Found problems: 1671
Let $x,y,z\in \mathbb{R}^+$ be such that $xy+yz+zx=x+y+z$. Prove the following inequality:
$\frac{1}{x^2+y+1}+\frac{1}{y^2+z+1}+\frac{1}{z^2+x+1}\leq 1$.
Suppose $a,b,c,d> 0$ and $x = \sqrt{a^2+b^2}, y = \sqrt{c^2+d^2}$. Prove that $xy \ge ac + bd$.
The numbers $2, 2, ..., 2$ are written on a blackboard (the number $2$ is repeated $n$ times). One step consists of choosing two numbers from the blackboard, denoting them as $a$ and $b$, and replacing them with $\sqrt{\frac{ab + 1}{2}}$.
$(a)$ If $x$ is the number left on the blackboard after $n - 1$ applications of the above operation, prove that $x \ge \sqrt{\frac{n + 3}{n}}$.
$(b)$ Prove that there are infinitely many numbers for which the equality holds and infinitely many for which the inequality is strict.
Let $n \geq 2$ be a positive integer and $a_{1},\ldots , a_{n}$ be positive real numbers such that $a_{1}+...+a_{n}= 1$. Prove that:
\[\frac{a_{1}}{1+a_{2}+\cdots +a_{n}}+\cdots +\frac{a_{n}}{1+a_{1}+a_{2}+\cdots +a_{n-1}}\geq \frac{n}{2n-1}\]
Let $f$, $g$, $h : [a, b] \to \mathbb{R}$, three integrable functions such that:$$\int \limits_a^b fgdx=\int \limits_a^bghdx=\int \limits_a^bhfdx=\int \limits_a^bg^2dx\int \limits_a^bh^2dx=1$$Then$$\int \limits_a^bg^2dx=\int \limits_a^bh^2dx=1$$
For any real numbers sequence $\{x_n\}$ ,suppose that $\{y_n\}$ is a sequence such that:
$y_1=x_1, y_{n+1}=x_{n+1}-(\sum\limits_{i = 1}^{n} {x^2_i})^{ \frac{1}{2}}$ ${(n \ge 1})$ .
Find the smallest positive number $\lambda$ such that for any real numbers sequence $\{x_n\}$ and all positive integers $m$ , have $\frac{1}{m}\sum\limits_{i = 1}^{m} {x^2_i}\le\sum\limits_{i = 1}^{m} {\lambda^{m-i}y^2_i} .$
(High School Affiliated to Nanjing Normal University )
Let $A_1$, $A_2$, $\cdots$, $A_m$ be $m$ subsets of a set of size $n$. Prove that $$ \sum_{i=1}^{m} \sum_{j=1}^{m}|A_i|\cdot |A_i \cap A_j|\geq \frac{1}{mn}\left(\sum_{i=1}^{m}|A_i|\right)^3.$$
Prove for all positive reals a,b,c,d:
$ \frac{a\minus{}b}{b\plus{}c}\plus{}\frac{b\minus{}c}{c\plus{}d}\plus{}\frac{c\minus{}d}{d\plus{}a}\plus{}\frac{d\minus{}a}{a\plus{}b} \geq 0$
Prove that for all positive real numbers $a_1,a_2,\cdots a_n$ \[\frac{1}{\frac{1}{1+a_1}+\frac{1}{1+a_2}+\cdots +\frac{1}{1+a_n}}-\frac{1}{\frac{1}{a_1}+\frac{1}{a_2}+\cdots +\frac{1}{a_n}}\geq \frac{1}{n}\] When does the inequality hold?
Prove the following assertion: If $c_1,c_2,\ldots ,c_n\ (n\ge 2)$ are real numbers such that
\[ (n-1)(c_1^2+c_2^2+\cdots +c_n^2)=(c_1+c_2+\cdots + c_n)^2,\]
then either all these numbers are nonnegative or all these numbers are nonpositive.
Let $a, b, c \in \mathbb{R}^+$ such that $a + b + c = 3$. Prove that $$\sum_{cyc}\left(\frac{a^3+1}{a^2+1}\right)\geq\frac{1}{27}(\sqrt{ab}+\sqrt{bc}+\sqrt{ca})^4.$$
Prove that $c=10\sqrt{24}$ is the largest constant such that if there exist positive numbers $a_1,a_2,\ldots ,a_{17}$ satisfying:
\[\sum_{i=1}^{17}a_i^2=24,\ \sum_{i=1}^{17}a_i^3+\sum_{i=1}^{17}a_i<c \]
then for every $i,j,k$ such that $1\le 1<j<k\le 17$, we have that $x_i,x_j,x_k$ are sides of a triangle.
