Found problems: 1671
Let $a,b,c$ positive reals such that $a^2+b^2+c^2=1$. Prove that $$\frac{a+b}{\sqrt{ab+1}}+\frac{b+c}{\sqrt{bc+1}}+\frac{c+a}{\sqrt{ac+1}}\le 3$$
Let $x,y$ be positive real numbers such that $x+y=1$.
Prove that$\frac{x}{x^2+y^3}+\frac{y}{x^3+y^2}\leq2(\frac{x}{x+y^2}+\frac{y}{x^2+y})$.
Let $a$ and $b$ be two real numbers, with $0<a,b<1$. Prove that
\[\sqrt{ab^2+a^2b}+\sqrt{(1-a)(1-b)^2+(1-a)^2(1-b)}<\sqrt{2}\]
Solve the following equation in $\mathbb{R}^+$ :
\[\left\{\begin{matrix}
\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=2010\\
x+y+z=\frac{3}{670}
\end{matrix}\right.\]
Let $A$ be a finite set, and $A_1,A_2,\cdots, A_n$ are subsets of $A$ with the following conditions:
(1) $|A_1|=|A_2|=\cdots=|A_n|=k$, and $k>\frac{|A|}{2}$;
(2) for any $a,b\in A$, there exist $A_r,A_s,A_t\,(1\leq r<s<t\leq n)$ such that
$a,b\in A_r\cap A_s\cap A_t$;
(3) for any integer $i,j\, (1\leq i<j\leq n)$, $|A_i\cap A_j|\leq 3$.
Find all possible value(s) of $n$ when $k$ attains maximum among all possible systems $(A_1,A_2,\cdots, A_n,A)$.
Some cells of a $10 \times 10$ are colored blue. A set of six cells is called [i]gremista[/i] when the cells are the intersection of three rows and two columns, or two rows and three columns, and are painted blue. Determine the greatest value of $n$ for which it is possible to color $n$ chessboard cells blue such that there is not a [i]gremista[/i] set.
Let $a_1$, $a_2$, $a_3$, $a_4$, and $a_5$ be real numbers satisfying
\begin{align*}
a_1a_2 + a_2a_3 + a_3a_4 + a_4a_5 + a_5a_1 & = 20,\\
a_1a_3 + a_2a_4 + a_3a_5 + a_4a_1 + a_5a_2 & = 22.
\end{align*}Then the smallest possible value of $a_1^2 + a_2^2 + a_3^2 + a_4^2 + a_5^2$ can be expressed as $m + \sqrt{n}$, where $m$ and $n$ are positive integers. Compute $100m + n$.
[i]Proposed by Ankan Bhattacharya[/i]
Prove that $\frac{1}{x+y+1}-\frac{1}{(x+1)(y+1)}<\frac{1}{11}$ for all positive $x$ and $y$.
Assume $ a_{1} \ge a_{2} \ge \dots \ge a_{107} > 0 $ satisfy $ \sum\limits_{k=1}^{107}{a_{k}} \ge M $ and $ b_{107} \ge b_{106} \ge \dots \ge b_{1} > 0 $ satisfy $ \sum\limits_{k=1}^{107}{b_{k}} \le M $. Prove that for any $ m \in \{1,2, \dots, 107\} $, the arithmetic mean of the following numbers $$ \frac{a_{1}}{b_{1}}, \frac{a_{2}}{b_{2}}, \dots, \frac{a_{m}}{b_{m}} $$ is greater than or equal to $ \frac{M}{N} $
Find all functions $ f: \mathbb{Q} \rightarrow \mathbb{Q}$ such that:
$ f(x\plus{}f(y))\equal{}y\plus{}f(x)$ for all $ x,y \in \mathbb{Q}$.
Find all functions $f: \mathbb R \to \mathbb R$ such that for all reals $x$ and $y$,
\[f(x+y)+f(x)f(y)=f(xy)+f(x)+f(y).\]
Let \( \mathbb{R} \) be the set of real numbers. Determine all functions \( f: \mathbb{R} \to \mathbb{R} \) such that, for any real numbers \( x \) and \( y \),
\[
f(x^2 y - y) = f(x)^2 f(y) + f(x)^2 - 1.
