The numbers $ a_i $, $ b_i $, $ c_i $, $ d_i $ satisfy the conditions $ 0\leq c_i \leq a_i \leq b_i \leq d_i $ and $ a_i+b_i = c_i+d_i $ for $ i=1,2 ,\ldots,n$. Prove that $$ \prod_{i=1}^n a_i + \prod_{i=1}^n b_i \leq \prod_{i=1}^n c_i + \prod_{i=1}^n d_i$$

Let $a, b, c$ be positive real numbers such that $a^2+b^2+c^2+(a+b+c)^2\leq4$. Prove that \[\frac{ab+1}{(a+b)^2}+\frac{bc+1}{(b+c)^2}+\frac{ca+1}{(c+a)^2}\geq 3.\]

Let $f: [0,1] \to (0, \infty)$ be an integrable function such that $f(x)f(1-x) = 1$ for all $x\in [0,1]$. Prove that $\int_0^1f(x)dx \geq 1$.

Let be a tetahedron $ OABC $ with $ OA\perp OB\perp OC\perp OA. $ Show that $$ OH\le r\left( 1+\sqrt 3 \right) , $$ where $ H $ is the orthocenter of $ ABC $ and $ r $ is radius of the inscribed spere of $ OABC. $ [i]Valentin Vornicu[/i]

Given a function $f$ such that $f(x)\le x\forall x\in\mathbb{R}$ and $f(x+y)\le f(x)+f(y)\forall \{x,y\}\in\mathbb{R}$, prove that $f(x)=x\forall x\in\mathbb{R}$.

Let $ABC$ be a triangle and $D$ the midpoint of the side $BC$. Define points $E$ and $F$ on $AC$ and $B$, respectively, such that $DE=DF$ and $\angle EDF =\angle BAC$. Prove that $$DE\geq \frac {AB+AC} 4.$$

For $j=1,2,3$ let $x_{j},y_{j}$ be non-zero real numbers, and let $v_{j}=x_{j}+y_{j}$.Suppose that the following statements hold: $x_{1}x_{2}x_{3}=-y_{1}y_{2}y_{3}$ $x_{1}^{2}+x_{2}^{2}+x_{3}^{2}=y_{1}^{2}+y_{2}^{2}+y_{3}^2$ $v_{1},v_{2},v_{3}$ satisfy triangle inequality $v_{1}^{2},v_{2}^{2},v_{3}^{2}$ also satisfy triangle inequality. Prove that exactly one of $x_{1},x_{2},x_{3},y_{1},y_{2},y_{3}$ is negative.

Let $x_1,x_2,\cdots,x_n$ $(n\geq2)$ be a non-decreasing monotonous sequence of positive numbers such that $x_1,\frac{x_2}{2},\cdots,\frac{x_n}{n}$ is a non-increasing monotonous sequence .Prove that \[ \frac{\sum_{i=1}^{n} x_i }{n\left (\prod_{i=1}^{n}x_i \right )^{\frac{1}{n}}}\le \frac{n+1}{2\sqrt[n]{n!}}\]

If $ab+\sqrt{ab+1}+\sqrt{a^2+b}\sqrt{a+b^2}=0$, find the value of $b\sqrt{a^2+b}+a\sqrt{b^2+a}$

For all integers $x$ and $y$, let $a_{x, y}$ be a real number. Suppose that $a_{0, 0} = 0$. Suppose that only a finite number of the $a_{x, y}$ are nonzero. Prove that \[ \sum_{x = -\infty}^\infty \sum_{y = -\infty}^{\infty} a_{x,y} ( a_{x, 2x + y} + a_{x + 2y, y} ) \le \sqrt{3} \sum_{x = -\infty}^\infty \sum_{y = -\infty}^{\infty} a_{x, y}^2 \, . \]

Let $a, b, c$ be real numbers satisfying the condition $18ab + 9ca + 29bc = 1$. Find the minimum value of the expression $T = 42a^2 + 34b^2 + 43c^2$.

