Show that the four perpendiculars dropped from the midpoints of the sides of a cyclic quadrilateral to the respective opposite sides are concurrent. [b]Note by Darij:[/b] A [i]cyclic quadrilateral [/i]is a quadrilateral inscribed in a circle.

[url=http://www.mathlinks.ro/Forum/viewtopic.php?t=107262]IMO 2007 HKTST 1[/url] Problem 5 The sequence $\{a_{n}\}$ is defined by $a_{1}=0$ and $(n+1)^{3}a_{n+1}=2n^{2}(2n+1)a_{n}+2(3n+1)$ for all integers $\geq 1$. Show that infintely many members of the sequence are positive integers.

A bag contains $8$ green candies and $4$ red candies. You randomly select one candy at a time to eat. If you eat five candies, there are relatively prime positive integers $m$ and $n$ so that $\frac{m}{n}$ is the probability that you do not eat a green candy after you eat a red candy. Find $m+n$.

Let $D$ be the set of real numbers excluding $-1$. Find all functions $f: D \to D$ such that for all $x,y \in D$ satisfying $x \neq 0$ and $y \neq -x$, the equality $$(f(f(x))+y)f \left(\frac{y}{x} \right)+f(f(y))=x$$ holds.

If $x,y,z$ satisfy the system of equations \[xy+yz+zx=23\] \[\frac{y}{x+y}+\frac{z}{y+z}+\frac{x}{z+x}=-1\] \[\frac{z^2x}{x+y}+\frac{x^2y}{y+z}+\frac{y^2z}{z+x}=202\] Find the value of $x^2+y^2+z^2$. [i]Proposed by Harry Kim[/i]

Let $ABCD$ be a convex quadrilateral. Let $E,F,G,H$ be the midpoints of $AB,BC,CD,DA$ respectively. If $AC,BD,EG,FH$ concur at a point $O,$ prove that $ABCD$ is a parallelogram.

Let $ABC$ be an acute triangle with $AB < AC$ and inscribed in the circle $(O)$. Denote $I$ as the incenter of $ABC$ and $D$, $E$ as the intersections of $AI$ with $BC$, $(O)$ respectively. Take a point $K$ on $BC$ such that $\angle AIK = 90^o$ and $KA$, $KE$ meet $(O)$ again at M,N respectively. The rays $ND$, $NI$ meet the circle $(O)$ at $Q$,$P$. 1. Prove that the quadrilateral $MPQE$ is a kite. 2. Take $J$ on $IO$ such that $AK \perp AJ$. The line through $I$ and perpendicular to $OI$ cuts $BC$ at $R$ ,cuts $EK$ at $S$ .Prove that $OR \parallel JS$.

Let us connect consecutive vertices of a regular heptagon inscribed in a unit circle by connected subsets (of the plane of the circle) of diameter less than $ 1$. Show that every continuum (in the plane of the circle) of diameter greater than $ 4$, containing the center of the circle, intersects one of these connected sets. [i]M. Bognar[/i]

Let $p(x)=x^4-5773x^3-46464x^2-5773x+46$. Determine the sum of $\arctan$-s of its real roots.

Solve the equation $ \sin x\sin 2x\cdots\sin nx+\cos x\cos 2x\cdots\cos nx =1, $ for $ n\in\mathbb{N} $ and $ x\in\mathbb{R} . $

Find the number of integers $n$ for which $n^2+10n<2008$.

Let's take a $ABCD$ rectangle of paper; the side $AB$ measures $5$ cm and the side $BC$ measures $9$ cm. We do three folds: 1.We take the $AB$ side on the $BC$ side and call $P$ the point on the $BC$ side that coincides with $A$. A right trapezoid $BCDQ$ is then formed. 2. We fold so that $B$ and $Q$ coincide. A $5$-sided polygon $RPCDQ$ is formed. 3. We fold again by matching $D$ with $C$ and $Q$ with $P$. A new right trapezoid $RPCS$. After making these folds, we make a cut perpendicular to $SC$ by its midpoint $T$, obtaining the right trapezoid $RUTS$. Calculate the area of the figure that appears as we unfold the last trapezoid $RUTS$.

(a) Prove that from $2007$ given positive integers, one of them can be chosen so the product of the remaining numbers is expressible in the form $a^2 - b^2$ for some positive integers $a$ and $b$. (2) (b) One of $2007$ given positive integers is $2006$. Prove that if there is a unique number among them such that the product of the remaining numbers is expressible in the form $a^2 - b^2$ for some positive integers $a$ and $b$, then this unique number is $2006$. (2)

Let $a$ be a given natural number and $x_1, x_2, x_3, ...$ the sequence with $x_n = \frac{n}{n+a}$ ($n \in N^*$ ). Prove that for every $n \in N^*$ , the term $x_n$ can be represented as the product of two terms of this sequence , and determine the number of representations depending on $n$ and $a$.

