Let $a,b,c$ be real numbers greater than $1$. Prove the inequality $$\log_a\left(\frac{b^2}{ac}-b+ac\right)\log_b\left(\frac{c^2}{ab}-c+ab\right)\log_c\left(\frac{a^2}{bc}-a+bc\right)\ge1.$$

For any permutation $p$ of set $\{1, 2, \ldots, n\}$, define $d(p) = |p(1) - 1| + |p(2) - 2| + \ldots + |p(n) - n|$. Denoted by $i(p)$ the number of integer pairs $(i, j)$ in permutation $p$ such that $1 \leqq < j \leq n$ and $p(i) > p(j)$. Find all the real numbers $c$, such that the inequality $i(p) \leq c \cdot d(p)$ holds for any positive integer $n$ and any permutation $p.$

For a real number $ x$, let $ \|x \|$ denote the distance between $ x$ and the closest integer. Let $ 0 \leq x_n <1 \; (n\equal{}1,2,\ldots)\ ,$ and let $ \varepsilon >0$. Show that there exist infinitely many pairs $ (n,m)$ of indices such that $ n \not\equal{} m$ and \[ \|x_n\minus{}x_m \|< \min \left( \varepsilon , \frac{1}{2|n\minus{}m|} \right).\] [i]V. T. Sos[/i]

Consider pairs of the sequences of positive real numbers \[a_1\geq a_2\geq a_3\geq\cdots,\qquad b_1\geq b_2\geq b_3\geq\cdots\] and the sums \[A_n = a_1 + \cdots + a_n,\quad B_n = b_1 + \cdots + b_n;\qquad n = 1,2,\ldots.\] For any pair define $c_n = \min\{a_i,b_i\}$ and $C_n = c_1 + \cdots + c_n$, $n=1,2,\ldots$. (1) Does there exist a pair $(a_i)_{i\geq 1}$, $(b_i)_{i\geq 1}$ such that the sequences $(A_n)_{n\geq 1}$ and $(B_n)_{n\geq 1}$ are unbounded while the sequence $(C_n)_{n\geq 1}$ is bounded? (2) Does the answer to question (1) change by assuming additionally that $b_i = 1/i$, $i=1,2,\ldots$? Justify your answer.

For any three nonnegative reals $a$, $b$, $c$, prove that $\left|ca-ab\right|+\left|ab-bc\right|+\left|bc-ca\right|\leq\left|b^{2}-c^{2}\right|+\left|c^{2}-a^{2}\right|+\left|a^{2}-b^{2}\right|$. [i]Generalization.[/i] For any $n$ nonnegative reals $a_{1}$, $a_{2}$, ..., $a_{n}$, prove that $\sum_{i=1}^{n}\left|a_{i-1}a_{i}-a_{i}a_{i+1}\right|\leq\sum_{i=1}^{n}\left|a_{i}^{2}-a_{i+1}^{2}\right|$. Here, the indices are cyclic modulo $n$; this means that we set $a_{0}=a_{n}$ and $a_{n+1}=a_{1}$. darij

Let $a, b, c$ be positive reals so that $a^2+b^2+c^2=1$. Find the minimum value of $S=1/a^2+1/b^2+1/c^2-2(a^3+b^3+c^3)/abc$

Bernado and Silvia play the following game. An integer between 0 and 999, inclusive, is selected and given to Bernado. Whenever Bernado receives a number, he doubles it and passes the result to Silvia. Whenever Silvia receives a number, she adds 50 to it and passes the result to Bernado. The winner is the last person who produces a number less than 1000. Let $N$ be the smallest initial number that results in a win for Bernado. What is the sum of the digits of $N$? $\textbf{(A)}\ 7 \qquad\textbf{(B)}\ 8 \qquad\textbf{(C)}\ 9 \qquad\textbf{(D)}\ 10 \qquad\textbf{(E)}\ 11$

Let $ n$ be a positive integer and let $ a_1,a_2,\ldots,a_n$ be positive real numbers such that: \[ \sum^n_{i \equal{} 1} a_i \equal{} \sum^n_{i \equal{} 1} \frac {1}{a_i^2}. \] Prove that for every $ i \equal{} 1,2,\ldots,n$ we can find $ i$ numbers with sum at least $ i$.

For each positive integer $k$ greater than $1$, find the largest real number $t$ such that the following hold: Given $n$ distinct points $a^{(1)}=(a^{(1)}_1,\ldots, a^{(1)}_k)$, $\ldots$, $a^{(n)}=(a^{(n)}_1,\ldots, a^{(n)}_k)$ in $\mathbb{R}^k$, we define the score of the tuple $a^{(i)}$ as \[\prod_{j=1}^{k}\#\{1\leq i'\leq n\textup{ such that }\pi_j(a^{(i')})=\pi_j(a^{(i)})\}\] where $\#S$ is the number of elements in set $S$, and $\pi_j$ is the projection $\mathbb{R}^k\to \mathbb{R}^{k-1}$ omitting the $j$-th coordinate. Then the $t$-th power mean of the scores of all $a^{(i)}$'s is at most $n$. Note: The $t$-th power mean of positive real numbers $x_1,\ldots,x_n$ is defined as \[\left(\frac{x_1^t+\cdots+x_n^t}{n}\right)^{1/t}\] when $t\neq 0$, and it is $\sqrt[n]{x_1\cdots x_n}$ when $t=0$. [i]Proposed by Cheng-Ying Chang and usjl[/i]

