\]
[i]Marko Radovanović, Serbia[/i]
For each integer $n\geqslant2$, determine the largest real constant $C_n$ such that for all positive real numbers $a_1, \ldots, a_n$ we have
\[\frac{a_1^2+\ldots+a_n^2}{n}\geqslant\left(\frac{a_1+\ldots+a_n}{n}\right)^2+C_n\cdot(a_1-a_n)^2\mbox{.}\]
[i](4th Middle European Mathematical Olympiad, Team Competition, Problem 2)[/i]
For each positive integer $n$, find the largest real number $C_n$ with the following property. Given any $n$ real-valued functions $f_1(x), f_2(x), \cdots, f_n(x)$ defined on the closed interval $0 \le x \le 1$, one can find numbers $x_1, x_2, \cdots x_n$, such that $0 \le x_i \le 1$ satisfying
\[|f_1(x_1)+f_2(x_2)+\cdots f_n(x_n)-x_1x_2\cdots x_n| \ge C_n\]
[i]Marko Radovanović, Serbia[/i]
Is there a sequence $ a_1,a_2,\ldots$ of positive reals satisfying simoultaneously the following inequalities for all positive integers $ n$:
a) $ a_1\plus{}a_2\plus{}\ldots\plus{}a_n\le n^2$
b) $ \frac1{a_1}\plus{}\frac1{a_2}\plus{}\ldots\plus{}\frac1{a_n}\le2008$?
[color=darkred]Let $\mathcal{C}$ be the set of integrable functions $f\colon [0,1]\to\mathbb{R}$ such that $0\le f(x)\le x$ for any $x\in [0,1]$ . Define the function $V\colon\mathcal{C}\to\mathbb{R}$ by
\[V(f)=\int_0^1f^2(x)\ \text{d}x-\left(\int_0^1f(x)\ \text{d}x\right)^2\ ,\ f\in\mathcal{C}\ .\]
Determine the following two sets:
[list][b]a)[/b] $\{V(f_a)\, |\, 0\le a\le 1\}$ , where $f_a(x)=0$ , if $0\le x\le a$ and $f_a(x)=x$ , if $a<x\le 1\, ;$
[b]b)[/b] $\{V(f)\, |\, f\in\mathcal{C}\}\ .$[/list] [/color]
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}}.$
Is there a sequence $ a_1,a_2,\ldots$ of positive reals satisfying simoultaneously the following inequalities for all positive integers $ n$:
a) $ a_1\plus{}a_2\plus{}\ldots\plus{}a_n\le n^2$
b) $ \frac1{a_1}\plus{}\frac1{a_2}\plus{}\ldots\plus{}\frac1{a_n}\le2008$?
For positive real numbers $a,b$ and $c$ we have $a+b+c=3$. Prove
$\frac{a}{1+(b+c)^2}+\frac{b}{1+(a+c)^2}+\frac{c}{1+(a+b)^2}\le \frac{3(a^2+b^2+c^2)}{a^2+b^2+c^2+12abc}$.
[i]proposed by Mohammad Ahmadi[/i]
For two given positive integers $ m,n > 1$, let $ a_{ij} (i = 1,2,\cdots,n, \; j = 1,2,\cdots,m)$ be nonnegative real numbers, not all zero, find the maximum and the minimum values of $ f$, where
\[ f = \frac {n\sum_{i = 1}^{n}(\sum_{j = 1}^{m}a_{ij})^2 + m\sum_{j = 1}^{m}(\sum_{i= 1}^{n}a_{ij})^2}{(\sum_{i = 1}^{n}\sum_{j = 1}^{m}a_{ij})^2 + mn\sum_{i = 1}^{n}\sum_{j=1}^{m}a_{ij}^2}. \]
$x_{1}, x_{2}, \cdots, x_{n}$ are positive numbers such that $\sum_{i=1}^{n}x_{i}= 1$. Prove that \[\left( \sum_{i=1}^{n}\sqrt{x_{i}}\right) \left( \sum_{i=1}^{n}\frac{1}{\sqrt{1+x_{i}}}\right) \leq \frac{n^{2}}{\sqrt{n+1}}\]
$a$,$b$,$c$ are real. What is the highest value of $a+b+c$ if $a^2+4b^2+9c^2-2a-12b+6c+2=0$
$x_{1}, x_{2}, \cdots, x_{n}$ are positive numbers such that $\sum_{i=1}^{n}x_{i}= 1$. Prove that \[\left( \sum_{i=1}^{n}\sqrt{x_{i}}\right) \left( \sum_{i=1}^{n}\frac{1}{\sqrt{1+x_{i}}}\right) \leq \frac{n^{2}}{\sqrt{n+1}}\]
Let $ x,y,z$ be positive real numbers,satisfying equality $ x^{2}\plus{}y^{2}\plus{}z^{2}\equal{}25$. Find the minimal possible value of the expression $ \frac{xy}{z} \plus{} \frac{yz}{x} \plus{} \frac{zx}{y}$.
