Determine all positive roots of the equation $ x^x = \frac{1}{\sqrt{2}}.$

Prove that for an arbitrary prime $p \ge 3$ the number of positive integers $n$, for which $p | n! +1$ does not exceed $cp^{2/3}$, where c is a constant that does not depend on $p$.

In regular tetrahedron $ABCD$, $E\in AB,F\in CD$, satisfying: $\frac{|AE|}{|EB|}=\frac{|CF|}{|FD|}=\lambda(\lambda\in R_+)$. Note that $f(\lambda)=\alpha_{\lambda}+\beta_{\lambda}$, where $\alpha_{\lambda}=<EF,AC>,\alpha_{\lambda}=<EF,BD>$. $\text{(A)}$ $f(\lambda)$ increases in $(0,+\infty)$ $\text{(B)}$ $f(\lambda)$ decreases in $(0,+\infty)$ $\text{(C)}$ $f(\lambda)$ increases in $(0,1)$, decreases in $(1,+\infty)$ $\text{(D)}$ $f(\lambda)$ is a fixed value in $(0,+\infty)$

If $ p(x)$ is a polynomial, denote by $ p^n(x)$ the polynomial $ p(p(...(p(x))..)$, where $ p$ is iterated $ n$ times. Prove that the polynomial $ p^{2003}(x)\minus{}2p^{2002}(x)\plus{}p^{2001}(x)$ is divisible by $ p(x)\minus{}x$

Suppose there are $n$ points on the plane, no three of which are collinear. Draw $n-1$ non-intersecting segments (except possibly at endpoints) between pairs of points, such that it is possible to travel between any two points by travelling along the segments. Such a configuration of points and segments is called a [i]network[/i]. Given a network, we may assign labels from $1$ to $n-1$ to each segment such that each segment gets a different label. Define a [i]spin[/i] as the following operation: $\bullet$ Choose a point $v$ and rotate the labels of its adjacent segments clockwise. Formally, let $e_1,e_2,\cdots,e_k$ be the segments which contain $v$ as an endpoint, sorted in clockwise order (it does not matter which segment we choose as $e_1$). Then, the label of $e_{i+1}$ is replaced with the label of $e_{i}$ simultaneously for all $1 \le i \le k$. (where $e_{k+1}=e_{1}$) A network is [i]nontrivial[/i] if there exists at least $2$ points with at least $2$ adjacent segments each. A network is [i]versatile[/i] if any labeling of its segments can be obtained from any initial labeling using a finite amount of spins. Find all integers $n \ge 5$ such that any nontrivial network with $n$ points is versatile. [i]Proposed by Yeoh Zi Song[/i]

With $\sigma (n)$ we denote the sum of natural divisors of the natural number $n$. Prove that, if $n$ is the product of different prime numbers of the form $2^k-1$ for $k \in \mathbb{N}$($Mersenne's$ prime numbers) , than $\sigma (n)=2^m$, for some $m \in \mathbb{N}$. Is the inverse statement true?

How many real numbers $x$ are solutions to the following equation? \[ |x-1| = |x-2| + |x-3| \]

Let $P$ be a point in the plane. Prove that it is possible to draw three rays with origin in $P$ with the following property: for every circle with radius $r$ containing $P$ in its interior, if $P_1$, $P_2$ and $P_3$ are the intersection points of the three rays with the circle, then \[|PP_1|+|PP_2|+|PP_3|\leq 3r.\]

A mushroom is called [i]bad [/i] if it contains at least $10$ worms. A basket contains $90$ bad and $10$ good mushrooms. Can all mushrooms become good after some worms crawl from bad mushrooms to good ones? [hide=original wording]Гриб называется плохим, если в нем не менее 10 червей. В лукошке 90 плохих и 10 хороших грибов. Могут ли все грибы стать хорошими после того, как некоторые черви переползут из плохих грибов в хорошие?[/hide]

Five identical empty buckets of $2$-liter capacity stand at the vertices of a regular pentagon. Cinderella and her wicked Stepmother go through a sequence of rounds: At the beginning of every round, the Stepmother takes one liter of water from the nearby river and distributes it arbitrarily over the five buckets. Then Cinderella chooses a pair of neighbouring buckets, empties them to the river and puts them back. Then the next round begins. The Stepmother goal's is to make one of these buckets overflow. Cinderella's goal is to prevent this. Can the wicked Stepmother enforce a bucket overflow? [i]Proposed by Gerhard Woeginger, Netherlands[/i]

Consider a variable point $P$ inside a given triangle $ABC$. Let $D$, $E$, $F$ be the feet of the perpendiculars from the point $P$ to the lines $BC$, $CA$, $AB$, respectively. Find all points $P$ which minimize the sum \[ {BC\over PD}+{CA\over PE}+{AB\over PF}. \]

Prove that for any positive integer $n$, the number \[ S_n = {2n+1\choose 0}\cdot 2^{2n}+{2n+1\choose 2}\cdot 2^{2n-2}\cdot 3 +\cdots + {2n+1 \choose 2n}\cdot 3^n \] is the sum of two consecutive perfect squares. [i]Dorin Andrica[/i]

You can inscribe a circle in the pentagon $ABCDE$. It is also known that $\angle ABC = \angle BAE = \angle CDE = 90^o$. Find the measure of the angle $ADB$.

