[b]i.)[/b] Consider a circle $K$ with diameter $AB;$ with circle $L$ tangent to $AB$ and to $K$ and with a circle $M$ tangent to circle $K,$ circle $L$ and $AB.$ Calculate the ration of the area of circle $K$ to the area of circle $M.$ [b]ii.)[/b] In triangle $ABC, AB = AC$ and $\angle CAB = 80^{\circ}.$ If points $D,E$ and $F$ lie on sides $BC, AC$ and $AB,$ respectively and $CE = CD$ and $BF = BD,$ then find the size of $\angle EDF.$

In a triangle $ABC$, the points $A_1, B_1, C_1$ are symmetric to $A, B,C$ with respect to $B,C, A$, respectively. Given the points $A_1, B_1,C_1$ reconstruct the triangle $ABC$.

3. Show that there do not exist positive integers $a$, $b$, $c$ and $d$, pairwise co-prime, such that $ab+cd$, $ac+bd$ and $ad+bc$ are odd divisors of the number $(a+b-c-d)(a-b+c-d)(a-b-c+d)$.

Two identically oriented equilateral triangles, $ABC$ with center $S$ and $A'B'C$, are given in the plane. We also have $A' \neq S$ and $B' \neq S$. If $M$ is the midpoint of $A'B$ and $N$ the midpoint of $AB'$, prove that the triangles $SB'M$ and $SA'N$ are similar.

Let $m$ be a positive integer, and let $T$ denote the set of all subsets of $\{1, 2, \dots, m\}$. Call a subset $S$ of $T$ $\delta$-[I]good[/I] if for all $s_1, s_2\in S$, $s_1\neq s_2$, $|\Delta (s_1, s_2)|\ge \delta m$, where $\Delta$ denotes the symmetric difference (the symmetric difference of two sets is the set of elements that is in exactly one of the two sets). Find the largest possible integer $s$ such that there exists an integer $m$ and $ \frac{1024}{2047}$-good set of size $s$.

A line in the plane of a triangle $ABC$ intersects the sides $AB$ and $AC$ respectively at points $X$ and $Y$ such that $BX = CY$ . Find the locus of the center of the circumcircle of triangle $XAY .$

[asy]pair A=(-2,0), O=origin, C=(2,0); path X=Arc(O,2,0,180), Y=Arc((-1,0),1,180,0), Z=Arc((1,0),1,180,0), N=X..Y..Z..cycle; filldraw(N, black, black); draw(reflect(A,C)*N); draw(A--C, dashed); label("A",A,W); label("C",C,E); label("O",O,SE); dot((-1,0)); dot(O); dot((1,0)); label("1",(-1,0),NE); label("1",(1,0),NW);[/asy] The large circle has diameter $ \overline{AC}$. The two small circles have their centers on $ \overline{AC}$ and just touch at $ O$, the center of the large circle. If each small circle has radius $ 1$, what is the value of the ratio of the area of the shaded region to the area of one of the small circles? \[ \textbf{(A)}\ \text{between }\frac{1}{2} \text{ and }1 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ \text{between 1 and }\frac{3}{2} \qquad \textbf{(D)}\ \text{between }\frac{3}{2} \text{ and }2 \\ \textbf{(E)}\ \text{cannot be determined from the information given} \]

A polygon is called concave if it has at least one angle strictly greater than $180^{\circ}$. What is the maximum number of symmetries that an 11-sided concave polygon can have? (A [i]symmetry[/i] of a polygon is a way to rotate or reflect the plane that leaves the polygon unchanged.)

Within a group of $ 2009$ people, every two people has exactly one common friend. Find the least value of the difference between the person with maximum number of friends and the person with minimum number of friends.

In a small pond there are eleven lily pads in a row labeled $0$ through $10$. A frog is sitting on pad $1$. When the frog is on pad $N$, $0<N<10$, it will jump to pad $N-1$ with probability $\frac{N}{10}$ and to pad $N+1$ with probability $1-\frac{N}{10}$. Each jump is independent of the previous jumps. If the frog reaches pad $0$ it will be eaten by a patiently waiting snake. If the frog reaches pad $10$ it will exit the pond, never to return. What is the probability that the frog will escape being eaten by the snake? $ \textbf {(A) } \frac{32}{79} \qquad \textbf {(B) } \frac{161}{384} \qquad \textbf {(C) } \frac{63}{146} \qquad \textbf {(D) } \frac{7}{16} \qquad \textbf {(E) } \frac{1}{2} $

Let $ ABCD$ be an isosceles trapezium. Determine the geometric location of all points $ P$ such that \[ |PA| \cdot |PC| \equal{} |PB| \cdot |PD|.\]

A game is played by $n$ girls ($n \geq 2$), everybody having a ball. Each of the $\binom{n}{2}$ pairs of players, is an arbitrary order, exchange the balls they have at the moment. The game is called nice [b]nice[/b] if at the end nobody has her own ball and it is called [b]tiresome[/b] if at the end everybody has her initial ball. Determine the values of $n$ for which there exists a nice game and those for which there exists a tiresome game.