For each positive integer $n$, find the largest real number $C_n$ with the following property. Given any $n$ real-valued functions $f_1(x), f_2(x), \cdots, f_n(x)$ defined on the closed interval $0 \le x \le 1$, one can find numbers $x_1, x_2, \cdots x_n$, such that $0 \le x_i \le 1$ satisfying
\[|f_1(x_1)+f_2(x_2)+\cdots f_n(x_n)-x_1x_2\cdots x_n| \ge C_n\]
[i]Marko Radovanović, Serbia[/i]
Let $a,b,c \in \mathbb {R}^{+}$ and $abc=1$ prove that:
$\frac {a+b}{(a+b+1)^2}+\frac {b+c}{(b+c+1)^2}+\frac {c+a}{(c+a+1)^2} \geq \frac {2}{a+b+c}$
$\triangle ABC$ has semiperimeter $s$ and area $F$ . A square $P QRS$ with side length $x$ is inscribed in $ABC$ with $P$ and $Q$ on $BC$, $R$ on $AC$, and $S$ on $AB$. Similarly, $y$ and $z$ are the sides of squares two vertices of which lie on $AC$ and $AB$, respectively. Prove that
\[\frac 1x +\frac 1y + \frac 1z \le \frac{s(2+\sqrt3)}{2F}\]
Let $ a,b,c$ be positive real numbers such that $ ab\plus{}bc\plus{}ca\equal{}3$. Prove that
\[ \frac 1{1\plus{}a^2(b\plus{}c)} \plus{} \frac 1{1\plus{}b^2(c\plus{}a)} \plus{} \frac 1 {1\plus{}c^2(a\plus{}b) } \leq \frac 3 {1\plus{}2abc} .\]
Let $ a$, $ b$, $ c$ be positive real numbers such that $ a \plus{} b \plus{} c \equal{} 1$. Prove inequality:
\[ \frac{1}{bc \plus{} a \plus{} \frac{1}{a}} \plus{} \frac{1}{ac \plus{} b \plus{} \frac{1}{b}} \plus{} \frac{1}{ab \plus{} c \plus{} \frac{1}{c}} \leqslant \frac{27}{31}.\]
Let $n \ge 2$ be a fixed even integer. We consider polynomials of the form
\[P(x) = x^n + a_{n-1}x^{n-1} + \cdots + a_1x + 1\]
with real coefficients, having at least one real roots. Find the least possible value of $a^2_1 + a^2_2 + \cdots + a^2_{n-1}$.
For two given positive integers $ m,n > 1$, let $ a_{ij} (i = 1,2,\cdots,n, \; j = 1,2,\cdots,m)$ be nonnegative real numbers, not all zero, find the maximum and the minimum values of $ f$, where
\[ f = \frac {n\sum_{i = 1}^{n}(\sum_{j = 1}^{m}a_{ij})^2 + m\sum_{j = 1}^{m}(\sum_{i= 1}^{n}a_{ij})^2}{(\sum_{i = 1}^{n}\sum_{j = 1}^{m}a_{ij})^2 + mn\sum_{i = 1}^{n}\sum_{j=1}^{m}a_{ij}^2}. \]