Let $ ABCD$ be a cyclic quadrilateral and $ O$ be the intersection of diagonal $ AC$ and $ BD$. The circumcircles of triangle $ ABO$ and the triangle $ CDO$ intersect at $ K$. Let $ L$ be a point such that the triangle $ BLC$ is similar to $ AKD$ (in that order). Prove that if $ BLCK$ is a convex quadrilateral, then it has an incircle.

Let $ ABCD $ be a regular tetahedron and $ M,N $ be middlepoints for $ AD, $ respectively, $ BC. $ Through a point $ P $ that is on segment $ MN, $ passes a plane perpendicular on $ MN, $ and meets the sides $ AB,AC,CD,BD $ of the tetahedron at $ E,F,G, $ respectively, $ H. $ [b]a)[/b] Prove that the perimeter of the quadrilateral $ EFGH $ doesn't depend on $ P. $ [b]b)[/b] Determine the maximum area of $ EFGH $ (depending on a side of the tetahedron).

A circle passing through the endpoints of the leg AB of an isosceles triangle ABC intersects the base BC in point P. A line tangent to the circle in point B intersects the circumcircle of ABC in point Q. Prove that P lies on line AQ if and only if AQ and BC are perpendicular.

A set $M$ of natural numbers is called a [i]spectrum[/i] if there is a first-order language $L$ and a sentence $\phi$ over $L$ such that: $$M = \{ n \mid \phi \text{ has a model containing exactly $n$ elements}\}$$ For example, consider a sentence $\phi = \exists e . (\forall x. x = e)$ in a first order language with no relation symbol, no function symbol, and no constant symbol. The formula $\phi$ only admits a model containing exactly 1 element. Therefore, the set $\{1\}$ is a spectrum.\\ Show that: [list=1] [*] Every finite subset of $\mathbb{N} \setminus \{0\} $ is a spectrum [/*] [*] The set of even numbers, i.e., $\{2k \mid k \in \mathbb{N}\}$ is a spectrum [/*] [*] For any fixed $m \geq 1$, the set of numbers greater than $0$ that are divisible by $m$, i.e., $\{m\cdot k \mid k \in \mathbb{N}\}$ is a spectrum[/*] [/list]

How many seven-digit numbers are multiples of $388$ and end in $388$?

Evaluate \[{\int_0^\frac{\pi}{2}} \frac{\sin 2005x}{\sin x}dx\]

Let $X$ be the set of all even positive integers $n$ such that the measure of the angle of some regular polygon is $n$ degrees. Find the number of elements in $X$.

Let $ABC$ be a triangle where $AC > AB$ and $I$ is its incentre. Let $M$ be the midpoint of arc $BAC$. Let $MI$ meet $AB$ at $T$. Let the incircle of $ABC$ meet $BC$ at $D$. Show that $B,I,C,T$ are concyclic if and only if $CD =3BD$.

Let $O_1$ be a point in the exterior of the circle $\omega$ of center $O$ and radius $R$ , and let $O_1N$ , $O_1D$ be the tangent segments from $O_1$ to the circle. On the segment $O_1N$ consider the point $B$ such that $BN=R$ .Let the line from $B$ parallel to $ON$ intersect the segment $O_1D$ at $C$ . If $A$ is a point on the segment $O_1D$ other than $C$ so that $BC=BA=a$ , and if the incircle of the triangle $ABC$ has radius $r$ , then find the area of $\triangle ABC$ in terms of $a ,R ,r$.

Let $ABC$ be a triangle with $AB=AC$, and let $M$ be the midpoint of $BC$. Let $P$ be a point such that $PB<PC$ and $PA$ is parallel to $BC$. Let $X$ and $Y$ be points on the lines $PB$ and $PC$, respectively, so that $B$ lies on the segment $PX$, $C$ lies on the segment $PY$, and $\angle PXM=\angle PYM$. Prove that the quadrilateral $APXY$ is cyclic.

The kingdom of Anisotropy consists of $n$ cities. For every two cities there exists exactly one direct one-way road between them. We say that a [i]path from $X$ to $Y$[/i] is a sequence of roads such that one can move from $X$ to $Y$ along this sequence without returning to an already visited city. A collection of paths is called [i]diverse[/i] if no road belongs to two or more paths in the collection. Let $A$ and $B$ be two distinct cities in Anisotropy. Let $N_{AB}$ denote the maximal number of paths in a diverse collection of paths from $A$ to $B$. Similarly, let $N_{BA}$ denote the maximal number of paths in a diverse collection of paths from $B$ to $A$. Prove that the equality $N_{AB} = N_{BA}$ holds if and only if the number of roads going out from $A$ is the same as the number of roads going out from $B$. [i]Proposed by Warut Suksompong, Thailand[/i]

All the grids of a $m\times n$ chess board ($m,n\geq 3$), are colored either with red or with blue. Two adjacent grids (having a common side) are called a "good couple" if they have different colors. Suppose there are $S$ "good couples". Explain how to determine whether $S$ is odd or even. Is it prescribed by some specific color grids? Justify your answers.