Find the product of all real $x$ for which \[ 2^{3x+1} - 17 \cdot 2^{2x} + 2^{x+3} = 0. \]

For $ a>1$, find $ \lim_{n\to\infty} \int_0^a \frac{e^x}{1\plus{}x^n}dx.$

Initially one have the number $0$ in each cell of the table $29 \times 29$. A [i]moviment[/i] is when one choose a sub-table $5 \times 5$ and add $+1$ for every cell of this sub-table. Find the maximum value of $n$, where after $1000$ [i]moviments[/i], there are $4$ cells such that your centers are vertices of a square and the sum of this $4$ cells is at least $n$. [b]Note:[/b] A square does not, necessarily, have your sides parallel with the sides of the table.

Find the least positive integer $N$ such that the set of $1000$ consecutive integers beginning with $1000 \cdot N$ contains no square of an integer.

It is known that among several banknotes of pairwise distinct face values (which are positive integers) there are exactly $N{}$ fakes. In a single test, a detector determines the sum of the face values of all real banknotes in an arbitrary set we have selected. Prove that by using the detector $N{}$ times, all fake banknotes can be identified, if a) $N=2$ and b) $N=3$. [i]Proposed by S. Tokarev[/i]

Given a parallelogram $ABCD$ in which $\angle ABD=90^\circ$ and $\angle CBD=45^\circ$. Point $E$ lies on segment $AD$ such that $BC=CE$. Determine the measure of angle $BCE$.

Find all positive integer $n$ such that for all positive integers $ x $, $ y $, $ n \mid x^n-y^n \Rightarrow n^2 \mid x^n-y^n $.

Let $\mathbb{N}$ denote the set of all positive integers. An ordered pair $(a;b)$ of numbers $a,b\in\mathbb{N}$ is called [i]interesting[/i], if for any $n\in\mathbb{N}$ there exists $k\in\mathbb{N}$ such that the number $a^k+b$ is divisible by $2^n$. Find all [i]interesting[/i] ordered pairs of numbers.

Let $a, b, c$ be positive real number such that $a + b + c \ge \frac{1}{a}+ \frac{1}{b}+ \frac{1}{c}$ . Prove that $ \frac{a}{b}+ \frac{b}{c}+ \frac{c}{a}\ge \frac{1}{ab}+ \frac{1}{bc}+ \frac{1}{ca}$ .

Aibek wrote 6 letters to 6 different person.[b][u]In how many ways[/u][/b] can he send the letters to them,such that no person gets his letter.

Let $n$ be a given integer such that $n\ge 2$. Find the smallest real number $\lambda$ with the following property: for any real numbers $x_1,x_2,\ldots ,x_n\in [0,1]$ , there exists integers $\varepsilon_1,\varepsilon_2,\ldots ,\varepsilon_n\in\{0,1\}$ such that the inequality $$\left\vert \sum^j_{k=i} (\varepsilon_k-x_k)\right\vert\le \lambda$$holds for all pairs of integers $(i,j)$ where $1\le i\le j\le n$.

Define two functions $g(x),\ f(x)\ (x\geq 0)$ by $g(x)=\int_0^x e^{-t^2}dt,\ f(x)=\int_0^1 \frac{e^{-(1+s^2)x}}{1+s^2}ds.$ Now we know that $f'(x)=-\int_0^1 e^{-(1+s^2)x}ds.$ (1) Find $f(0).$ (2) Show that $f(x)\leq \frac{\pi}{4}e^{-x}\ (x\geq 0).$ (3) Let $h(x)=\{g(\sqrt{x})\}^2$. Show that $f'(x)=-h'(x).$ (4) Find $\lim_{x\rightarrow +\infty} g(x)$ Please solve the problem without using Double Integral or Jacobian for those Japanese High School Students who don't study them.

Let $a$,$b$,$c$ be positive real numbers and $a+b+c+2=abc$. Prove that \[\frac{a}{b+1}+\frac{b}{c+1}+\frac{c}{a+1}\geq{2}. \]

Let $ABCDE$ be a convex pentagon with $CD= DE$ and $\angle EDC \ne 2 \cdot \angle ADB$. Suppose that a point $P$ is located in the interior of the pentagon such that $AP =AE$ and $BP= BC$. Prove that $P$ lies on the diagonal $CE$ if and only if area $(BCD)$ + area $(ADE)$ = area $(ABD)$ + area $(ABP)$. (Hungary)

There are several discs whose radii are no more that $1$, and whose centers all lie on a segment with length ${l}$. Prove that the union of all the discs has a perimeter not exceeding $4l+8$. [i]Proposed by Morteza Saghafian - Iran[/i]

