In convex quadrilateral $ABCD$ we have $AB=15$, $BC=16$, $CD=12$, $DA=25$, and $BD=20$. Let $M$ and $\gamma$ denote the circumcenter and circumcircle of $\triangle ABD$. Line $CB$ meets $\gamma$ again at $F$, line $AF$ meets $MC$ at $G$, and line $GD$ meets $\gamma$ again at $E$. Determine the area of pentagon $ABCDE$.

Find all $n \in N$ for which the value of the expression $x^n+y^n+z^n$ is constant for all $x,y,z \in R$ such that $x+y+z=0$ and $xyz=1$. D. Bazylev

Quadrilateral $ABCD$ is inscribed in a circle of radius $6$. If $\angle BDA = 40^o$ and $AD = 6$, what is the measure of $\angle BAD$ in degrees?

Let $a,~b,$ and $x$ be positive real numbers distinct from one. Then \[4(\log_ax)^2+3(\log_bx)^2=8(\log_ax)(\log_bx)\] $\textbf{(A) }\text{for all values of }a,~b,\text{ and }x\qquad$ $\textbf{(B) }\text{if and only if }a=b^2\qquad$ $\textbf{(C) }\text{if and only if }b=a^2\qquad$ $\textbf{(D) }\text{if and only if }x=ab\qquad$ $ \textbf{(E) }\text{for none of these}$

Let $x>1$ be a real number which is not an integer. For each $n\in\mathbb{N}$, let $a_n=\lfloor x^{n+1}\rfloor - x\lfloor x^n\rfloor$. Prove that the sequence $(a_n)$ is not periodic.

Lesha put numbers from 1 to $22^2$ into cells of $22\times 22$ board. Can Oleg always choose two cells, adjacent by the side or by vertex, the sum of numbers in which is divisible by 4?

Let $ABCDEF$ be a convex hexagon with sides not parallel and tangent to a circle $\Gamma$ at the midpoints $P$, $Q$, $R$ of the sides AB, $CD$, $EF$ respectively. $\Gamma$ is tangent to $BC$, $DE$ and $FA$ at the points $X, Y, Z$ respectively. Line $AB$ intersects lines $EF$ and $CD$ at points $M$ and $N$ respectively. Lines $MZ$ and $NX$ intersect at point $K$. Let $ r$ be the line joining the center of $\Gamma$ and point $K$. Prove that the intersection of $PY$ and $QZ$ lies on the line $ r$.

There are $40$ members of jury, that want to choose problem for contest. There are list with $30$ problems. They want to find such problem, that can be solved at least half members , but not all. Every member solved $26$ problems, and every two members solved different sets of problems. Prove, that they can find problem for contest.

Find all functions $f : \mathbb{N}\rightarrow{\mathbb{N}}$ such that for all positive integers $m$ and $n$ the number $f(m)+n-m$ is divisible by $f(n)$.

Let $n\ge2$ be an integer. A regular $(2n+1)-gon$ is divided in to $2n-1$ triangles by diagonals which do not meet except at the vertices. Prove that at least three of these triangles are isosceles.

Find the largest integer $n$ which equals the product of its leading digit and the sum of its digits.

There are $20$ cities in the County of Newton. Their populations are shown in the bar chart below. The average population of all the cities is indicated by the horizontal dashed line. Which of the following is closest to the total population of all $20$ cities? [asy] // made by SirCalcsALot size(300); pen shortdashed=linetype(new real[] {6,6}); // axis draw((0,0)--(0,9300), linewidth(1.25)); draw((0,0)--(11550,0), linewidth(1.25)); for (int i = 2000; i < 9000; i = i + 2000) { draw((0,i)--(11550,i), linewidth(0.5)+1.5*grey); label(string(i), (0,i), W); } for (int i = 500; i < 9300; i=i+500) { draw((0,i)--(150,i),linewidth(1.25)); if (i % 2000 == 0) { draw((0,i)--(250,i),linewidth(1.25)); } } int[] data = {8750, 3800, 5000, 2900, 6400, 7500, 4100, 1400, 2600, 1470, 2600, 7100, 4070, 7500, 7000, 8100, 1900, 1600, 5850, 5750}; int data_length = 20; int r = 550; for (int i = 0; i < data_length; ++i) { fill(((i+1)*r,0)--((i+1)*r, data[i])--((i+1)*r,0)--((i+1)*r, data[i])--((i+1)*r,0)--((i+1)*r, data[i])--((i+2)*r-100, data[i])--((i+2)*r-100,0)--cycle, 1.5*grey); draw(((i+1)*r,0)--((i+1)*r, data[i])--((i+1)*r,0)--((i+1)*r, data[i])--((i+1)*r,0)--((i+1)*r, data[i])--((i+2)*r-100, data[i])--((i+2)*r-100,0)); } draw((0,4750)--(11450,4750),shortdashed); label("Cities", (11450*0.5,0), S); label(rotate(90)*"Population", (0,9000*0.5), 10*W); [/asy] $\textbf{(A) }65{,}000 \qquad \textbf{(B) }75{,}000 \qquad \textbf{(C) }85{,}000 \qquad \textbf{(D) }95{,}000 \qquad \textbf{(E) }105{,}000$

