Find the sum of the smallest 20 positive real solutions of the equation $\sin x=\cos 2x .$

Find the nondecreasing functions $ f:[0,1]\rightarrow\mathbb{R} $ that satisfy $$ \left| \int_0^1 f(x)e^{nx} dx\right|\le 2008 , $$ for any nonnegative integer $ n. $ [i]Mihai Piticari[/i]

Triangle $ABC$ is isosceles ($AB=AC$). From $A$, we draw a line $\ell$ parallel to $BC$. $P,Q$ are on perpendicular bisectors of $AB,AC$ such that $PQ\perp BC$. $M,N$ are points on $\ell$ such that angles $\angle APM$ and $\angle AQN$ are $\frac\pi2$. Prove that \[\frac{1}{AM}+\frac1{AN}\leq\frac2{AB}\]

On the coordinate plane, draw a set of points $M(x,y)$, the coordinates of which satisfy the inequality $$\log_{x+y} (x^2+y^2) \le 1.$$

Let $a_1,a_2,a_3,\ldots,a_{250}$ be real numbers such that $a_1=2$ and $$a_{n+1}=a_n+\frac{1}{a_n^2}$$ for every $n=1,2, \ldots, 249$. Let $x$ be the greatest integer which is less than $$\frac{1}{a_1}+\frac{1}{a_2}+\ldots+\frac{1}{a_{250}}$$ How many digits does $x$ have? [i]Proposed by Miroslav Marinov, Bulgaria[/i]

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

For a positive integer $n$, $a_1, a_2, \cdots a_k$ are all positive integers without repetition that are not greater than $n$ and relatively prime to $n$. If $k>8$, prove the following. $$\sum_{i=1}^k |a_i-\frac{n}{2}|<\frac{n(k-4)}{2}$$

Let $x_1,x_2,x_3$ be three positive real roots of the equation $x^3+ax^2+bx+c=0$ $(a,b,c\in R)$ and $x_1+x_2+x_3\leq 1. $ Prove that $$a^3(1+a+b)-9c(3+3a+a^2)\leq 0$$

Let $a,b$ be positive real numbers.Prove that $(1+a)^{8}+(1+b)^{8}\geq 128ab(a+b)^{2}$.

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 $f(x)=x^2-6x+5$. On the plane $(x, y)$ draw a set of points $M(x, y)$ whose coordinates satisfy the inequalities $$\begin{cases} f(x)+f(y)\le 0 \\ f(x)-f(y)\ge 0 \end{cases}$$

Prove that if $x,y,z >1$ and $\frac 1x +\frac 1y +\frac 1z = 2$, then \[\sqrt{x+y+z} \geq \sqrt{x-1}+\sqrt{y-1}+\sqrt{z-1}.\]

All the prime numbers are written in order, $p_1 = 2, p_2 = 3, p_3 = 5, ...$ Find all pairs of positive integers $a$ and $b$ with $a - b \geq 2$, such that $p_a - p_b$ divides $2(a-b)$.

Let $a,b,c $ be real positive numbers such that $abc(a+b+c)=3$ Prove that $(a+b)(b+c)(c+a) \geq 8$

if $x,y,z>0$ are postive real numbers such that $x^{2}+y^{2}+z^{2}=x^{2}y^{2}+y^{2}z^{2}+z^{2}x^{2}$ prove that \[((x-y)(y-z)(z-x))^{2}\leq 2((x^{2}-y^{2})^{2}+(y^{2}-z^{2})^{2}+(z^{2}-x^{2})^{2})\]

If $a,b,c,d$ are real numbers with $b < c < d$, prove that $(a+b+c+d)^2 > 8(ac+bd)$.

Let be three positive real numbers $ a,b,c $ whose sum is $ 1. $ Prove: $$ 0\le\sum_{\text{cyc}} \log_a\frac{(abc)^a}{a^2+b^2+c^2} $$

If $k$ and $n$ are positive integers, prove the inequality $$\frac{1}{kn} +\frac{1}{kn+1} +...+\frac{1}{(k+1)n-1} \ge n \left(\sqrt[n]{\frac{k+1}{k}}-1\right)$$

Suppose that $1985$ points are given inside a unit cube. Show that one can always choose $32$ of them in such a way that every (possibly degenerate) closed polygon with these points as vertices has a total length of less than $8 \sqrt 3.$

Prove the following theorem: If $n > 2$ is a natural number, $a_1, ..., a_n$ are positive real numbers and becomes $\sum_{i=1}^n a_i = s$, then the following holds $$\sum_{i=1}^n \frac{a_i}{s - a_i} \ge \frac{n}{n - 1}$$

A number of robots are placed on the squares of a finite, rectangular grid of squares. A square can hold any number of robots. Every edge of each square of the grid is classified as either passable or impassable. All edges on the boundary of the grid are impassable. You can give any of the commands up, down, left, or right. All of the robots then simultaneously try to move in the specified direction. If the edge adjacent to a robot in that direction is passable, the robot moves across the edge and into the next square. Otherwise, the robot remains on its current square. You can then give another command of up, down, left, or right, then another, for as long as you want. Suppose that for any individual robot, and any square on the grid, there is a finite sequence of commands that will move that robot to that square. Prove that you can also give a finite sequence of commands such that all of the robots end up on the same square at the same time.

Show that if natural numbers $a,b, p,q,r,s$ satisfy the conditions $$qr- ps = 1 \,\,\,\,\, and \,\,\,\,\, \frac{p}{q}<\frac{a}{b}<\frac{r}{s},$$ then $b \ge q+s.$

For all $x,y,z\in \mathbb{R}\backslash \{1\}$, such that $xyz=1$, prove that \[ \frac{x^2}{(x-1)^2}+\frac{y^2}{(y-1)^2}+\frac{z^2}{(z-1)^2}\ge 1 \]

Let $S$ be the total area of a tetrahedron whose edges have lengths $a,b,c,d, e, f$ . Prove that $S \le \frac{\sqrt3}{6} (a^2 +b^2 +...+ f^2)$

Let $n \ge 1$ and $x_1, \ldots, x_n \ge 0$. Prove that $$ (x_1 + \frac{x_2}{2} + \ldots + \frac{x_n}{n}) (x_1 + 2x_2 + \ldots + nx_n) \le \frac{(n+1)^2}{4n} (x_1 + x_2 + \ldots + x_n)^2 .$$