Found problems: 1671
Let $x,y,z$ be real numbers such that $0<x,y,z<1$. Find the minimum value of: $$\frac {xyz(x+y+z)+(xy+yz+zx)(1-xyz)}{xyz\sqrt {1-xyz}}$$
Let $ p$ be the product of two consecutive integers greater than 2. Show that there are no integers $ x_1, x_2, \ldots, x_p$ satisfying the equation
\[ \sum^p_{i \equal{} 1} x^2_i \minus{} \frac {4}{4 \cdot p \plus{} 1} \left( \sum^p_{i \equal{} 1} x_i \right)^2 \equal{} 1
\]
[b]OR[/b]
Show that there are only two values of $ p$ for which there are integers $ x_1, x_2, \ldots, x_p$ satisfying
\[ \sum^p_{i \equal{} 1} x^2_i \minus{} \frac {4}{4 \cdot p \plus{} 1} \left( \sum^p_{i \equal{} 1} x_i \right)^2 \equal{} 1
\]
Let $f : \left[ 0, 1 \right] \to \mathbb R$ be a continuous function and $g : \left[ 0, 1 \right] \to \left( 0, \infty \right)$.
Prove that if $f$ is increasing, then
\[\int_{0}^{t}f(x) g(x) \, dx \cdot \int_{0}^{1}g(x) \, dx \leq \int_{0}^{t}g(x) \, dx \cdot \int_{0}^{1}f(x) g(x) \, dx .\]
Let $\mathbb{R}$ be the set of real numbers. Determine all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that, for any real numbers $x$ and $y$, \[ f(f(x)f(y)) + f(x+y) = f(xy). \]
[i]Proposed by Dorlir Ahmeti, Albania[/i]
Given that $a,b,c,d,e$ are real numbers such that
$a+b+c+d+e=8$,
$a^2+b^2+c^2+d^2+e^2=16$.
Determine the maximum value of $e$.
Let $f : \mathbb R \to \mathbb R$ be a function such that $f(1)=1$ and
\[f(x+y)=f(x)+f(y)\]
And for all $x \in \mathbb R / \{0\}$ we have $f\left( \frac 1x \right) = \frac{1}{f(x)}.$ Find all such functions $f.$
Let $0<a\leq 1$ be a real number and let $a\leq a_{i}\leq\frac{1}{a_{i}}\forall i=\overline{1,1996}$ are real numbers. Prove that for any nonnegative real numbers $k_{i}(i=1,2,...,1996)$ such that $\sum_{i=1}^{1996}k_{i}=1$ we have $(\sum_{i=1}^{1996}k_{i}a_{i})(\sum_{i=1}^{1996}\frac{k_{i}}{a_{i}})\leq (a+\frac{1}{a})^{2}$.
Given is a real number $a \in (0,1)$ and positive reals $x_0, x_1, \ldots, x_n$ such that $\sum x_i=n+a$ and $\sum \frac{1}{x_i}=n+\frac{1}{a}$. Find the minimal value of $\sum x_i^2$.
Find all increasing functions $f$ from the nonnegative integers to the integers satisfying $f(2)=7$ and \[ f(mn) = f(m) + f(n) + f(m)f(n) \] for all nonnegative integers $m$ and $n$.
Let $a, b, c$ and $d$ be positive real numbers. Prove that
$$\frac{a - b}{b + c}+\frac{b - c}{c + d}+\frac{c - d}{d + a} +\frac{d - a}{a + b } \ge 0 $$
On a circumference of a unit radius, take points $A$ and $B$ such that section $AB$ has length one. $C$ can be any point on the longer arc of the circle between $A$ and $B$. How do we take $C$ to make the perimeter of the triangle $ABC$ as large as possible?
With the conditions $a,b,c\in\mathbb{R^+}$ and $a+b+c=1$, prove that \[\frac{7+2b}{1+a}+\frac{7+2c}{1+b}+\frac{7+2a}{1+c}\geq\frac{69}{4}\]
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 $n\geq2$.