$ABC$ is an acute triangle with orthocenter $H$. Points $D$ and $E$ lie on segment $BC$. Circumcircle of $\triangle BHC$ instersects with segments $AD$,$AE$ at $P$ and $Q$, respectively. Prove that if $BD^2+CD^2=2DP\cdot DA$ and $BE^2+CE^2=2EQ\cdot EA$, then $BP=CQ$.

A unit square $ABCD$ with centre $O$ is rotated about $O$ by an angle $\alpha$. Compute the common area of the two squares.

A composite positive integer is a product $ab$ with $a$ and $b$ not necessarily distinct integers in $\{2,3,4,\dots\}$. Show that every composite positive integer is expressible as $xy+xz+yz+1$, with $x,y,z$ positive integers.

Consider points $A, B, C$. Draw a line through $A$ so that the sum of distances from $B$ and $C$ to this line is equal to the length of a given segment.

Determine all pairs $(a, b)$ of positive real numbers with $a \neq 1$ such that \[\log_a b < \log_{a+1} (b + 1).\]

Let $\mathcal{S}$ be the set of permutations of $\{1,2,\ldots,6\}$, and let $\mathcal{T}$ be the set of permutations of $\mathcal{S}$ that preserve compositions: i.e., if $F\in\mathcal{T}$ then \[F(f_2\circ f_1)=F(f_2)\circ F(f_1)\] for all $f_1,f_2\in\mathcal{S}$. Find the number of elements $F\in\mathcal{T}$ such that if $f\in\mathcal{S}$ satisfies $f(1)=2$ and $f(2)=1$, then $(F(f))(1)=2$ and $(F(f))(2)=1$.

Consider $n \geq 2$ distinct points in the plane $A_1,A_2,...,A_n$ . Color the midpoints of the segments determined by each pair of points in red. What is the minimum number of distinct red points?

At a party, a group of $20$ people needs to be seated at $4$ tables. The seating arrangement is called [i]successful [/i] if any two people at the same table are friends. It turned out that successful seating arrangements exist. In a successful seating arrangement, exactly $5$ people sit at each table. What is the greatest possible number of pairs of friends in this companies?

An irregular octahedron has eight faces that are equilateral triangles of side length $2$. However, instead of each vertex having four "neighbors" (vertices that share an edge with it) like in a regular octahedron, for this octahedron, two of the vertices have exactly three neighbors, two of the vertices have exactly four neighbors, and two of the vertices have exactly five neighbors. Compute the volume of this octahedron. [i]Proposed by Connor Gordon[/i]

Find all positive integers $A,B$ satisfying the following properties: (i) $A$ and $B$ has same digit in decimal. (ii) $2\cdot A\cdot B=\overline{AB}$ (Here $\cdot$ denotes multiplication, $\overline{AB}$ denotes we write $A$ and $B$ in turn. For example, if $A=12,B=34$, then $\overline{AB}=1234$)

Prove that for every convex function $ f(x)$ defined on the interval $ \minus{}1\leq x \leq 1$ and having absolute value at most $ 1$, there is a linear function $ h(x)$ such that \[ \int_{\minus{}1}^1|f(x)\minus{}h(x)|dx\leq 4\minus{}\sqrt{8}.\] [L. Fejes-Toth]

Consider the sequence $1, 2, 1, 2, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 1,...$ Find $n$ such that the first $n$ terms sum up to $2010$.

Let $ABC$ be an acute angled triangle. Let $M$ be the midpoint of $BC$, and let $BE$ and $CF$ be the altitudes of the triangle. Let $D \ne M$ be a point on the circumcircle of the triangle $EFM$ such that $DE = DF$. Prove that $AD \perp BC$.