[b]a)[/b] Do there exist $2$-element subsets $A_1,A_2,A_3,...$ of natural numbers such that each natural number appears in exactly one of these sets and also for each natural number $n$, sum of the elements of $A_n$ equals $1391+n$? [b]b)[/b] Do there exist $2$-element subsets $A_1,A_2,A_3,...$ of natural numbers such that each natural number appears in exactly one of these sets and also for each natural number $n$, sum of the elements of $A_n$ equals $1391+n^2$? [i]Proposed by Morteza Saghafian[/i]

Let $C$ and $D$ be two points on the semicricle with diameter $AB$ such that $B$ and $C$ are on distinct sides of the line $AD$. Denote by $M$, $N$ and $P$ the midpoints of $AC$, $BD$ and $CD$ respectively. Let $O_A$ and $O_B$ the circumcentres of the triangles $ACP$ and $BDP$. Show that the lines $O_AO_B$ and $MN$ are parallel.

Let $ABC$ be a triangle with $AB = 26$, $BC = 51$, and $CA = 73$, and let $O$ be an arbitrary point in the interior of $\vartriangle ABC$. Lines $\ell_1$, $\ell_2$, and $\ell_3$ pass through $O$ and are parallel to $\overline{AB}$, $\overline{BC}$, and $\overline{CA}$, respectively. The intersections of $\ell_1$, $\ell_2$, and $\ell_3$ and the sides of $\vartriangle ABC$ form a hexagon whose area is $A$. Compute the minimum value of $A$.

The figure below contains two squares which share an edge, one with side length $200$ units and the other with side length $289$ units. The figure is divided into a whole number of regions, each with an equal whole number area but not necessarily of the same shape. Given that there is more than one region and each region has an area greater than $1$, find the sum of the number of regions and the area of each region. [asy] size(4cm); draw((0,0)--(200,0)--(200,200)--(0,200)--cycle); label("$200$",(0,0)--(200,0)); label("$289$",(200,0)--(489,0)); draw((200,0)--(489,0)--(489,289)--(200,289)--cycle); [/asy] $\textbf{(A) } 704\qquad\textbf{(B) } 874\qquad\textbf{(C) } 924\qquad\textbf{(D) } 978\qquad\textbf{(E) } 1028$

Find all integer solutions to $m^{m+n} = n^{12}, n^{m+n} = m^3$.

A tennis tournament has at least three participants. Every participant plays exactly one match against every other participant. Moreover, every participant wins at least one of the matches he plays. (Draws do not occur in tennis matches.) Show that there are three participants $A, B $ and $C$ for which the following holds: $A$ wins against $B, B$ wins against $C$, and $C$ wins against $A$.

Island Hopping Holidays offer short holidays to $64$ islands, labeled Island $i, 1 \le i \le 64$. A guest chooses any Island $a$ for the first night of the holiday, moves to Island $b$ for the second night, and finally moves to Island $c$ for the third night. Due to the limited number of boats, we must have $b \in T_a$ and $c \in T_b$, where the sets $T_i$ are chosen so that (a) each $T_i$ is non-empty, and $i \notin T_i$, (b) $\sum^{64}_{i=1} |T_i| = 128$, where $|T_i|$ is the number of elements of $T_i$. Exhibit a choice of sets $T_i$ giving at least $63\cdot 64 + 6 = 4038$ possible holidays. Note that c = a is allowed, and holiday choices $(a, b, c)$ and $(a',b',c')$ are considered distinct if $a \ne a'$ or $b \ne b'$ or $c \ne c'$.

Are there $14$ consecutive positive integers, each of which has a divisor other than $1$ and not exceeding $11$?

For what value of $x$ does the following determinant have the value $2013$? \[\left|\begin{array}{ccc}5+x & 8 & 2+x \\ 1 & 1+x & 3 \\ 2 & 1 & 2\end{array}\right|\]

Let the function $f(x,y,t)=\frac{x^2-y^2}{2}-\frac{(x-yt)^2}{1-t^2}$ for all real values $x,y$ and $t\not=\pm1$ a) Evaluate $f(2,0,3)$ and $f(0,2,3)$. b) Show that $f(x,y,0)=f(y,x,0)$ for any values of $(x,y)$. c) Show that $f(x,y,t)=f(y,x,t)$ for any values of $(x,y)$ and $t\not=\pm1$. d) Given $$g(x,y,s)=\frac{(x^2-y^2)(1+\sin(s))}{2} -\frac{(x-y\sin(s))^2}{1-\sin(s)}$$ for all real values $x,y$ and $s\not=\frac{\pi}{2}+2\pi k$, where $k$ is an integer number, show that $g(x,y,s)=g(y,x,s)$ for any values of $(x,y)$ and $s$ in the domain of $g(x,y,s)$.

For how many integers $N$ between $1$ and $1990$ is the improper fraction $\frac{N^2+7}{N+4}$ not in lowest terms? $\text{(A)} \ 0 \qquad \text{(B)} \ 86 \qquad \text{(C)} \ 90 \qquad \text{(D)} \ 104 \qquad \text{(E)} \ 105$