The maximum value of $k$ such that the enequality $\sqrt{x-3}+\sqrt{6-x}\geq k$ has a real solution is $\text{(A)}\sqrt6-\sqrt3\qquad\text{(B)}\sqrt3\qquad\text{(C)}\sqrt3+\sqrt6\qquad\text{(D)}\sqrt6$

For positive real numbers$a,b,c$such that $ab+ac+bc=1$ prove that: $\prod\limits_{cyc} (\sqrt{bc}+\frac{1}{2a+\sqrt{bc}}) \ge 8abc$

Let $n$ be a natural number and $f(n) = 2n - 1995 \lfloor \frac{n}{1000} \rfloor$($\lfloor$ $\rfloor$ denotes the floor function). 1. Show that if for some integer $r$: $f(f(f...f(n)...))=1995$ (where the function $f$ is applied $r$ times), then $n$ is multiple of $1995$. 2. Show that if $n$ is multiple of 1995, then there exists r such that:$f(f(f...f(n)...))=1995$ (where the function $f$ is applied $r$ times). Determine $r$ if $n=1995.500=997500$

A lattice point in an $xy$-coordinate system is any point $(x,y)$ where both $x$ and $y$ are integers. The graph of $y=mx+2$ passes through no lattice point with $0<x \le 100$ for all $m$ such that $\frac{1}{2}<m<a$. What is the maximum possible value of $a$? $ \textbf{(A)}\ \frac{51}{101} \qquad \textbf{(B)}\ \frac{50}{99} \qquad \textbf{(C)}\ \frac{51}{100} \qquad \textbf{(D)}\ \frac{52}{101} \qquad \textbf{(E)}\ \frac{13}{25} $

Let $0\leq x_1\leq x_2\leq \cdots \leq x_n\leq 1 $ $(n\geq 2).$ Prove that $$\sqrt[n]{x_1x_2 \cdots x_n}+ \sqrt[n]{(1-x_1)(1-x_2)\cdots (1-x_n)}\leq \sqrt[n]{1-(x_1- x_n)^2}.$$

Prove that the following inequality is valid for the positive $x$: $$2^{x^{1/12}}+ 2^{x^{1/4}} \ge 2^{1 + x^{1/6} }$$

In a convex quadrilateral of area $1$, the sum of the lengths of all sides and diagonals is not less than $4+\sqrt 8$. Prove this.

Given integer $n\geq 2$ and positive real number $a$, find the smallest real number $M=M(n,a)$, such that for any positive real numbers $x_1,x_2,\cdots,x_n$ with $x_1 x_2\cdots x_n=1$, the following inequality holds: \[\sum_{i=1}^n \frac {1}{a+S-x_i}\leq M\] where $S=\sum_{i=1}^n x_i$.

(i) Solve the simultaneous inequalities, $x<\frac{1}{4x}$ and $x<0$; i.e. find a single inequality equivalent to the two simultaneous inequalities. (ii) What is the greatest integer that satisfies both inequalities $4x+13 < 0$ and $x^{2}+3x > 16$. (iii) Give a rational number between $11/24$ and $6/13$. (iv) Express 100000 as a product of two integers neither of which is an integral multiple of 10. (v) Without the use of logarithm tables evaluate \[\frac{1}{\log_{2}36}+\frac{1}{\log_{3}36}.\]

Find all the functions $f:[0,1]\rightarrow \mathbb{R}$ for which we have: \[|x-y|^2\le |f(x)-f(y)|\le |x-y|,\] for all $x,y\in [0,1]$.

The numbers $A_1 , A_2 ,... , A_n$ are given. Prove, without calculating derivatives, that the value of $X$ that minimizes the sum $(X - A_1)^2 + (X -A_2)^2 + ...+ (X - A_n)^2$ is precisely the arithmetic mean of the given numbers.

$\angle{A}$ is the least angle in $\Delta{ABC}$. Point $D$ is on the arc $BC$ from the circumcircle of $\Delta{ABC}$. The perpendicular bisectors of the segments $AB,AC$ intersect the line $AD$ at $M,N$, respectively. Point $T$ is the meet point of $BM,CN$. Suppose that $R$ is the radius of the circumcircle of $\Delta{ABC}$. Prove that: \[ BT+CT\leq{2R}. \]

Find all $n \in \mathbb N$ such that $n = \varphi (n) + 402$, where $\varphi$ denotes the Euler phi function.

Let real numbers $ a,b,c$. Prove that $ \sqrt{2(a^2\plus{}b^2)}\plus{}\sqrt{2(b^2\plus{}c^2)}\plus{}\sqrt{2(c^2\plus{}a^2)}\ge \sqrt{3(a\plus{}b)^2\plus{}3(b\plus{}c)^2\plus{}3(c\plus{}a)^2}$.