Let $p$ be an odd prime. If $g_{1}, \cdots, g_{\phi(p-1)}$ are the primitive roots $\pmod{p}$ in the range $1<g \le p-1$, prove that \[\sum_{i=1}^{\phi(p-1)}g_{i}\equiv \mu(p-1) \pmod{p}.\]

In convex quadrilateral $ABCD$, let $M$ and $N$ denote the midpoints of sides $AD$ and $BC$, respectively. On sides $AB$ and $CD$ are points $K$ and $L$, respectively, such that $\angle MKA=\angle NLC$. Prove that if lines $BD$, $KM$, and $LN$ are concurrent, then \[ \angle KMN = \angle BDC\qquad\text{and}\qquad\angle LNM=\angle ABD.\]

In a plane a circle with radius $R$ and center $w$ and a line $\Lambda$ are given. The distance between $w$ and $\Lambda$ is $d, d > R$. The points $M$ and $N$ are chosen on $\Lambda$ in such a way that the circle with diameter $MN$ is externally tangent to the given circle. Show that there exists a point $A$ in the plane such that all the segments $MN$ are seen in a constant angle from $A.$

In triangle ABC, let $P$ and $Q$ be two interior points such that $\angle ABP = \angle QBC$ and $\angle ACP = \angle QCB$. Point $D$ lies on segment $BC$. Prove that $\angle APB + \angle DPC = 180^\circ$ if and only if $\angle AQC + \angle DQB = 180^\circ$.

Let $f$ be a function from the set of integers to the set of positive integers. Suppose that, for any two integers $m$ and $n$, the difference $f(m) - f(n)$ is divisible by $f(m- n)$. Prove that, for all integers $m$ and $n$ with $f(m) \leq f(n)$, the number $f(n)$ is divisible by $f(m)$. [i]Proposed by Mahyar Sefidgaran, Iran[/i]

Let $A_1A_2A_3A_4A_5$ be a regular pentagon with side length 1. The sides of the pentagon are extended to form the 10-sided polygon shown in bold at right. Find the ratio of the area of quadrilateral $A_2A_5B_2B_5$ (shaded in the picture to the right) to the area of the entire 10-sided polygon. [asy] size(8cm); defaultpen(fontsize(10pt)); pair A_2=(-0.4382971011,5.15554989475), B_4=(-2.1182971011,-0.0149584477027), B_5=(-4.8365942022,8.3510997895), A_3=(0.6,8.3510997895), B_1=(2.28,13.521608132), A_4=(3.96,8.3510997895), B_2=(9.3965942022,8.3510997895), A_5=(4.9982971011,5.15554989475), B_3=(6.6782971011,-0.0149584477027), A_1=(2.28,3.18059144705); filldraw(A_2--A_5--B_2--B_5--cycle,rgb(.8,.8,.8)); draw(B_1--A_4^^A_4--B_2^^B_2--A_5^^A_5--B_3^^B_3--A_1^^A_1--B_4^^B_4--A_2^^A_2--B_5^^B_5--A_3^^A_3--B_1,linewidth(1.2)); draw(A_1--A_2--A_3--A_4--A_5--cycle); pair O = (A_1+A_2+A_3+A_4+A_5)/5; label("$A_1$",A_1, 2dir(A_1-O)); label("$A_2$",A_2, 2dir(A_2-O)); label("$A_3$",A_3, 2dir(A_3-O)); label("$A_4$",A_4, 2dir(A_4-O)); label("$A_5$",A_5, 2dir(A_5-O)); label("$B_1$",B_1, 2dir(B_1-O)); label("$B_2$",B_2, 2dir(B_2-O)); label("$B_3$",B_3, 2dir(B_3-O)); label("$B_4$",B_4, 2dir(B_4-O)); label("$B_5$",B_5, 2dir(B_5-O)); [/asy]

A finite set of integers is called [i]bad[/i] if its elements add up to $2010$. A finite set of integers is a [i]Benelux-set[/i] if none of its subsets is bad. Determine the smallest positive integer $n$ such that the set $\{502, 503, 504, . . . , 2009\}$ can be partitioned into $n$ Benelux-sets. (A partition of a set $S$ into $n$ subsets is a collection of $n$ pairwise disjoint subsets of $S$, the union of which equals $S$.) [i](2nd Benelux Mathematical Olympiad 2010, Problem 1)[/i]

When $n$ standard 6-sided dice are rolled, the probability of obtaining a sum of 1994 is greater than zero and is the same as the probability of obtaining a sum of $S$. The smallest possible value of $S$ is $ \textbf{(A)}\ 333 \qquad\textbf{(B)}\ 335 \qquad\textbf{(C)}\ 337 \qquad\textbf{(D)}\ 339 \qquad\textbf{(E)}\ 341 $

A triangle $ T$ is contained inside a point-symmetrical polygon $ M.$ The triangle $ T'$ is the mirror image of the triangle $ T$ with the reflection at one point $ P$, which inside the triangle $ T$ lies. Prove that at least one of the vertices of the triangle $ T'$ lies in inside or on the boundary of the polygon $ M.$

We are given a finite collection of segments in the plane, of total length 1. Prove that there exists a line $ l$ such that the sum of the lengths of the projections of the given segments to the line $ l$ is less than $ \frac{2}{\pi}.$

For any two nonnegative integers $n$ and $k$ satisfying $n\geq k$, we define the number $c(n,k)$ as follows: - $c\left(n,0\right)=c\left(n,n\right)=1$ for all $n\geq 0$; - $c\left(n+1,k\right)=2^{k}c\left(n,k\right)+c\left(n,k-1\right)$ for $n\geq k\geq 1$. Prove that $c\left(n,k\right)=c\left(n,n-k\right)$ for all $n\geq k\geq 0$.

Find the sum of the digits of all numerals in the sequence $1,2,3,4,\cdots ,10000$. $\textbf{(A) }180,001\qquad\textbf{(B) }154,756\qquad\textbf{(C) }45,001\qquad\textbf{(D) }154,755\qquad \textbf{(E) }270,001$

Show that given any 9 points inside a square of side 1 we can always find 3 which form a triangle with area less than $\frac 18$. [i]Bulgaria[/i]