Point $X$ is taken inside a regular $n$-gon of side length $a$. Let $h_1,h_2,...,h_n$ be the distances from $X$ to the lines defined by the sides of the $n$-gon. Prove that $\frac{1}{h_1}+\frac{1}{h_2}+...+\frac{1}{h_n}>\frac{2\pi}{a}$
Find maximal value of the function $f(x)=8-3\sin^2 (3x)+6 \sin (6x)$
Let $a,\ b$ and $c$ be positive real numbers so that $a+b+c=1$. Show that
$$a\sqrt{a^2+6bc}+b\sqrt{b^2+6ac}+c\sqrt{c^2+6ab}\leq\frac{3\sqrt{2}}{4}$$
Let $f,g:[0,1]\to(0,+\infty)$ be two continuous functions such that $f$ and $\frac gf$ are increasing. Prove that
$$\int^1_0\frac{\int^x_0f(t)\text dt}{\int^x_0g(t)\text dt}\text dx\le2\int^1_0\frac{f(t)}{g(t)}\text dt.$$
Let $a,b,c$ positive reals such that $a+b+c=3$. Show that following expression's minimum value is $2$.
$$\frac{\sqrt a +\sqrt b +\sqrt c}{ab+bc+ca} + \frac{1}{1+2\sqrt {ab}} + \frac {1}{1+ 2\sqrt {bc}} + \frac{1}{1+ 2\sqrt {ca}}$$
Find all functions $f: \mathbb{R}^+ \to \mathbb{R}^+$ such that
$$(z + 1)f(x + y) = f(xf(z) + y) + f(yf(z) + x),$$
for all positive real numbers $x, y, z$.
[i]Fajar Yuliawan, Indonesia[/i]
Let $a,b,c$ be positive real numbers with $a^2+b^2+c^2 \geq 3$. Prove that:
$$\frac{(a+1)(b+2)}{(b+1)(b+5)}+\frac{(b+1)(c+2)}{(c+1)(c+5)}+\frac{(c+1)(a+2)}{(a+1)(a+5)} \geq \frac{3}{2}.$$
Let $p, q, r$ be positive real numbers and $n$ a natural number. Show that if $pqr = 1$, then \[ \frac{1}{p^n+q^n+1} + \frac{1}{q^n+r^n+1} + \frac{1}{r^n+p^n+1} \leq 1. \]
Let $n \geq 4$ be an integer. Find all positive real solutions to the following system of $2n$ equations:
\begin{align*}
a_{1} &=\frac{1}{a_{2 n}}+\frac{1}{a_{2}}, & a_{2}&=a_{1}+a_{3}, \\
a_{3}&=\frac{1}{a_{2}}+\frac{1}{a_{4}}, & a_{4}&=a_{3}+a_{5}, \\
a_{5}&=\frac{1}{a_{4}}+\frac{1}{a_{6}}, & a_{6}&=a_{5}+a_{7} \\
&\vdots & &\vdots \\
a_{2 n-1}&=\frac{1}{a_{2 n-2}}+\frac{1}{a_{2 n}}, & a_{2 n}&=a_{2 n-1}+a_{1}
\end{align*}
Find all functions $f: \mathbb{R} \to \mathbb{R}$ such that$$f(xf(y)-y) + f(xy-x) + f(x+y) = 2xy$$for all $x, y \in \mathbb{R}$.
Find all functions $f:\mathbb{R}^+\to\mathbb{R}^+$ such that whenever $a>b>c>d>0$ and $ad=bc$,
\[f(a+d)+f(b-c)=f(a-d)+f(b+c).\]
[i]Calvin Deng.[/i]
In a contest, there are $m$ candidates and $n$ judges, where $n\geq 3$ is an odd integer. Each candidate is evaluated by each judge as either pass or fail. Suppose that each pair of judges agrees on at most $k$ candidates. Prove that \[{\frac{k}{m}} \geq {\frac{n-1}{2n}}. \]