Found problems: 1671
Let $f_1,f_2,\ldots,f_n : [0,1]\to (0,\infty)$ be $n$ continuous functions, $n\geq 1$, and let $\sigma$ be a permutation of the set $\{1,2,\ldots, n\}$. Prove that \[ \prod^n_{i=1} \int^1_0 \frac{ f_i^2(x) }{ f_{\sigma(i)}(x) } dx \geq \prod^n_{i=1} \int^1_0 f_i(x) dx. \]
Santa Claus has at least $n$ gifts for $n$ children. For $i\in\{1,2, ... , n\}$, the $i$-th child considers $x_i > 0$ of these items to be desirable. Assume that
\[\dfrac{1}{x_1}+\cdots+\dfrac{1}{x_n}\le1.\]
Prove that Santa Claus can give each child a gift that this child likes.
Let $ x$ and $ y$ positive real numbers such that $ (1\plus{}x)(1\plus{}y)\equal{}2$. Show that $ xy\plus{}\frac{1}{xy}\geq\ 6$
Prove that \[ \frac{a}{b+2c+3d} +\frac{b}{c+2d+3a} +\frac{c}{d+2a+3b}+ \frac{d}{a+2b+3c} \geq \frac{2}{3} \] for all positive real numbers $a,b,c,d$.
Let $a_1, a_2, \ldots, a_n$ be positive real numbers. Prove the inequality
\[\binom n2 \sum_{i<j} \frac{1}{a_ia_j} \geq 4 \left( \sum_{i<j} \frac{1}{a_i+a_j} \right)^2\]
Let $a,b,c,d$ be positive reals such taht $a+b+c+d=1$.
Prove that: \[6(a^{3}+b^{3}+c^{3}+d^{3})\geq a^{2}+b^{2}+c^{2}+d^{2}+\frac{1}{8}.\]
Let $ a,\ b$ be real numbers satisfying $ \int_0^1 (ax\plus{}b)^2dx\equal{}1$.
Determine the values of $ a,\ b$ for which $ \int_0^1 3x(ax\plus{}b)\ dx$ is maximized.
Find all functions $f(x)$ with the domain of all positive real numbers, such that for any positive numbers $x$ and $y$, we have $f(x^y)=f(x)^{f(y)}$.
Find all polynomials $ f\in\mathbb Z[x]$ such that for each $ a,b,x\in\mathbb N$
\[ a\plus{}b\plus{}c|f(a)\plus{}f(b)\plus{}f(c)\]
For positive real numbers $a,b,c$ with $abc=1$ prove that $\left(a+\frac{1}{b}\right)^{2}+\left(b+\frac{1}{c}\right)^{2}+\left(c+\frac{1}{a}\right)^{2}\geq 3(a+b+c+1)$
Let $f:[0,\infty)\to\mathbb{R}$ be a strictly decreasing continuous function such that $\lim_{x\to\infty}f(x)=0.$ Prove that $\displaystyle\int_0^{\infty}\frac{f(x)-f(x+1)}{f(x)}\,dx$ diverges.
Let $a, b, c$ be non-negative reals which satisfy $a+b+c=1$. Prove that $\frac{\sqrt{a}}{b+1}+\frac{\sqrt{b}}{c+1}+\frac{\sqrt{c}}{a+1}>\frac{1}{2}(\sqrt{a}+\sqrt{b}+\sqrt{c})$
Find all functions $ f: \mathbb{R}^{ \plus{} }\to\mathbb{R}^{ \plus{} }$ satisfying $ f\left(x \plus{} f\left(y\right)\right) \equal{} f\left(x \plus{} y\right) \plus{} f\left(y\right)$ for all pairs of positive reals $ x$ and $ y$. Here, $ \mathbb{R}^{ \plus{} }$ denotes the set of all positive reals.
[i]Proposed by Paisan Nakmahachalasint, Thailand[/i]
If the non-negative reals $x,y,z$ satisfy $x^2+y^2+z^2=x+y+z$. Prove that
$$\displaystyle\frac{x+1}{\sqrt{x^5+x+1}}+\frac{y+1}{\sqrt{y^5+y+1}}+\frac{z+1}{\sqrt{z^5+z+1}}\geq 3.$$
When does the equality occur?
