Let $A$ and $B$ be points on a circle $k$. Suppose that an arc $k'$ of another circle, $\ell$, connects $A$ with $B$ and divides the area inside the circle $k$ into two equal parts. Prove that arc $k'$ is longer than the diameter of $k$.

$(BUL 5)$ Let $Z$ be a set of points in the plane. Suppose that there exists a pair of points that cannot be joined by a polygonal line not passing through any point of $Z.$ Let us call such a pair of points unjoinable. Prove that for each real $r > 0$ there exists an unjoinable pair of points separated by distance $r.$

In tetrahedron $ABCD$, let $V$ be the volume of the tetrahedron and $R$ the radius of the sphere that it tangent to all four faces of the tetrahedron. Let $P$ be the surface area of the tetrahedron. Prove that $$r=\frac{3V}{P}.$$ [i]Proposed by CaptainFlint.[/i]

In triangle $ABC$, $\angle A = 90^\circ$ and $\angle C = 70^\circ$. $F$ is point on $AB$ such that $\angle ACF = 30^\circ$, and $E$ is a point on $CA$ such that $\angle CF E = 20^\circ$. Prove that $BE$ bisects $\angle B$.

The figure below, composed of four regular pentagons with a side length of $1$, was glued in space as follows. First, it was folded along the broken sections, by combining the bold sections, and then formed in such a way that colored sections formed a square. Find the length of the segment $AB$ created in this way. [img]https://cdn.artofproblemsolving.com/attachments/0/7/bddad6449f74cbc7ea2623957ef05b3b0d2f19.png[/img]

[b]p1.[/b] Let $2^k$ be the largest power of $2$ dividing $30! = 30 \cdot 29 \cdot 28 ... 2 \cdot 1$. Find $k$. [b]p2.[/b] Let $d(n)$ be the total number of digits needed to write all the numbers from $1$ to $n$ in base $10$, for example, $d(5) = 5$ and $d(20) = 31$. Find $d(2012)$. [b]p3.[/b] Jim and TongTong play a game. Jim flips $10$ coins and TongTong flips $11$ coins, whoever gets the most heads wins. If they get the same number of heads, there is a tie. What is the probability that TongTong wins? [b]p4.[/b] There are a certain number of potatoes in a pile. When separated into mounds of three, two remain. When divided into mounds of four, three remain. When divided into mounds of five, one remain. It is clear there are at least $150$ potatoes in the pile. What is the least number of potatoes there can be in the pile? [b]p5.[/b] Call an ordered triple of sets $(A, B, C)$ nice if $|A \cap B| = |B \cap C| = |C \cap A| = 2$ and $|A \cap B \cap C| = 0$. How many ordered triples of subsets of $\{1, 2, · · · , 9\}$ are nice? [b]p6.[/b] Brett has an $ n \times n \times n$ cube (where $n$ is an integer) which he dips into blue paint. He then cuts the cube into a bunch of $ 1 \times 1 \times 1$ cubes, and notices that the number of un-painted cubes (which is positive) evenly divides the number of painted cubes. What is the largest possible side length of Brett’s original cube? Note that $\lfloor x\rfloor$ denotes the largest integer less than or equal to $x$. [b]p7.[/b] Choose two real numbers $x$ and $y$ uniformly at random from the interval $[0, 1]$. What is the probability that $x$ is closer to $1/4$ than $y$ is to $1/2$? [b]p8. [/b] In triangle $ABC$, we have $\angle BAC = 20^o$ and $AB = AC$. $D$ is a point on segment $AB$ such that $AD = BC$. What is $\angle ADC$, in degree. [b]p9.[/b] Let $a, b, c, d$ be real numbers such that $ab + c + d = 2012$, $bc + d + a = 2010$, $cd + a + b = 2013$, $da + b + c = 2009$. Find $d$. [b]p10. [/b]Let $\theta \in [0, 2\pi)$ such that $\cos \theta = 2/3$. Find $\sum_{n=0}^{\infty}\frac{1}{2^n}\cos(n \theta)$ PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Regular hexagon $ABCDEF$ has side length $2$ and center $O$. The point $P$ is defined as the intersection of $AC$ and $OB$. Find the area of quadrilateral $OPCD$.

