Let $ABC$ be an acute scalene triangle, and let $A_1, B_1, C_1$ be the feet of the altitudes from $A, B, C$. Let $A_2$ be the intersection of the tangents to the circle $ABC$ at $B, C$ and define $B_2, C_2$ similarly. Let $A_2A_1$ intersect the circle $A_2B_2C_2$ again at $A_3$ and define $B_3, C_3$ similarly. Show that the circles $AA_1A_3, BB_1B_3$, and $CC_1C_3$ all have two common points, $X_1$ and $X_2$ which both lie on the Euler line of the triangle $ABC$. [i]United Kingdom, Joe Benton[/i]

There are integers $v,w,x,y,z$ and real numbers $0\le \theta < \theta' \le \pi$ such that $$\cos 3\theta = \cos 3\theta' = v^{-1}, \qquad w+x\cos \theta + y\cos 2\theta = z\cos \theta'.$$ Given that $z\ne 0$ and $v$ is positive, find the sum of the $4$ smallest possible values of $v$. [i]Proposed by Vijay Srinivasan[/i]

The positive integer formed after writing $k$ consecutive positive integers from smallest to largest is called a $k-\text{continuous}$ number. For example $99100101$ is a $3-\text{continuous}$ number. Prove that: for $\forall N$, $k\in\mathbb Z^+$, there must be a $k-\text{continuous}$ number that can be divisible by $N$.

Let $ABCD$ be a square and $M,N$ points on sides $AB, BC$ respectively such that $\angle MDN = 45^{\circ}$. If $R$ is the midpoint of $MN$ show that $RP =RQ$ where $P,Q$ are points of intersection of $AC$ with the lines $MD, ND$.

Let $a$, $b$, $c$, $d$, $e$ be distinct positive integers such that $a^4+b^4=c^4+d^4=e^5$. Show that $ac+bd$ is a composite number.

Let $G$ be a graph, not containing $K_4$ as a subgraph and $|V(G)|=3k$ (I interpret this to be the number of vertices is divisible by 3). What is the maximum number of triangles in $G$?

With the use of three different weights, namely 1 lb., 3 lb., and 9 lb., how many objects of different weights can be weighed, if the objects is to be weighed and the given weights may be placed in either pan of the scale? $ \textbf{(A)}\ 15 \qquad\textbf{(B)}\ 13\qquad\textbf{(C)}\ 11\qquad\textbf{(D)}\ 9\qquad\textbf{(E)}\ 7 $

The circles $k_1$ and $k_2$ intersect at points $M$ and $N$. The line $\ell$ intersects with the circle $k_1$ at points $A$ and $C$ and with circle $k_2$ at points $B$ and $D$, so that points $A, B, C$ and $D$ are on the line $\ell$ in that order. Let $X$ be a point on line $MN$ such that the point $M$ is between points $X$ and $N$. Lines $AX$ and $BM$ intersect at point $P$ and lines $DX$ and $CM$ intersect at point $Q$. Prove that $PQ \parallel \ell $.

A paper rectangle $ABCD$ has $AB = 8$ and $BC = 6$. After corner $B$ is folded over diagonal $AC$, what is $BD$?

The combination to open a safe is a five-digit number. different, randomly selected from $2$ to $9$. To open the box strong, you also need a key that is labeled with the number $410639104$, which is the sum of all combinations that do not open the box. What is the combination that opens the safe?

Find two consecutive integers with the property that the sums of their digits are each divisible by $2006$.

Let $H$ be an orhocenter of an acute triangle $ABC$ and $M$ midpoint of side $BC$. If $D$ and $E$ are foots of perpendicular of $H$ on internal and external angle bisector of angle $\angle BAC$, prove that $M$, $D$ and $E$ are collinear

Let $S$ be a finite set, and let $\mathcal{A}$ be the set of all functions from $S$ to $S$. Let $f$ be an element of $\mathcal{A}$, and let $T=f(S)$ be the image of $S$ under $f$. Suppose that $f\circ g\circ f\ne g\circ f\circ g$ for every $g$ in $\mathcal{A}$ with $g\ne f$. Show that $f(T)=T$.

