Let $ABCD$ be a cyclic quadrangle whose midpoints of diagonals $AC$ and $BD$ are $F$ and $G$, respectively. a) Prove the following implication: if the bisectors of angles at $B$ and $D$ of the quadrangle intersect at diagonal $AC$ then $\frac14 \cdot |AC| \cdot |BD| = | AG| \cdot |BF| \cdot |CG| \cdot |DF|$. b) Does the converse implication also always hold?

The circles ${{w} _ {1}}$ and ${{w} _ {2}}$ intersect at points $P$ and $Q$. Let $AB$ and $CD$ be parallel diameters of circles ${ {w} _ {1}}$ and ${{w} _ {2}} $, respectively. In this case, none of the points $A, B, C, D$ coincides with either $P$ or $Q$, and the points lie on the circles in the following order: $A, B, P, Q$ on the circle ${{w} _ {1} }$ and $C, D, P, Q$ on the circle ${{w} _ {2}} $. The lines $AP$ and $BQ$ intersect at the point $X$, and the lines $CP$ and $DQ$ intersect at the point $Y, X \ne Y$. Prove that all lines $XY$ for different diameters $AB$ and $CD$ pass through the same point or are all parallel. (Serdyuk Nazar)

Two circles intersect at two points $A$ and $B$. A line $\ell$ which passes through the point $A$ meets the two circles again at the points $C$ and $D$, respectively. Let $M$ and $N$ be the midpoints of the arcs $BC$ and $BD$ (which do not contain the point $A$) on the respective circles. Let $K$ be the midpoint of the segment $CD$. Prove that $\measuredangle MKN = 90^{\circ}$.

Let $ ABC $ be an acute triangle having $ AB<AC, $ let $ M $ be the midpoint of the segment $ BC, D$ be a point on the segment $ AM, E $ be a point on the segment $ BD $ and $ F $ on the line $ AB $ such that $ EF $ is parallel to $ BC, $ and such that $ AE $ and $ DF $ pass through the orthocenter of $ ABC. $ Prove that the interior bisectors of $ \angle BAC $ and $ \angle BDC, $ together with $ BC $ are concurrent. [i]Vlad Robu[/i]

On the sides $AB$ and $BC$ of triangle $ABC$, points $M$ and $K$ are taken, respectively, so that $S_{KMC} + S_{KAC}=S_{ABC}$. Prove that all such lines $MK$ pass through one point.

Let $ABC$ be a triangle with circumcircle $\Omega$, circumcenter $O$ and orthocenter $H$. Let $S$ lie on $\Omega$ and $P$ lie on $BC$ such that $\angle ASP=90^\circ$, line $SH$ intersects the circumcircle of $\triangle APS$ at $X\neq S$. Suppose $OP$ intersects $CA,AB$ at $Q,R$, respectively, $QY,RZ$ are the altitude of $\triangle AQR$. Prove that $X,Y,Z$ are collinear. [i]Proposed by Shuang-Yen Lee[/i]

Let $ABCD$ a cyclic convex quadrilateral. There is a line $l$ parallel to $DC$ containing $A$. Let $P$ a point on $l$ closer to $A$ than to $B$. Let $B'$ the reflection of $B$ over the midpoint of $AD$. Prove that $\angle B'AP = \angle BAC$

In trapezoid $ABCD$, sides $AB$ and $CD$ are parallel, and diagonal $BD$ and side $AD$ have equal length. If $m\angle DBC=110^\circ$ and $m\angle CBD =30^\circ$, then $m \angle ADB=$ $\text{(A)}\ 80^\circ \qquad \text{(B)}\ 90^\circ \qquad \text{(C)}\ 100^\circ \qquad \text{(D)}\ 110^\circ \qquad \text{(E)}\ 120^\circ$

The $100$ vertices of a prism, whose base is a $50$-gon, are labeled with numbers $1, 2, 3, \ldots, 100$ in any order. Prove that there are two vertices, which are connected by an edge of the prism, with labels differing by not more than $48$. Note: In all the triangles the three vertices do not lie on a straight line.