Prove that if $0<x<\frac{\pi}{2}$ and $n>m$, where $n$,$m$ are natural numbers, \[ 2 \left| \sin^n x - \cos^n x \right| \le 3 \left| \sin^m x - \cos^m x \right|.\]
Let $x_1,x_2,\dots,x_{2023}$ be pairwise different positive real numbers such that
\[a_n=\sqrt{(x_1+x_2+\dots+x_n)\left(\frac{1}{x_1}+\frac{1}{x_2}+\dots+\frac{1}{x_n}\right)}\]
is an integer for every $n=1,2,\dots,2023.$ Prove that $a_{2023} \geqslant 3034.$
Let $\triangle ABC$ be an acute triangle, $H$ is its orthocenter. $\overrightarrow{AH},\overrightarrow{BH},\overrightarrow{CH}$ intersect $\triangle ABC$'s circumcircle at $A',B',C'$ respectively. Find the range (minimum value and the maximum upper bound) of
$$\dfrac{AH}{AA'}+\dfrac{BH}{BB'}+\dfrac{CH}{CC'}$$
Prove that the fraction $ \dfrac{21n \plus{} 4}{14n \plus{} 3}$ is irreducible for every natural number $ n$.
Let $f : \mathbb R \to \mathbb R$ be a function such that
\[f(x+y)=f(x) \cdot f(y) \qquad \forall x,y \in \mathbb R\]
Suppose that $f(0) \neq 0$ and $f(0)$ exists and it is finite $(f(0) \neq \infty)$. Prove that $f$ has derivative in each point $x \in \mathbb R.$
Prove that for every prime $p>100$ and every integer $r$, there exist two integers $a$ and $b$ such that $p$ divides $a^2+b^5-r$.
Let $f : \{ 1, 2, 3, \dots \} \to \{ 2, 3, \dots \}$ be a function such that $f(m + n) | f(m) + f(n) $ for all pairs $m,n$ of positive integers. Prove that there exists a positive integer $c > 1$ which divides all values of $f$.
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 $n\geq2$.
Let $a,b,c$ be positive reals such that $a+b+c=ab+bc+ca$. Prove that \[ (a+b)^{ab-bc}(b+c)^{bc-ca}(c+a)^{ca-ab} \ge a^{ca}b^{ab}c^{bc}. \][i]Proposed by Sammy Luo[/i]
Let $n \ge 2$ be an integer. Let $x_1 \ge x_2 \ge ... \ge x_n$ and $y_1 \ge y_2 \ge ... \ge y_n$ be $2n$ real numbers such that $$0 = x_1 + x_2 + ... + x_n = y_1 + y_2 + ... + y_n $$ $$\text{and} \hspace{2mm} 1 =x_1^2 + x_2^2 + ... + x_n^2 = y_1^2 + y_2^2 + ... + y_n^2.$$ Prove that $$\sum_{i = 1}^n (x_iy_i - x_iy_{n + 1 - i}) \ge \frac{2}{\sqrt{n-1}}.$$
[i]Proposed by David Speyer and Kiran Kedlaya[/i]
Does there exist a function $f:\mathbb{Z}\to\mathbb{Z}$ such that $f(x+f(y))=f(x)-y$ for all integers $x$ and $y$?
Find all functions $f, g: \mathbb{R} \rightarrow \mathbb{R} $ satisfying the following conditions:
[list][*] $f$ is not a constant function and if $x \le y$ then $f(x)\le f(y)$
[*] For all real number $x$, $f(g(x))=g(f(x))=0$
[*] For all real numbers $x$ and $y$, $f(x)+f(y)+g(x)+g(y)=f(x+y)+g(x+y)$
[*] For all real numbers $x$ and $y$, $f(x)+f(y)+f(g(x)+g(y))=f(x+y)$
[/list]
In a group of $n\geq 4$ persons, every three who know each other have a common signal. Assume that these signals are not repeatad and that there are $m\geq 1$ signals in total. For any set of four persons in which there are three having a common signal, the fourth person has a common signal with at most one of them. Show that there three persons who have a common signal, such that the number of persons having no signal with anyone of them does not exceed $[n+3-\frac{18m}{n}]$.