Let $ a\geq 2$ be a real number; with the roots $ x_{1}$ and $ x_{2}$ of the equation $ x^2\minus{}ax\plus{}1\equal{}0$ we build the sequence with $ S_{n}\equal{}x_{1}^n \plus{} x_{2}^n$. [b]a)[/b]Prove that the sequence $ \frac{S_{n}}{S_{n\plus{}1}}$, where $ n$ takes value from $ 1$ up to infinity, is strictly non increasing. [b]b)[/b]Find all value of $ a$ for the which this inequality hold for all natural values of $ n$ $ \frac{S_{1}}{S_{2}}\plus{}\cdots \plus{}\frac{S_{n}}{S_{n\plus{}1}}>n\minus{}1$

Let $A, B, C$ be any three real numbers. Prove that there exists a number $\nu$ such that $$An^2 + Bn+ < n!$$ for every natural number $n > \nu.$

Let $a_1,a_2,a_3,a_4$ be the sides of an arbitrary quadrilateral of perimeter $2s$. Prove that \[ \sum\limits^4_{i=1} \dfrac 1{a_i+s} \leq \dfrac 29\sum\limits_{1\leq i<j\leq 4} \dfrac 1{ \sqrt { (s-a_i)(s-a_j)}}. \] When does the equality hold?

Let $x,y,z$ is positive. Solve: $\begin{cases}{x\left( {6 - y} \right) = 9}\\ {y\left( {6 - z} \right) = 9}\\ {z\left( {6 - x} \right) = 9}\end{cases}$

Solve in $(\mathbb{R}_{+}^{*})^{4}$ the following system : $\left\{\begin{matrix} x+y+z+t=4\\ \frac{1}{x}+\frac{1}{y}+\frac{1}{z}+\frac{1}{t}=5-\frac{1}{xyzt} \end{matrix}\right.$

Show that \[ \sum_{cyc} \sqrt[4]{\frac{(a^2+b^2)(a^2-ab+b^2)}{2}} \leq \frac{2}{3}(a^2+b^2+c^2)\left(\frac{1}{a+b}+\frac{1}{b+c}+\frac{1}{c+a}\right) \] for all positive real numbers $a, \: b, \: c.$

The lengths of the sides of a triangle are prime numbers of centimeters. Prove that its area cannot be an integer number of square centimeters.

Let $n \geq 2$ be a fixed integer. Find the least constant $C$ such the inequality \[\sum_{i<j} x_{i}x_{j} \left(x^{2}_{i}+x^{2}_{j} \right) \leq C \left(\sum_{i}x_{i} \right)^4\] holds for any $x_{1}, \ldots ,x_{n} \geq 0$ (the sum on the left consists of $\binom{n}{2}$ summands). For this constant $C$, characterize the instances of equality.

Assume that the bisecting plane of the dihedral angle at edge $AB$ of the tetrahedron $ABCD$ meets the edge $CD$ at point $E$. Denote by $S_1, S_2, S_3$, respectively the areas of the triangles $ABC, ABE$, and $ABD$. Prove that no tetrahedron exists for which $S_1, S_2, S_3$ (in this order) form an arithmetic or geometric progression.

Given $a$, $b$, and $c$ be positive real numbers with $a+b+c=1$, prove that \[\sqrt{a+bc}+\sqrt{b+ca}+\sqrt{c+ab}\le2\]

The 2010 positive numbers $a_1, a_2, \ldots , a_{2010}$ satisfy the inequality $a_ia_j \le i+j$ for all distinct indices $i, j$. Determine, with proof, the largest possible value of the product $a_1a_2\ldots a_{2010}$.

If $ x,y$ are positive real numbers with sum $ 2a$, prove that : $ x^3y^3(x^2\plus{}y^2)^2 \leq 4a^{10}$ When does equality hold ? Babis

Suppose $S=\{1,2,3,...,2017\}$,for every subset $A$ of $S$,define a real number $f(A)\geq 0$ such that: $(1)$ For any $A,B\subset S$,$f(A\cup B)+f(A\cap B)\leq f(A)+f(B)$; $(2)$ For any $A\subset B\subset S$, $f(A)\leq f(B)$; $(3)$ For any $k,j\in S$,$$f(\{1,2,\ldots,k+1\})\geq f(\{1,2,\ldots,k\}\cup \{j\});$$ $(4)$ For the empty set $\varnothing$, $f(\varnothing)=0$. Confirm that for any three-element subset $T$ of $S$,the inequality $$f(T)\leq \frac{27}{19}f(\{1,2,3\})$$ holds.

Let $n \ge 1$ be a fixed positive integer. We consider all the sets $S$ which consist of sub-sequences of the sequence $0, 1,2, ..., n$ satisfying the following conditions: i) If $(a_i)_{i=0}^k$ belongs to $S$, then $a_0 = 0$, $a_k = n$ and $a_{i+1} - a_i \le 2$ for all $0 \le i \le k - 1$. ii) If $(a_i)_{i=0}^k$ and $(b_j)^h_{j=0}$ both belong to $S$, then there exist $0 \le i_0 \le k - 1$ and $0 \le j_0 \le h - 1$ such that $a_{i_0} = b_{j_0}$ and $a_{i_0+1} = b_{j_0+1}$. Find the maximum value of $|S|$ (among all the above-mentioned sets $S$).

a) Find all real numbers $k$ such that there exists a positive constant $c_k$ satisfying $$(x^2 + 1)(y^2 + 1)(z^2 + 1) \ge c_k(x + y + z)^k$$ for all positive real numbers. b) Given the numbers $k$ found, determine the largest number $c_k$.

For $a,b \in [1;\infty)$ show that \[ab\leq e^{a-1}+b\ln b\]