On a circle $k$, we marked four points $(A, B, C, D)$ and drew pairwise their connecting segments. We denoted angles as seen on the diagram. We know that $\alpha_1 : \alpha_2 = 2 : 5$, $\beta_1 : \beta_2 = 7 : 11$, and $\gamma_1 : \gamma_2 = 10 : 3$. If $\delta_1 : \delta_2 = p : q$, where $p$ and $q$ are coprime positive integers, then what is $p$? [img]https://cdn.artofproblemsolving.com/attachments/c/e/b532dd164a7cf99cea7b3b7d98f81796622da5.png[/img]

Let $ABCD$ be a parallelogram with at least an angle not equal to $90^o$ and $k$ the circumcircle of the triangle $ABC$. Let $E$ be the diametrically opposite point of $B$. Show that the circumcircle of the triangle $ADE$ and $k$ have the same radius.

In a triangle $ABC$, with $AB\neq AC$ and $A\neq 60^{0},120^{0}$, $D$ is a point on line $AC$ different from $C$. Suppose that the circumcentres and orthocentres of triangles $ABC$ and $ABD$ lie on a circle. Prove that $\angle ABD=\angle ACB$.

Let $\triangle ABC$ and $\triangle C'B'A$ be two congruent triangles ( with this order and orient. ). Define point $M$ as the midpoint of segment $AB$ and suppose that the extension of $CB'$ from $B'$ passes trough $M$ , if $F$ be a point on the smaller arc $MC$ of circumcircle of triangle $\triangle BMC$ such that $\angle FB'A=90$ and $\angle C'CB' \neq 90$ , then prove that $\angle B'C'C=\angle CAF$. [i]Proposed by Alireza Dadgarnia[/i]

We define a relation between subsets of $\mathbb R ^n$. $A \sim B\Longleftrightarrow$ we can partition $A,B$ in sets $A_1,\dots,A_n$ and $B_1,\dots,B_n$(i.e $\displaystyle A=\bigcup_{i=1} ^n A_i,\ B=\bigcup_{i=1} ^n B_i, A_i\cap A_j=\emptyset,\ B_i\cap B_j=\emptyset$) and $A_i\simeq B_i$. Say the the following sets have the relation $\sim$ or not ? a) Natural numbers and composite numbers. b) Rational numbers and rational numbers with finite digits in base 10. c) $\{x\in\mathbb Q|x<\sqrt 2\}$ and $\{x\in\mathbb Q|x<\sqrt 3\}$ d) $A=\{(x,y)\in\mathbb R^2|x^2+y^2<1\}$ and $A\setminus \{(0,0)\}$

Points $X$ and $Y$ are chosen inside an acute triangle $ABC$ so that: $$\angle AXB = \angle CYB = 180^\circ - \angle ABC, \text{ } \angle ABX = \angle CBY$$ Show that the points $X$ and $Y$ are equidistant from the center of the circumscribed circle of $\triangle ABC$. [i]Proposed by Anton Trygub[/i]

Let $ABC$ be a scalene acute triangle, where $AL$ $(L \in BC)$ is the internal bisector of $\angle BAC$ and $M$ is the midpoint of $BC$. Let the internal bisectors of $\angle AMB$ and $\angle CMA$ intersect $AB$ and $AC$ in $P$ and $Q$, respectively. Prove that the circumcircle of $APQ$ is tangent to $BC$ if and only if $L$ belongs to it.

Let a circle $ (O)$ with diameter $ AB$. A point $ M$ move inside $ (O)$. Internal bisector of $ \widehat{AMB}$ cut $ (O)$ at $ N$, external bisector of $ \widehat{AMB}$ cut $ NA,NB$ at $ P,Q$. $ AM,BM$ cut circle with diameter $ NQ,NP$ at $ R,S$. Prove that: median from $ N$ of triangle $ NRS$ pass over a fix point.