Let $ABCD$ be a convex quadrilateral, with $\measuredangle ABC > 90^\circ$, $\measuredangle CDA > 90^\circ$ and $\measuredangle DAB = \measuredangle BCD$. Let $E$, $F$ and $G$ be the reflections of $A$ with respect to the lines $BC$, $CD$ and $DB$. Finally, let the line $BD$ meet the line segment $AE$ at a point $K$, and the line segment $AF$ at a point $L$. Prove that the circumcircles of the triangles $BEK$ and $DFL$ are tangent to each other at $G$.

Triangle $ ABC$ is a right triangle with $ \angle ACB$ as its right angle, $ m\angle ABC \equal{} 60^\circ$, and $ AB \equal{} 10$. Let $ P$ be randomly chosen inside $ \triangle ABC$, and extend $ \overline{BP}$ to meet $ \overline{AC}$ at $ D$. What is the probability that $ BD > 5\sqrt2$? [asy]import math; unitsize(4mm); defaultpen(fontsize(8pt)+linewidth(0.7)); dotfactor=4; pair A=(10,0); pair C=(0,0); pair B=(0,10.0/sqrt(3)); pair P=(2,2); pair D=extension(A,C,B,P); draw(A--C--B--cycle); draw(B--D); dot(P); label("A",A,S); label("D",D,S); label("C",C,S); label("P",P,NE); label("B",B,N);[/asy] $ \textbf{(A)}\ \frac {2 \minus{} \sqrt2}{2} \qquad \textbf{(B)}\ \frac {1}{3} \qquad \textbf{(C)}\ \frac {3 \minus{} \sqrt3}{3} \qquad \textbf{(D)}\ \frac {1}{2} \qquad \textbf{(E)}\ \frac {5 \minus{} \sqrt5}{5}$

In a triangle $ABC$, the incircle centered at $I$ is tangent to the sides $BC, AC$ and $AB$ at $D, E$ and $F$, respectively. Let $X, Y$ and $Z$ be the feet of the perpendiculars drawn from $A, B$ and $C$ to a line $\ell$ passing through $I$. Prove that $DX, EY$ and $FZ$ are concurrent.

Points $X$ and $Y$ are the midpoints of arcs $AB$ and $BC$ of the circumscribed circle of triangle $ABC$. Point $T$ lies on side $AC$. It turned out that the bisectors of the angles $ATB$ and $BTC$ pass through points $X$ and $Y$ respectively. What angle $B$ can be in triangle $ABC$?

Let $ ABC $ be an acute triangle having $ AB<AC, I $ be its incenter, $ D,E,F $ be intersection of the incircle with $ BC, CA, $ respectively, $ AB, X $ be the middle of the arc $ BAC, $ which is an arc of the circumcicle of it, $ P $ be the projection of $ D $ on $ EF $ and $ Q $ be the projection of $ A $ on $ ID. $ [b]a)[/b] Show that $ IX $ and $ PQ $ are parallel. [b]b)[/b] If the circle of diameter $ AI $ intersects the circumcircle of $ ABC $ at $ Y\neq A, $ prove that $ XQ $ intersects $ PI $ at $ Y. $

Point $O$ is the foot of the altitude of a quadrilateral pyramid. A sphere with center $O$ is tangent to all lateral faces of the pyramid. Points $A,B,C,D$ are taken on successive lateral edges so that segments $AB$, $BC$, and $CD$ pass through the three corresponding tangency points of the sphere with the faces. Prove that the segment $AD$ passes through the fourth tangency point

Let be a line $ d: 3x+4y-5=0 $ on a Cartesian plane. We mark with $ \mathcal{L} $ de locus of the planar points $ P $ such that the distance from $ P $ to $ d $ is double the distance from $ P $ to the origin. Let be $ B_{\lambda } ,C_{\lambda }\in\mathcal{L} $ such that $ C_{\lambda } -B_{\lambda } +\lambda =0. $ Find the locus of the middlepoints of the segments $ B_{\lambda }C_{\lambda }, $ if $ \lambda\in\mathbb{R} $ is variable.

