$2024$ positive real numbers with sum $1$ are arranged on a circle. It is known that any two adjacent numbers differ at least in $2$ times. For each pair of adjacent numbers, the smaller one was subtracted from the larger one, and then all these differences were added together. What is the smallest possible value of this resulting sum? [i]Proposed by Oleksiy Masalitin[/i]

Given real numbers $x,y,z$ which satisfies $$|x+y+z|+|xy+yz+zx|+|xyz| \le 1$$ Show that $max\{ |x|,|y|,|z|\} \le 1$.

Find the largest positive real $ k$, such that for any positive reals $ a,b,c,d$, there is always: \[ (a\plus{}b\plus{}c) \left[ 3^4(a\plus{}b\plus{}c\plus{}d)^5 \plus{} 2^4(a\plus{}b\plus{}c\plus{}2d)^5 \right] \geq kabcd^3\]

Let $x$, $y$ be real numbers, such that $x^2 + x \leq y$. Prove that $y^2 + y \geq x$.

Let $ a$, $ b$, $ c$ be positive real numbers such that $ abc=1$. Prove that $ \frac{1}{1+b+c}+\frac{1}{1+c+a}+\frac{1}{1+a+b}\leq\frac{1}{2+a}+\frac{1}{2+b}+\frac{1}{2+c}$.

Let $x_1, x_2, . . . , x_n$ be positive numbers. Prove that \[ \sum\limits_{i=1}^n \dfrac{x_i ^2}{x_i ^2+x_{i+1}x_{i+2}} \leq n-1 \] Where $x_{n+1}=x_1$ and $x_{n+2}=x_2$.

Suppose that $p, q \in \mathbb{N}$ satisfy the inequality \[\exp(1)\cdot( \sqrt{p+q}-\sqrt{q})^{2}<1.\] Show that $\ln \left(1+\frac{p}{q}\right)$ is irrational.

Find all $x$ for which the inequality holds $$\sqrt{7+8x-16x^2} \ge 2^{\cos^2 \pi x}+2^{\sin ^2 \pi x}$$

Let $a,b,c$ be sides of a triangle. Show that $a^3+b^3+3abc>c^3$.

Let $n \geqslant 3$ be an integer, and let $x_1,x_2,\ldots,x_n$ be real numbers in the interval $[0,1]$. Let $s=x_1+x_2+\ldots+x_n$, and assume that $s \geqslant 3$. Prove that there exist integers $i$ and $j$ with $1 \leqslant i<j \leqslant n$ such that \[2^{j-i}x_ix_j>2^{s-3}.\]

Let $x$ and $y$ be positive real numbers. 1. Prove: if $x^3 - y^3 \ge 4x$, then $x^2 > 2y$. 2. Prove: if $x^5 - y^3 \ge 2x$, then $x^3 \ge 2y$.

Let $n > 3$ be an integer. Prove that there exist positive integers $x_1,..., x_n$ in geometric progression and positive integers $y_1,..., y_n$ in arithmetic progression such that $x_1<y_1<x_2<y_2<...<x_n<y_n$

Let $ 0 < x <y$ be real numbers. Let $ H\equal{}\frac{2xy}{x\plus{}y}$ , $ G\equal{}\sqrt{xy}$ , $ A\equal{}\frac{x\plus{}y}{2}$ , $ Q\equal{}\sqrt{\frac{x^2\plus{}y^2}{2}}$ be the harmonic, geometric, arithmetic and root mean square (quadratic mean) of $ x$ and $ y$. As generally known $ H<G<A<Q$. Arrange the intervals $ [H,G]$ , $ [G,A]$ and $ [A,Q]$ in ascending order by their length.

Let $ f: \mathbb R \rightarrow \mathbb R$ be a function such that $ f(tx_1\plus{}(1\minus{}t)x_2)\leq tf(x_1)\plus{}(1\minus{}t)f(x_2)$ for all $ x_1 , x_2 \in \mathbb R$ and $ t\in (0,1)$. Show that $ \sum_{k\equal{}1}^{2003}f(a_{k\plus{}1})a_k \geq \sum_{k\equal{}1}^{2003}f(a_k)a_{k\plus{}1}$ for all real numbers $ a_1,a_2,...,a_{2004}$ such that $ a_1\geq a_2\geq ... \geq a_{2003}$ and $ a_{2004}\equal{}a_1$

Let $ x,y,z$ be nonnegative real numbers such that $ x \plus{} y \plus{} z \equal{} 2$. Prove that $ x^{2}y^{2} \plus{} y^{2}z^{2} \plus{} z^{2}x^{2} \plus{} xyz\leq 1$. When does the equality occur?

