Solve the equation (for positive $x$) $$x^x=\frac{1}{\sqrt2}$$

Let $n>1$ be a positive integer and $\mathcal S$ be the set of $n^{\text{th}}$ roots of unity. Suppose $P$ is an $n$-variable polynomial with complex coefficients such that for all $a_1,\ldots,a_n\in\mathcal S$, $P(a_1,\ldots,a_n)=0$ if and only if $a_1,\ldots,a_n$ are all different. What is the smallest possible degree of $P$? [i]Adam Ardeishar and Michael Ren[/i]

Given $x,y,z$, three different integers. Prove that $$(x-y)^5+(y-z)^5+(z-x)^5$$ is divisible by $$5(x-y)(y-z)(z-x)$$

$a, b, c, d$ are integers such that $ad + bc$ divides each of $a, b, c$ and $d$. Prove that $ad + bc =\pm 1$

The circle inscribed in the triangle $ABC$ touches the sides $AB$ and $AC$ at the points $K$ and $L$ , respectively. The angle bisectors from $B$ and $C$ intersect the altitude of the triangle from the vertex $A$ at the points $Q$ and $R$ , respectively. Prove that one of the points of intersection of the circles circumscribed around the triangles $BKQ$ and $CPL$ lies on $BC$.

One hundred natural numbers whose greatest common divisor is $1$ are arranged around a circle. An allowed operation is to add to a number the greatest common divisor of its two neighhbors. Prove that we can make all the numbers pairwise copirme in a finite number of moves.

Let $a,b,c,d$ be distinct integers such that $$(x-a)(x-b)(x-c)(x-d) -4=0$$ has an integer root $r.$ Show that $4r=a+b+c+d.$

Show that \[\cos(56^{\circ}) \cdot \cos(2 \cdot 56^{\circ}) \cdot \cos(2^2\cdot 56^{\circ})\cdot . . . \cdot \cos(2^{23}\cdot 56^{\circ}) = \frac{1}{2^{24}} .\]

Let $x,y,z$ be real numbers with $|x|,|y|,|z| > 2$. What is the smallest possible value of $|xyz+2(x+y+z)|$ ?

Let $S$ be a set of positive real numbers with five elements such that for any distinct $a,~ b,~ c$ in $S$, the number $ab + bc + ca$ is rational. Prove that for any $a$ and $b$ in $S$, $\tfrac{a}{b}$ is a rational number.

In how many ways can $1 + 2 + \cdots + 2007$ be expressed as a sum of consecutive positive integers?

Prove that $\sqrt 3$ is irrational.

Find the largest $n$ for which there exists a sequence $(a_0, a_1, \ldots, a_n)$ of non-zero digits such that, for each $k$, $1 \le k \le n$, the $k$-digit number $\overline{a_{k-1} a_{k-2} \ldots a_0} = a_{k-1} 10^{k-1} + a_{k-2} 10^{k-2} + \cdots + a_0$ divides the $(k+1)$-digit number $\overline{a_{k} a_{k-1}a_{k-2} \ldots a_0}$. P.S.: This is basically the same problem as http://www.artofproblemsolving.com/Forum/viewtopic.php?f=57&t=548550.

Antonette gets $ 70\%$ on a 10-problem test, $ 80\%$ on a 20-problem test and $ 90\%$ on a 30-problem test. If the three tests are combined into one 60-problem test, which percent is closest to her overall score? $ \textbf{(A)}\ 40 \qquad \textbf{(B)}\ 77 \qquad \textbf{(C)}\ 80 \qquad \textbf{(D)}\ 83 \qquad \textbf{(E)}\ 87$

There are 2008 congruent circles on a plane such that no two are tangent to each other and each circle intersects at least three other circles. Let $ N$ be the total number of intersection points of these circles. Determine the smallest possible values of $ N$.

We define a binary operation $\star$ in the plane as follows: Given two points $A$ and $B$ in the plane, $C = A \star B$ is the third vertex of the equilateral triangle ABC oriented positively. What is the relative position of three points $I, M, O$ in the plane if $I \star (M \star O) = (O \star I)\star M$ holds?

Let $h_1$ and $h_2$ be the altitudes of a triangle drawn to the sides with length $5$ and $2\sqrt 6$, respectively. If $5+h_1 \leq 2\sqrt 6 + h_2$, what is the third side of the triangle? $ \textbf{(A)}\ 5 \qquad\textbf{(B)}\ 7 \qquad\textbf{(C)}\ 2\sqrt 6 \qquad\textbf{(D)}\ 3\sqrt 6 \qquad\textbf{(E)}\ 5\sqrt 3 $

An airline company is planning to introduce a network of connections between the ten different airports of Sawubonia. The airports are ranked by priority from first to last (with no ties). We call such a network [i]feasible[/i] if it satisfies the following conditions: [list] [*] All connections operate in both directions [*] If there is a direct connection between two airports A and B, and C has higher priority than B, then there must also be a direct connection between A and C.[/list] Some of the airports may not be served, and even the empty network (no connections at all) is allowed. How many feasible networks are there?