Let $n$ be a natural number divisible by $3$. We have a $n \times n$ table and each square is colored either black or white. Suppose that for all $m \times m$ sub-tables from the table ($m > 1$), the number of black squares is not more than white squares. Find the maximum number of black squares.

Let $N{}$ be the number of positive integers with $10$ digits $\overline{d_9d_8\cdots d_0}$ in base $10$ (where $0\le d_i\le9$ for all $i$ and $d_9>0$) such that the polynomial \[d_9x^9+d_8x^8+\cdots+d_1x+d_0\] is irreducible in $\Bbb Q$. Prove that $N$ is even. (A polynomial is irreducible in $\Bbb Q$ if it cannot be factored into two non-constant polynomials with rational coefficients.)

4. Given is an integer $n\geq 1$. Find out the number of possible values of products $k \cdot m$, where $k,m$ are integers satisfying $n^{2}\leq k \leq m \leq (n+1)^{2}$.

Let $A'B'C'$ denote the reflection of scalene and acute triangle $ABC$ across its Euler-line. Let $P$ be an arbitrary point of the nine-point circle of $ABC$. For every point $X$, let $p(X)$ denote the reflection of $X$ across $P$. [b]a)[/b] Let $e_{AB}$ denote the line connecting the orthogonal projection of $A$ to line $BB'$ and the orthogonal projection of $B$ to line $AA'$. Lines $e_{BC}$ and $e_{CA}$ are defined analogously. Prove that these three lines are concurrent (and denote their intersection by $K$). [b]b)[/b] Prove that there are two choices of $P$ such that lines $Ap(A')$, $Bp(B')$ and $Cp(C')$ are concurrent, and the four points $p(A)p(A')\cap BC$, $p(B)p(B')\cap CA$, $p(C)p(C')\cap AB$, and $K$ are collinear. [i]Proposed by Áron Bán-Szabó, Budapest[/i]

Solve the system $\begin {cases} 16x^3 + 4x = 16y + 5 \\ 16y^3 + 4y = 16x + 5 \end{cases}$

If $x_{1}, x_{2}, . . . , x_{n}$ are positive numbers, prove the inequality $\frac{x_{1}^{3}}{x_{1}^{2}+x_{1}x_{2}+x_{2}^{2}}+\frac{x_{2}^{3}}{x_{2}^{2}+x_{2}x_{3}+x_{3}^{2}}+...+\frac{x_{n}^{3}}{x_{n}^{2}+x_{n}x_{1}+x_{1}^{2}}\geq\frac{x_{1}+x_{2}+...+x_{n}}{3}$.

Prove that for every positive integer $m$ we can find a finite set $S$ of points in the plane, such that given any point $A$ of $S$, there are exactly $m$ points in $S$ at unit distance from $A$.

Inside an acute triangle $ABC$ is a point $P$ that is not the circumcenter. Prove that among the segments $AP$, $BP$ and $CP$, at least one is longer and at least one is shorter than the circumradius of $ABC$.

Let $a,b,c,d$ be odd integers such that $0<a<b<c<d$ and $ad=bc$. Prove that if $a+d=2^k$ and $b+c=2^m$ for some integers $k$ and $m$, then $a=1$.

The number of $ x$-intercepts on the graph of $ y \equal{} \sin(1/x)$ in the interval $ (0.0001,0.001)$ is closest to $ \textbf{(A)}\ 2900 \qquad \textbf{(B)}\ 3000 \qquad \textbf{(C)}\ 3100 \qquad \textbf{(D)}\ 3200 \qquad \textbf{(E)}\ 3300$

Prove that any triangle can be cut by at most into $3$ parts, from which an isosceles triangle is formed.

Let $a,\ b$ are real numbers such that $a+b=1$. Find the minimum value of the following integral. \[\int_{0}^{\pi}(a\sin x+b\sin 2x)^{2}\ dx \]