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

One considers the set $$A = \left\{ n \in N^* \big | 1 < \sqrt{1 + \sqrt{n}} < 2 \right\}$$ a) Find the set $A$. b) Find the set of numbers $n \in A$ such that $$\sqrt{n} \cdot \left| 1-\sqrt{1 + \sqrt{n}}\right| <1 ?$$

Let $\alpha$ be an arbitrary positive real number. Determine for this number $\alpha$ the greatest real number $C$ such that the inequality$$\left(1+\frac{\alpha}{x^2}\right)\left(1+\frac{\alpha}{y^2}\right)\left(1+\frac{\alpha}{z^2}\right)\geq C\left(\frac{x}{z}+\frac{z}{x}+2\right)$$ is valid for all positive real numbers $x, y$ and $z$ satisfying $xy + yz + zx =\alpha.$ When does equality occur? [i](Proposed by Walther Janous)[/i]

Prove that \[\frac{a^3}{a^2+ab+b^2}+\frac{b^3}{b^2+bc+c^2}+\frac{c^3}{c^2+ca+a^2}\geq \frac{a+b+c}{3}\] for positive reals $a,b,c$ (S Tokarev)

Let $P$ be an arbitrary point inside the triangle $ABC$. Let $A_1, B_1$ and $C_1$ denote the intersection points of the straight lines $AP, BP$ and $CP$, respectively, with the sides $BC, CA$ and $AB$. We order the areas of the triangles $AB_1C_1,A_1BC_1,A_1B_1C$. Denote the smaller by $S_1$, the middle by $S_2$, and the larger by $S_3$. Prove that $\sqrt{S_1 S_2} \le S \le \sqrt{S_2 S_3}$ ,where $S$ is the area of the triangle $A_1B_1S_1$.

Let $a,b,c$ be positive real numbers. Prove that: $\left (a+b+c \right )\left ( \frac{1}{a}+\frac{1}{b}+\frac{1}{c} \right ) \geq 9+3\sqrt[3]{\frac{(a-b)^2(b-c)^2(c-a)^2}{a^2b^2c^2}}$

Given a positive integer $n \geq 3$, let $C_{n}$ be the collection of all $n$-tuples $a=(a_{1},a_{2},...,a_{n})$ of nonnegative reals $a_i$, $i=1,...,n$, such that $a_{1}+a_{2}+...+a_{n}=1$. For $k \in \left \{ 1,...,n-1 \right \}$ and $a \in C_{n}$, consider the sum set $\sigma_{k}(a) = \left \{a_{1}+...+a_{k},a_{2}+...+a_{k+1},...,a_{n-k+1}+...+a_{n} \right \}$. Show the following. (a) There exist $m_k=\max\{\min\sigma_k(a):a\in\mathcal{C}_n\}$ and $M_k=\min\{\max\sigma_k(a):a\in\mathcal{C}_n\}$. (b) It holds that $\displaystyle{1\leq\sum_{k=1}^{n-1}(\frac{1}{M_k}-\frac{1}{m_k})\leq n-2}$. Moreover, on the left side, equality is attained only for finitely many values of $n$, whereas on the right side, equality holds for infinitely values of $n$.

Suppose $n$ is a perfect square. Consider the set of all numbers which is the product of two numbers, not necessarily distinct, both of which are at least $n$. Express the $n-$th smallest number in this set in terms of $n$.

$a,b,c,d$ are positive numbers such that $\sum_{cyc} \frac{1}{ab} =1$. Prove that : $abcd+16 \geq 8 \sqrt{(a+c)(\frac{1}{a} + \frac{1}{c})}+8\sqrt{(b+d)(\frac{1}{b}+\frac{1}{d})}$

Point $O$ is inside triangle $ABC$ such that $\angle AOB = \angle BOC = \angle COA = 120^\circ .$ Prove that \[\frac{AO^2}{BC}+\frac{BO^2}{CA}+\frac{CO^2}{AB} \geq \frac{AO+BO+CO}{\sqrt 3}.\]