The sum of $2025$ non-negative real numbers is $1$. Prove that they can be organized in a circle in such a way that the sum of all the $2025$ products of pairs of neighbouring numbers isn't greater than $\frac{1}{2025}$.

For $ x\in\mathbb{R} , $ determine the minimum of $ \sqrt{(x-1)^2+\left( x^2-5\right)^2} +\sqrt{(x+2)^2+\left( x^2+1 \right)^2} $ and the maximum of $ \sqrt{(x-1)^2+\left( x^2-5\right)^2} -\sqrt{(x+2)^2+\left( x^2+1 \right)^2} . $ [i]Vasile Pop[/i]

Let $n>3$ be a natural number . Prove that \[\frac{1}{3^3}+\frac{1}{4^3}+\cdots+\frac{1}{n^3}<\frac{1}{12}.\]

Prove inequality, with $a$ and $b$ nonnegative real numbers: $\frac{a+b}{1+a+b}\leq \frac{a}{1+a} + \frac{b}{1+b} \leq \frac{2(a+b)}{2+a+b}$

Let $ G=(V,E)$ be a simple graph. a) Let $ A,B$ be a subsets of $ E$, and spanning subgraphs of $ G$ with edges $ A,B,A\cup B$ and $ A\cap B$ have $ a,b,c$ and $ d$ connected components respectively. Prove that $ a+b\leq c+d$. We say that subsets $ A_1,A_2,\dots,A_m$ of $ E$ have $ (R)$ property if and only if for each $ I\subset\{1,2,\dots,m\}$ the spanning subgraph of $ G$ with edges $ \cup_{i\in I}A_i$ has at most $ n-|I|$ connected components. b) Prove that when $ A_1,\dots,A_m,B$ have $ (R)$ property, and $ |B|\geq2$, there exists an $ x\in B$ such that $ A_1,A_2,\dots,A_m,B\backslash\{x\}$ also have property $ (R)$. Suppose that edges of $ G$ are colored arbitrarily. A spanning subtree in $ G$ is called colorful if and only if it does not have any two edges with the same color. c) Prove that $ G$ has a colorful subtree if and only if for each partition of $ V$ to $ k$ non-empty subsets such as $ V_1,\dots,V_k$, there are at least $ k\minus{}1$ edges with distinct colors that each of these edges has its two ends in two different $ V_i$s. d) Assume that edges of $ K_n$ has been colored such that each color is repeated $ \left[\frac n2\right]$ times. Prove that there exists a colorful subtree. e) Prove that in part d) if $ n\geq5$ there is a colorful subtree that is non-isomorphic to $ K_{1,n-1}$. f) Prove that in part e) there are at least two non-intersecting colorful subtrees.

Prove that there exists a positive real number $C$ with the following property: for any integer $n\ge 2$ and any subset $X$ of the set $\{1,2,\ldots,n\}$ such that $|X|\ge 2$, there exist $x,y,z,w \in X$(not necessarily distinct) such that \[0<|xy-zw|<C\alpha ^{-4}\] where $\alpha =\frac{|X|}{n}$.

Determine the minimum value of the expression $2x^4 - 2x^2y^2 + y^4 - 8x^2 + 18$ where $x, y \in R$.

Find all $x, y, z \in \left (0, \frac{1}{2}\right )$ such that $$ \begin{cases} (3 x^{2}+y^{2}) \sqrt{1-4 z^{2}} \geq z; \\ (3 y^{2}+z^{2}) \sqrt{1-4 x^{2}} \geq x; \\ (3 z^{2}+x^{2}) \sqrt{1-4 y^{2}} \geq y. \end{cases} $$ [i]Proposed by Ngo Van Trang, Vietnam[/i]

Let $R$ and $r$ be the radii of the circles circumscribed and inscribed, respectively, in a triangle. Prove that $R \ge 2r$, and that $R = 2r$ only for an equilateral triangle.

Let $a_1, a_2, \ldots, a_n$ be reals such that for all $k=1,2, \ldots, n$, $na_k \geq a_1^2+a_2^2+ \ldots+a_k^2$. Prove that there exist at least $\frac{n} {10}$ indices $k$, such that $a_k \leq 1000$.

Let $ H(n)$ be the number of simply connected subsets with $ n$ hexagons in an infinite hexagonal network. Also let $ P(n)$ be the number of paths starting from a fixed vertex (that do not connect itself) with lentgh $ n$ in this hexagonal network. a) Prove that the limits \[ \alpha: \equal{}\lim_{n\rightarrow\infty}H(n)^{\frac1n}, \beta: \equal{}\lim_{n\rightarrow\infty}P(n)^{\frac1n}\]exist. b) Prove the following inequalities: $ \sqrt2\leq\beta\leq2$ $ \alpha\leq 12.5$ $ \alpha\geq3.5$ $ \alpha\leq\beta^4$