Does there exist a twice differentiable function $f:\mathbb R\to\mathbb R$ such that $f'(x)=f(x+1)-f(x)$ for all $x$ and $f''(0)\ne0$? Justify your answer.

A dealer bought $n$ radios for $d$ dollars, $d$ a positive integer. He contributed two radios to a community bazaar at half their cost. The rest he sold at a profit of $\$8$ on each radio sold. If the overall profit was $\$72$, then the least possible value of $n$ for the given information is: $\textbf{(A)}\ 18\qquad \textbf{(B)}\ 16\qquad \textbf{(C)}\ 15\qquad \textbf{(D)}\ 12\qquad \textbf{(E)}\ 11$

Let $ A_1,B_1,C_1 $ be points on the sides (excluding their endpoints) $ BC,CA,AB, $ respectively, of a triangle $ ABC, $ such that $ \angle A_1AB =\angle B_1BC=\angle C_1CA. $ Let $ A^* $ be the intersection of $ BB_1 $ with $ CC_1,B^* $ be the intersection of $ CC_1 $ with $ AA_1, $ and $ C^* $ be the intersection of $ AA_1 $ with $ BB_1. $ Denote with $ r_A,r_B,r_C $ the inradii of $ A^*BC,AB^*C,ABC^*, $ respectively. Prove that $$ \frac{r_A}{BC}=\frac{r_B}{CA}=\frac{r_C}{AB} $$ if and only if $ ABC $ is equilateral. [i]Daniel Văcărețu[/i]

Given is a directed graph with $28$ vertices, such that there do not exist vertices $u, v$, such that $u \rightarrow v$ and $v \rightarrow u$. Every $16$ vertices form a directed cycle. Prove that among any $17$ vertices, we can choose $15$ which form a directed cycle.

Let $M=\bigoplus_{i \in \mathbb{Z}} \mathbb{C}e_i$ be an infinite dimensional $\mathbb{C}$-vector space, and let $\text{End}(M)$ denote the $\mathbb{C}$-algebra of $\mathbb{C}$-linear endomorphisms of $M$. Let $A$ and $B$ be two commuting elements in $\text{End}(M)$ satisfying the following condition: there exist integers $m \le n<0<p \le q$ satisfying $\text{gd}(-m,p)=\text{gcd}(-n,q)=1$, and such that for every $j \in \mathbb{Z}$, one has \[Ae_j=\sum_{i=j+m}^{j+n} a_{i,j}e_i, \quad \text{with } a_{i,j} \in \mathbb{C}, a_{j+m,j}a_{j+n,j} \ne 0,\] \[Be_j=\sum_{i=j+p}^{j+q} b_{i,j}e_i, \quad \text{with } b_{i,j} \in \mathbb{C}, b_{j+p,j}b_{j+q,j} \ne 0.\] Let $R \subset \text{End}(M)$ be the $\mathbb{C}$-subalgebra generated by $A$ and $B$. Note that $R$ is commutative and $M$ can be regarded as an $R$-module. (a) Show that $R$ is an integral domain and is isomorphic to $R \cong \mathbb{C}[x,y]/f(x,y)$, where $f(x,y)$ is a non-zero polynomial such that $f(A,B)=0$. (b) Let $K$ be the fractional field of $R$. Show that $M \otimes_R K$ is a $1$-dimensional vector space over $K$.

Show that a line passing through the feet of two altitudes of an acute-angled triangle cuts off a similar triangle.

$ C_{1}$ and $ C_{2}$ be two externally tangent circles with diameter $ \left[AB\right]$ and $ \left[BC\right]$, with center $ D$ and $ E$, respectively. Let $ F$ be the intersection point of tangent line from A to $ C_{2}$ and tangent line from $ C$ to $ C_{1}$ (both tangents line on the same side of $ AC$). If $ \left|DB\right|\equal{}\left|BE\right|\equal{}\sqrt{2}$, find the area of triangle $ AFC$. $\textbf{(A)}\ \frac{7\sqrt{3} }{2} \qquad\textbf{(B)}\ \frac{9\sqrt{2} }{2} \qquad\textbf{(C)}\ 4\sqrt{2} \qquad\textbf{(D)}\ 2\sqrt{3} \qquad\textbf{(E)}\ 2\sqrt{2}$