\]
Find the smallest positive real number $k$ such that the following inequality holds $$\left|z_{1}+\ldots+z_{n}\right| \geqslant \frac{1}{k}\big(\left|z_{1}\right|+\ldots+\left|z_{n}\right|\big) .$$ for every positive integer $n \geqslant 2$ and every choice $z_{1}, \ldots, z_{n}$ of complex numbers with non-negative real and imaginary parts.
[Hint: First find $k$ that works for $n=2$. Then show that the same $k$ works for any $n \geqslant 2$.]
Show that
\[\left(a+2b+\dfrac{2}{a+1}\right)\left(b+2a+\dfrac{2}{b+1}\right)\geq 16\]
for all positive real numbers $a$ and $b$ such that $ab\geq 1$.
Find all functions $f$ from the reals to the reals such that \[ \left(f(x)+f(z)\right)\left(f(y)+f(t)\right)=f(xy-zt)+f(xt+yz) \] for all real $x,y,z,t$.
For pairwise distinct nonnegative reals $a,b,c$, prove that
$$\frac{a^2}{(b-c)^2}+\frac{b^2}{(c-a)^2}+\frac{c^2}{(b-a)^2}>2$$.
Suppose that a sequence $a_1,a_2,\ldots$ of positive real numbers satisfies \[a_{k+1}\geq\frac{ka_k}{a_k^2+(k-1)}\] for every positive integer $k$. Prove that $a_1+a_2+\ldots+a_n\geq n$ for every $n\geq2$.
Suppose that the function $h:\mathbb{R}^2\to\mathbb{R}$ has continuous partial derivatives and satisfies the equation
\[h(x,y)=a\frac{\partial h}{\partial x}(x,y)+b\frac{\partial h}{\partial y}(x,y)\]
for some constants $a,b.$ Prove that if there is a constant $M$ such that $|h(x,y)|\le M$ for all $(x,y)$ in $\mathbb{R}^2,$ then $h$ is identically zero.
let $0<a,b<\pi/2$. Show that
$\frac{5}{cos^2(a)}+\frac{5}{sin^2(a)sin^2(b)cos^2(b)} \geq 27cos(a)+36sin(a) $
Let $n$ be a positive integer. Positive numbers $a$, $b$, $c$ satisfy $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=1$. Find the greatest possible value of $$E(a,b,c)=\frac{a^{n}}{a^{2n+1}+b^{2n} \cdot c + b \cdot c^{2n}}+\frac{b^{n}}{b^{2n+1}+c^{2n} \cdot a + c \cdot a^{2n}}+\frac{c^{n}}{c^{2n+1}+a^{2n} \cdot b + a \cdot b^{2n}}$$
[url=http://www.mathlinks.ro/Forum/viewtopic.php?t=107262]IMO 2007 HKTST 1[/url]
Problem 1
Let $p,q,r$ and $s$ be real numbers such that $p^{2}+q^{2}+r^{2}-s^{2}+4=0$. Find the maximum value of $3p+2q+r-4|s|$.
Prove the following inequality.
\[\frac{1}{12}(\pi -6+2\sqrt{3})\leq \int_{\frac{\pi}{6}}^{\frac{\pi}{4}} \ln (1+\cos 2x) dx\leq \frac{1}{4}(2-\sqrt{3})\]
Find all functions $f: \mathbb {R}^+ \times \mathbb {R}^+ \to \mathbb {R}^+$ that satisfy the following conditions for all positive real numbers $x,y,z:$
$$f\left ( f(x,y),z \right )=x^2y^2f(x,z)$$
$$f\left ( x,1+f(x,y) \right ) \ge x^2 + xyf(x,x)$$
[i]Proposed by Mojtaba Zare, Ali Daei Nabi[/i]
[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 $ f: [0,1]\to\mathbb R$ be a derivable function, with a continuous derivative $ f'$ on $ [0,1]$. Prove that if $ f\left( \frac 12\right) \equal{} 0$, then
\[ \int^1_0 \left( f'(x) \right)^2 dx \geq 12 \left( \int^1_0 f(x) dx \right)^2.\]