Given a nonisosceles, nonright triangle ABC, let O denote the center of its circumscribed circle, and let $A_1$, $B_1$, and $C_1$ be the midpoints of sides BC, CA, and AB, respectively. Point $A_2$ is located on the ray $OA_1$ so that $OAA_1$ is similar to $OA_2A$. Points $B_2$ and $C_2$ on rays $OB_1$ and $OC_1$, respectively, are defined similarly. Prove that lines $AA_2$, $BB_2$, and $CC_2$ are concurrent, i.e. these three lines intersect at a point.

A parallelogram $ABCD$ is given. On the diagonal BD, a point $P$ is selected such that $AP = BD$ is satisfied. Point $Q$ is the midpoint of segment $CP$. Prove that $\angle BQD = 90^o$. [img]https://cdn.artofproblemsolving.com/attachments/2/0/4bc69ec0330e2afa6b560c56da5dd783b16efb.png[/img] .

Let $R{}$ and $r{}$ be the radii of the circumscribed and inscribed circles of the acute-angled triangle $ABC{}$ respectively. The point $M{}$ is the midpoint of its largest side $BC.$ The tangents to its circumscribed circle at $B{}$ and $C{}$ intersect at $X{}$. Prove that \[\frac{r}{R}\geqslant\frac{AM}{AX}.\]

Construct a trapezoid given the midpoints of the legs, the point of intersection of the diagonals and the foot of the perpendicular, drawn from this point on the larger base.

In a triangle $ABC$, let $I$ be its incenter; $Q$ the point at which the incircle touches the line $AC$; $E$ the midpoint of $AC$ and $K$ the orthocenter of triangle $BIC$. Prove that the line $KQ$ is perpendicular to the line $IE$.

Given a triangle $ABC$ with $\angle ACB = 90^{\circ}$. Let $\omega$ be the circumcircle of triangle $ABC$. The tangents of $\omega$ at $B$ and $C$ intersect at $P$. Let $M$ be the midpoint of $PB$. Line $CM$ intersects $\omega$ at $N$ and line $PN$ intersects $AB$ at $E$. Point $D$ is on $CM$ such that $ED \parallel BM$. Show that the circumcircle of $CDE$ is tangent to $\omega$.

Assume three circles mutually outside each other with the property that every line separating two of them have intersection with the interior of the third one. Prove that the sum of pairwise distances between their centers is at most $2\sqrt{2}$ times the sum of their radii. (A line separates two circles, whenever the circles do not have intersection with the line and are on different sides of it.) [color=#45818E]Note.[/color] Weaker results with $2\sqrt{2}$ replaced by some other $c$ may be awarded points depending on the value of $c>2\sqrt{2}$ [i]Proposed by Morteza Saghafian[/i]

Let $ABCD$ be a parallelogram and $l$ the line parallel to $AC$ which passes through $D$. Let $E$ and $F$ points on $l$ such that $DE=DF=DB$. Show that $EA,FC$ and $BD$ are concurrent.

In order for Mateen to walk a kilometer ($1000$m) in his rectangular backyard, he must walk the length $25$ times or walk its perimeter $10$ times. What is the area of Mateen's backyard in square meters? $\text{(A)}\ 40 \qquad \text{(B)}\ 200 \qquad \text{(C)}\ 400 \qquad \text{(D)}\ 500 \qquad \text{(E)}\ 1000$

Let $ABCDEF$ be a convex hexagon with $\angle A= \angle D$ and $\angle B=\angle E$ . Let $K$ and $L$ be the midpoints of the sides $AB$ and $DE$ respectively. Prove that the sum of the areas of triangles $FAK$, $KCB$ and $CFL$ is equal to half of the area of the hexagon if and only if \[\frac{BC}{CD}=\frac{EF}{FA}.\]