[i]Proposed by Dorlir Ahmeti, Albania[/i]
Let $a_1$, $a_2$, $\cdots$, $a_n$ be positive real numbers, and let $S_k$ be the sum of the products of $a_1$, $a_2$, $\cdots$, $a_n$ taken $k$ at a time. Show that
\[ S_k S_{n-k} \geq {n \choose k}^2 a_1 a_2 \cdots a_n \]
for $k = 1$, $2$, $\cdots$, $n - 1$.
The real numbers $a$, $b$, $c$ are such that $a+b+c+ab+bc+ca+abc \geq 7$. Prove that
$\sqrt{a^2+b^2+2}+\sqrt{b^2+c^2+2}+\sqrt{c^2+a^2+2} \geq 6$
Let $P$ be a polynomial with each root is real and each coefficient is either $1$ or $-1$. The degree of $P$ can be at most ?
$ \textbf{(A)}\ 5
\qquad\textbf{(B)}\ 4
\qquad\textbf{(C)}\ 3
\qquad\textbf{(D)}\ 2
\qquad\textbf{(E)}\ \text{None}
$
If $a$, $b$ ,$c$ are three positive real numbers such that $ab+bc+ca = 1$, prove that \[ \sqrt[3]{ \frac{1}{a} + 6b} + \sqrt[3]{\frac{1}{b} + 6c} + \sqrt[3]{\frac{1}{c} + 6a } \leq \frac{1}{abc}. \]
Let $\mathbb R^\ast$ denote the set of nonzero reals. Find all functions $f: \mathbb R^\ast \to \mathbb R^\ast$ satisfying \[ f(x^2+y)+1=f(x^2+1)+\frac{f(xy)}{f(x)} \] for all $x,y \in \mathbb R^\ast$ with $x^2+y\neq 0$.
[i]Proposed by Ryan Alweiss[/i]
Prove that for any three positive real numbers $ a,b,c, \frac{1}{a\plus{}b}\plus{}\frac{1}{b\plus{}c}\plus{}\frac{1}{c\plus{}a} \ge \frac{9}{2} \cdot \frac{1}{a\plus{}b\plus{}c}$.
The sides of a triangle have lengths $a,b,c$. Prove that:
\begin{align*}(-a+b+c)(a-b+c)\, +\, & (a-b+c)(a+b-c)+(a+b-c)(-a+b+c)\\ &\le\sqrt{abc}(\sqrt{a}+\sqrt{b}+\sqrt{c})\end{align*}
For every complex number $z$ different from 0 and 1 we define the following function
\[ f(z) := \sum \frac 1 { \log^4 z } \]
where the sum is over all branches of the complex logarithm.
a) Prove that there are two polynomials $P$ and $Q$ such that $f(z) = \displaystyle \frac {P(z)}{Q(z)} $ for all $z\in\mathbb{C}-\{0,1\}$.
b) Prove that for all $z\in \mathbb{C}-\{0,1\}$ we have
\[ f(z) = \frac { z^3+4z^2+z}{6(z-1)^4}. \]
Let $n\ge3$ and $a_1,a_2,...,a_n \in \mathbb{R^{+}}$, such that $\frac{1}{1+a_1^4} + \frac{1}{1+a_2^4} + ... + \frac{1}{1+a_n^4} = 1$. Prove that: $$a_1a_2...a_n \ge (n-1)^{\frac n4}$$
For a positive reals $ x_{1},...,x_{n} $ prove inequlity:
$ \frac{1}{x_{1}+1}+...+\frac{1}{x_{n}+1}\le \frac{n}{1+\frac{n}{\frac{1}{x_{1}}+...+\frac{1}{x_{n}}}}$
Let $ n$ be a positive integer. Consider
\[ S \equal{} \left\{ (x,y,z) \mid x,y,z \in \{ 0, 1, \ldots, n\}, x \plus{} y \plus{} z > 0 \right \}
\]
as a set of $ (n \plus{} 1)^{3} \minus{} 1$ points in the three-dimensional space. Determine the smallest possible number of planes, the union of which contains $ S$ but does not include $ (0,0,0)$.
[i]Author: Gerhard Wöginger, Netherlands [/i]