Let $BCDK$ be a convex quadrilateral such that $BC=BK$ and $DC=DK$. $A$ and $E$ are points such that $ABCDE$ is a convex pentagon such that $AB=BC$ and $DE=DC$ and $K$ lies in the interior of the pentagon $ABCDE$. If $\angle ABC=120^{\circ}$ and $\angle CDE=60^{\circ}$ and $BD=2$ then determine area of the pentagon $ABCDE$.

$ABCD$ is a fixed rhombus. Segment $PQ$ is tangent to the inscribed circle of $ABCD$, where $P$ is on side $AB$, $Q$ is on side $AD$. Show that, when segment $PQ$ is moving, the area of $\Delta CPQ$ is a constant.

Let the incircle $\gamma$ of triangle $ABC$ be tangent to $BA, BC$ at $D, E$, respectively. A tangent $t$ to $\gamma$ , distinct from the sidelines, intersects the line $AB$ at $M$. If lines $CM, DE$ meet at$ K$, prove that lines $AK,BC$ and $t$ are parallel or concurrent. Michel Bataille , France

[b]p1.[/b] Compute the sum of sharp angles at all five nodes of the star below. ( [url=http://www.math.msu.edu/~mshapiro/NewOlympiad/Olymp2009/10_12_2009.pdf]figure missing[/url] ) [b]p2.[/b] Arrange the integers from $1$ to $15$ in a row so that the sum of any two consecutive numbers is a perfect square. In how many ways this can be done? [b]p3.[/b] Prove that if $p$ and $q$ are prime numbers which are greater than $3$ then $p^2 -q^2$ is divisible by $ 24$. [b]p4.[/b] A city in a country is called Large Northern if comparing to any other city of the country it is either larger or farther to the North (or both). Similarly, a city is called Small Southern. We know that in the country all cities are Large Northern city. Show that all the cities in this country are simultaneously Small Southern. [b]p5.[/b] You have four tall and thin glasses of cylindrical form. Place on the flat table these four glasses in such a way that all distances between any pair of centers of the glasses' bottoms are equal. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

In a trapezoid, the length of one of the diagonals is equal to the sum of the lengths of the bases, and the angle between the diagonals is $60$ degrees. Prove that this trapezoid is isosceles.

[u]Round 5[/u] [b]p13.[/b] Let $\{a\} _{n\ge 1}$ be an arithmetic sequence, with $a_ 1 = 0$, such that for some positive integers $k$ and $x$ we have $a_{k+1} = {k \choose x}$, $a_{2k+1} ={k \choose {x+1}}$ , and $a_{3k+1} ={k \choose {x+2}}$. Let $\{b\}_{n\ge 1}$ be an arithmetic sequence of integers with $b_1 = 0$. Given that there is some integer $m$ such that $b_m ={k \choose x}$, what is the number of possible values of $b_2$? [b]p14.[/b] Let $A = arcsin \left( \frac14 \right)$ and $B = arcsin \left( \frac17 \right)$. Find $\sin(A + B) \sin(A - B)$. [b]p15.[/b] Let $\{f_i\}^{9}_{i=1}$ be a sequence of continuous functions such that $f_i : R \to Z$ is continuous (i.e. each $f_i$ maps from the real numbers to the integers). Also, for all $i$, $f_i(i) = 3^i$. Compute $\sum^{9}_{k=1} f_k \circ f_{k-1} \circ ... \circ f_1(3^{-k})$. [u]Round 6[/u] [b]p16.[/b] If $x$ and $y$ are integers for which $\frac{10x^3 + 10x^2y + xy^3 + y^4}{203}= 1134341$ and $x - y = 1$, then compute $x + y$. [b]p17.[/b] Let $T_n$ be the number of ways that n letters from the set $\{a, b, c, d\}$ can be arranged in a line (some letters may be repeated, and not every letter must be used) so that the letter a occurs an odd number of times. Compute the sum $T_5 + T_6$. [b]p18.[/b] McDonald plays a game with a standard deck of $52$ cards and a collection of chips numbered $1$ to $52$. He picks $1$ card from a fully shuffled deck and $1$ chip from a bucket, and his score is the product of the numbers on card and on the chip. In order to win, McDonald must obtain a score that is a positive multiple of $6$. If he wins, the game ends; if he loses, he eats a burger, replaces the card and chip, shuffles the deck, mixes the chips, and replays his turn. The probability that he wins on his third turn can be written in the form $\frac{x^2 \cdot y}{z^3}$ such that $x, y$, and $z$ are relatively prime positive integers. What is $x + y + z$? (NOTE: Use Ace as $1$, Jack as $11$, Queen as $12$, and King as $13$) [u]Round 7[/u] [b]p19.[/b] Let $f_n(x) = ln(1 + x^{2^n}+ x^{2^{n+1}}+ x^{3\cdot 2^n})$. Compute $\sum^{\infty}_{k=0} f_{2k} \left( \frac12 \right)$. [b]p20.[/b] $ABCD$ is a quadrilateral with $AB = 183$, $BC = 300$, $CD = 55$, $DA = 244$, and $BD = 305$. Find $AC$. [b]p21.[/b] Define $\overline{xyz(t + 1)} = 1000x + 100y + 10z + t + 1$ as the decimal representation of a four digit integer. You are given that $3^x5^y7^z2^t = \overline{xyz(t + 1)}$ where $x, y, z$, and t are non-negative integers such that $t$ is odd and $0 \le x, y, z,(t + 1) \le 9$. Compute$3^x5^y7^z$ PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c4h2782822p24445934]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Triangle $ABC$ is not isosceles. The incircle of $\triangle ABC$ touches the sides $BC$, $CA$, $AB$ in the points $K$, $L$, $M$. The parallel with $LM$ through $B$ meets $KL$ at $D$, the parallel with $LM$ through $C$ meets $KM$ at $E$. Prove that $DE$ passes through the midpoint of $\overline{LM}$.