Let $N$ be a positive integer. Kingdom Wierdo has $N$ castles, with at most one road between each pair of cities. There are at most four guards on each road. To cost down, the King of Wierdos makes the following policy: (1) For any three castles, if there are roads between any two of them, then any of these roads cannot have four guards. (2) For any four castles, if there are roads between any two of them, then for any one castle among them, the roads from it toward the other three castles cannot all have three guards. Prove that, under this policy, the total number of guards on roads in Kingdom Wierdo is smaller than or equal to $N^2$. [i]Remark[/i]: Proving that the number of guards does not exceed $cN^2$ for some $c > 1$ independent of $N$ will be scored based on the value of $c$. [i]Proposed by usjl[/i]

Placing $n \in {\mathbb N}$ circles with radius $1$ $unit$ inside a square with side $100$ $unit$ such that, whichever line segment with lenght $10$ $unit$ intersect at least one circle. Prove that $$n \geq 416$$

Consider a prism, not necessarily right, whose base is a rhombus $ABCD$ with side $AB = 5$ and diagonal $AC = 8$. A sphere of radius $r$ is tangent to the plane $ABCD$ at $C$ and tangent to the edges $AA_1$ , $BB _1$ and $DD_ 1$ of the prism. Calculate $r$ .

Given a figure, containing $16$ segments. You should prove that there is no curve, that intersect each segment exactly once. The curve may be not closed, may intersect itself, but it is not allowed to touch the segments or to pass through the vertices. [asy] draw((0,0)--(6,0)--(6,3)--(0,3)--(0,0)); draw((0,3/2)--(6,3/2)); draw((2,0)--(2,3/2)); draw((4,0)--(4,3/2)); draw((3,3/2)--(3,3)); [/asy]

Heather and Kyle need to mow a lawn and paint a room. If Heather does both jobs by herself, it will take her a total of nine hours. If Heather mows the lawn and, after she finishes, Kyle paints the room, it will take them a total of eight hours. If Kyle mows the lawn and, after he finishes, Heather paints the room, it will take them a total of seven hours. If Kyle does both jobs by himself, it will take him a total of six hours. It takes Kyle twice as long to paint the room as it does for him to mow the lawn. The number of hours it would take the two of them to complete the two tasks if they worked together to mow the lawn and then worked together to paint the room is a fraction $\tfrac{m}{n}$where $m$ and $n$ are relatively prime positive integers. Find $m + n$.

Let $f : \mathbb{R} \to \mathbb{R}$ be a continuous function and let $g : \mathbb{R} \to \mathbb{R}$ be arbitrary. Suppose that the Minkowski sum of the graph of $f$ and the graph of $g$ (i.e., the set $\{( x+y; f(x)+g(y) ) \mid x, y \in \mathbb{R}\}$) has Lebesgue measure zero. Does it follow then that the function $f$ is of the form $f(x) = ax + b$ with suitable constants $a, b \in \mathbb{R}$ ?

There are real numbers $x, y$ such that $x \ne 0$, $y \ne 0$, $xy + 1 \ne 0$ and $x + y \ne 0$. Suppose the numbers $x + \frac{1}{x} + y + \frac{1}{y}$ and $x^3+\frac{1}{x^3} + y^3 + \frac{1}{y^3}$ are rational. Prove that then the number $x^2+\frac{1}{x^2} + y^2 + \frac{1}{y^2}$ is also rational.

The function $\psi : \mathbb{N}\rightarrow\mathbb{N}$ is defined by $\psi (n)=\sum_{k=1}^n\gcd (k,n)$. $(a)$ Prove that $\psi (mn)=\psi (m)\psi (n)$ for every two coprime $m,n \in \mathbb{N}$. $(b)$ Prove that for each $a\in\mathbb{N}$ the equation $\psi (x)=ax$ has a solution.

In quadrilateral $ABCD$, $AB=2$, $AD=3$, $BC=CD=\sqrt7$, and $\angle DAB=60^\circ$. Semicircles $\gamma_1$ and $\gamma_2$ are erected on the exterior of the quadrilateral with diameters $\overline{AB}$ and $\overline{AD}$; points $E\neq B$ and $F\neq D$ are selected on $\gamma_1$ and $\gamma_2$ respectively such that $\triangle CEF$ is equilateral. What is the area of $\triangle CEF$?

Let $a, b, c $ be distinct positive integers. a) Prove that $a^2b^2 + a^2c^2 + b^2c^2 \ge 9$. b) if, moreover, $ab + ac + bc +3 = abc > 0,$ show that $$(a -1)(b -1)+(a -1)(c -1)+(b -1)(c -1) \ge 6.$$