Given positive reals $a,b,c,p,q$ satisfying $abc=1$ and $p \geq q$, prove that \[ p \left(a^2+b^2+c^2\right) + q\left( \frac{1}{a} + \frac{1}{b} + \frac{1}{c}\right) \geq (p+q) (a+b+c). \][i]Proposed by AJ Dennis[/i]

Determine the smallest real number $C$ such that the inequality \[ C(x_1^{2005} +x_2^{2005} + \cdots + x_5^{2005}) \geq x_1x_2x_3x_4x_5(x_1^{125} + x_2^{125}+ \cdots + x_5^{125})^{16} \] holds for all positive real numbers $x_1,x_2,x_3,x_4,x_5$.

Let $a,b,c\in\mathbb R^{n}, a+b+c=0$ and $\lambda>0$. Prove that \[\prod_{cycle}\frac{|a|+|b|+(2\lambda+1)|c|}{|a|+|b|+|c|}\geq(2\lambda+3)^{3}\]

With $\sigma (n)$ we denote the sum of natural divisors of the natural number $n$. Prove that, if $n$ is the product of different prime numbers of the form $2^k-1$ for $k \in \mathbb{N}$($Mersenne's$ prime numbers) , than $\sigma (n)=2^m$, for some $m \in \mathbb{N}$. Is the inverse statement true?

Find all monotonically increasing or monotonically decreasing functions $f: \mathbb{R}_+\to\mathbb{R}_+$ which satisfy the equation $f\left(xy\right)\cdot f\left(\frac{f\left(y\right)}{x}\right)=1$ for any two numbers $x$ and $y$ from $\mathbb{R}_+$. Hereby, $\mathbb{R}_+$ is the set of all positive real numbers. [i]Note.[/i] A function $f: \mathbb{R}_+\to\mathbb{R}_+$ is called [i]monotonically increasing[/i] if for any two positive numbers $x$ and $y$ such that $x\geq y$, we have $f\left(x\right)\geq f\left(y\right)$. A function $f: \mathbb{R}_+\to\mathbb{R}_+$ is called [i]monotonically decreasing[/i] if for any two positive numbers $x$ and $y$ such that $x\geq y$, we have $f\left(x\right)\leq f\left(y\right)$.

Let $a$ be a real number. Find the minimum value of $\int_0^1 |ax-x^3|dx$. How many solutions (including University Mathematics )are there for the problem? Any advice would be appreciated.

Find all real $x$ such that $$\sqrt{2+\frac{5}{2}\cos x}\leq\sin x.$$

For two arbitrary reals $x, y$ which are larger than $0$ and less than $1.$ Prove that$$\frac{x^2}{x+y}+\frac{y^2}{1-x}+\frac{(1-x-y)^2}{1-y}\geq\frac{1}{2}.$$

Let $x,y,z \ge 0$ be real numbers such that $(x+y-1)^2+(y+z-1)^2+(z+x-1)^2=27$. Find the maximum and minimum of $x^4+y^4+z^4$

Prove that for all positive integers $n$ the following inequality holds: $$ \frac{1}{1^2 +2020}+\frac{1}{2^2+2020} + \ldots + \frac{1}{n^2+2020} < \frac{1}{22} $$

Points $A', B', C'$ are chosen on the sides $BC, CA, AB$ of triangle $ABC$, respectively, so that $\frac{|BA'|}{|A'C|}=\frac{|CB'|}{|B'A|}=\frac{|AC'|}{|C'B|}$. The line which is parallel to line $B'C'$ and goes through point $A$ intersects the lines $AC$ and $AB$ at $P$ and $Q$, respectively. Prove that $\frac{|PQ|}{|B'C'|} \ge 2$

In an $n$-gon $A_{1}A_{2}\ldots A_{n}$, all of whose interior angles are equal, the lengths of consecutive sides satisfy the relation \[a_{1}\geq a_{2}\geq \dots \geq a_{n}. \] Prove that $a_{1}=a_{2}= \ldots= a_{n}$.

Is there a strictly increasing function $f:\mathbb{R}\to\mathbb{R}$ such that $f'(x)=f(f(x))$ for all $x?$

Let $S_n = \{1, \cdots, n\}$ and let $f$ be a function that maps every subset of $S_n$ into a positive real number and satisfies the following condition: For all $A \subseteq S_n$ and $x, y \in S_n, x \neq y, f(A \cup \{x\})f(A \cup \{y\}) \le f(A \cup \{x, y\})f(A)$. Prove that for all $A,B \subseteq S_n$ the following inequality holds: \[f(A) \cdot f(B) \le f(A \cup B) \cdot f(A \cap B)\]