Points $B,C$ vary on two fixed rays emanating from point $A$ such that $AB+AC$ is constant. Show that there is a point $D$, other than $A$, such that the circumcircle of triangle $ABC$ passes through $D$ for all possible choices of $B, C$.

The vertices of $\triangle ABC$ are labeled in counter-clockwise order, and its sides have lengths $CA = 2022$, $AB = 2023$, and $BC = 2024$. Rotate $B$ $90^\circ$ counter-clockwise about $A$ to get a point $B'$. Let $D$ be the orthogonal projection of $B'$ unto line $AC$, and let $M$ be the midpoint of line segment $BB'$. Then ray $BM$ intersects the circumcircle of $\triangle CDM$ at a point $N \neq M$. Compute $MN$. [i]Proposed by Thomas Lam[/i]

Let $AL$ and $BK$ be angle bisectors in the non-isosceles triangle $ABC$ ($L$ lies on the side $BC$, $K$ lies on the side $AC$). The perpendicular bisector of $BK$ intersects the line $AL$ at point $M$. Point $N$ lies on the line $BK$ such that $LN$ is parallel to $MK$. Prove that $LN = NA$.

Consider a line segment $[AB]$ with midpoint $M$ and perpendicular bisector $m$. For each point$ X \ne M$ on m consider we are the intersection point $Y$ of the line $BX$ with the bisector from the angle $\angle BAX$. As $X$ approaches $M$, then approaches $Y$ to a point of $[AB]$. Which? [img]https://cdn.artofproblemsolving.com/attachments/a/3/17d72a23011a9ec22deb20184717cc9c020a2b.png[/img] [hide=original wording]Beschouw een lijnstuk [AB] met midden M en middelloodlijn m. Voor elk punt X 6= M op m beschouwenwe het snijpunt Y van de rechte BX met de bissectrice van de hoek < BAX . Als X tot M nadert, dan nadert Y tot een punt van [AB]. Welk? [/hide]

For a given integer $n\geq2$, a pyramid of height $n$ if defined as a collection of $1^2+2^2+\dots+n^2$ stone cubes of equal size stacked in $n$ layers such that the cubes in the $k$-th layer form a square with sidelength $n+1-k$ and every cube (except for the ones in the bottom layer) rests on four cubes in the layer below. Some of the cubes are made of sandstone, some are made of granite. The top cube is made of granite, and to ensure the stability of the piramid, for each granite cube (except for the ones in the bottom layer), at least three out of four of the cubes supporting it have to be granite. What is the minimum possible number of granite cubes in such an arrangement?

The largest side of the triangle $ABC$ is equal to $1$ unit. Prove that , the circles centred at $A,B$ and $C$ wit radiuses $\frac{1}{\sqrt{3}}$ can compeletely cover the triangle $ABC$.

Let $ABC$ be an equilateral triangle. Denote by $D$ the midpoint of $\overline{BC}$, and denote the circle with diameter $\overline{AD}$ by $\Omega$. If the region inside $\Omega$ and outside $\triangle ABC$ has area $800\pi-600\sqrt3$, find the length of $AB$. [i]Proposed by Eugene Chen[/i]

Given is a fixed regular pentagon $ABCDE$ with side $1$. Let $M$ be an arbitrary point inside or on it. Let the distance from $M$ to the closest vertex be $r_1$, to the next closest be $r_2$ and so on, so that the distances from $M$ to the five vertices satisfy $r_1\le r_2\le r_3\le r_4\le r_5$. Find (a) the locus of $M$ which gives $r_3$ the minimum possible value, and (b) the locus of $M$ which gives $r_3$ the maximum possible value.

A rectangle with sides of lengths $a$ and $b$ ($a > b$) is cut into rightangled triangles so that any two of these triangles either have a common side, a common vertex or no common points. Moreover, any common side of two triangles is a leg of one of them and the hypotenuse of the other. Prove that $a > 2b$. (A Shapovalov)

Let $ABC$ be a scalene triangle with $\angle BCA = 90^{\circ}$, and let $D$ be the foot of the altitude from $C$. Let $X$ be a point in the interior of the segment $CD$. Let $K$ be the point on the segment $AX$ such that $BK = BC$. Similarly, let $L$ be the point on the segment $BX$ such that $AL = AC$. The circumcircle of triangle $DKL$ intersects segment $AB$ at a second point $T$ (other than $D$). Prove that $\angle ACT = \angle BCT$.

The length of the middle line of a trapezoid is $4$ and the angles at one of the bases are $40^\circ$ and $50^\circ$. Determine the lengths of the bases if the distance between their midpoints is $1$.

Let $ ABC$ be an equilateral triangle and $ P$ in its interior. The distances from $ P$ to the triangle's sides are denoted by $ a^2, b^2,c^2$ respectively, where $ a,b,c>0$. Find the locus of the points $ P$ for which $ a,b,c$ can be the sides of a non-degenerate triangle.

Two circles $ (O_1)$ and $ (O_2)$ touch externally at the point $C$ and internally at the points $A$ and $B$ respectively with another circle $(O)$. Suppose that the common tangent of $ (O_1)$ and $ (O_2)$ at $C$ meets $(O)$ at $P$ such that $PA=PB$. Prove that $PO$ is perpendicular to $AB$.

On the side $BC$ of the rectangle $ABCD$, a point $P{}$ is marked so that $\angle APD = 90^\circ$. On the straight line $AD$, points $Q{}$ and $R{}$ are selected outside the segment $AD$ such that $AQ = BP$ and $CP = DR$. The circle $\omega$ passes through the points $Q, D$ and the circumcenter of the triangle $PDQ$. The circle $\gamma$ passes through the points $A, R$ and the circumcenter of the triangle $APR$. Prove that the radius of one of the circles touching the line $AD$ and the circles $\omega$ and $\gamma$ is $2AB$.

Find the area $S$ of the region enclosed by the curve $y=\left|x-\frac{1}{x}\right|\ (x>0)$ and the line $y=2$.

In triangle $\vartriangle ABC$, $AB = 5$, $BC = 7$, and $CA = 8$. Let $E$ and $F$ be the feet of the altitudes from $B$ and $C$, respectively, and let $M$ be the midpoint of $BC$. The area of triangle $MEF$ can be expressed as $\frac{a \sqrt{b}}{c}$ for positive integers $a$, $b$, and $c$ such that the greatest common divisor of $a$ and $c$ is $1$ and $b$ is not divisible by the square of any prime. Compute $a + b + c$.

Let $A B C$ be a triangle in the $x y$ plane, where $B$ is at the origin $(0,0)$. Let $B C$ be produced to $D$ such that $B C: C D=1: 1, C A$ be produced to $E$ such that $C A: A E=1: 2$ and $A B$ be produced to $F$ such that $A B: B F=1: 3$. Let $G(32,24)$ be the centroid of the triangle $A B C$ and $K$ be the centroid of the triangle $D E F$. Find the length $G K$.

A regular 14-gon with side $a$ is inscribed in a circle of radius one. Prove \[ \frac{2-a}{2 \cdot a} > \sqrt{3 \cdot \cos \left( \frac{\pi}{7} \right)}. \]