What is the volume of tetrahedron $ABCD$ with edge lengths $AB=2, AC=3, AD=4, BC=\sqrt{13}, BD=2\sqrt{5},$ and $CD=5$? $\textbf{(A) }3 \qquad \textbf{(B) }2\sqrt{3} \qquad \textbf{(C) }4 \qquad \textbf{(D) }3\sqrt{3} \qquad \textbf{(E) }6$

[b]p1.[/b] Find the smallest positive integer $N$ such that $N!$ is a multiple of $10^{2010}$. [b]p2.[/b] An equilateral triangle $T$ is externally tangent to three mutually tangent unit circles, as shown in the diagram. Find the area of $T$. [b]p3. [/b]The polynomial $p(x) = x^3 + ax^2 + bx + c$ has the property that the average of its roots, the product of its roots, and the sum of its coefficients are all equal. If $p(0) = 2$, find $b$. [b]p4.[/b] A regular pentagon $P = A_1A_2A_3A_4A_5$ and a square $S = B_1B_2B_3B_4$ are both inscribed in the unit circle. For a given pentagon $P$ and square $S$, let $f(P, S)$ be the minimum length of the minor arcs AiBj , for $1 \le i \le 5$ and $1 \le j \le 4$. Find the maximum of $f(P, S)$ over all pairs of shapes. [b]p5.[/b] Let $ a, b, c$ be three three-digit perfect squares that together contain each nonzero digit exactly once. Find the value of $a + b + c$. [b]p6. [/b]There is a big circle $P$ of radius $2$. Two smaller circles $Q$ and $R$ are drawn tangent to the big circle $P$ and tangent to each other at the center of the big circle $P$. A fourth circle $S$ is drawn externally tangent to the smaller circles $Q$ and $R$ and internally tangent to the big circle $P$. Finally, a tiny fifth circle $T$ is drawn externally tangent to the $3$ smaller circles $Q, R, S$. What is the radius of the tiny circle $T$? [b]p7.[/b] Let $P(x) = (1 +x)(1 +x^2)(1 +x^4)(1 +x^8)(...)$. This infinite product converges when $|x| < 1$. Find $P\left( \frac{1}{2010}\right)$. [b]p8.[/b] $P(x)$ is a polynomial of degree four with integer coefficients that satisfies $P(0) = 1$ and $P(\sqrt2 + \sqrt3) = 0$. Find $P(5)$. [b]p9.[/b] Find all positive integers $n \ge 3$ such that both roots of the equation $$(n - 2)x^2 + (2n^2 - 13n + 38)x + 12n - 12 = 0$$ are integers. [b]p10.[/b] Let $a, b, c, d, e, f$ be positive integers (not necessarily distinct) such that $$a^4 + b^4 + c^4 + d^4 + e^4 = f^4.$$ Find the largest positive integer $n$ such that $n$ is guaranteed to divide at least one of $a, b, c, d, e, f$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

The corner of a unit cube is chopped off such that the cut runs through the three vertices adjacent to the vertex of the chosen corner. What is the height of the cube when the freshly-cut face is placed on a table?

The circles $\Gamma_1$ and $\Gamma_2$ intersect at $D$ and $P$. The common tangent line of the two circles closest to point $D$ touches $\Gamma_1$ in A and $\Gamma_2$ in $B$. The line $AD$ intersects $\Gamma_2$ for the second time in $C$. Let $M$ be the midpoint of line segment $BC$. Prove that $\angle DPM = \angle BDC$.

Peter is a pentacrat and spends his time drawing pentagrams. With the abbreviation $\left|XYZ\right|$ for the area of an arbitrary triangle $XYZ$, he notes that any convex pentagon $ABCDE$ satisfies the equality $\left|EAC\right|\cdot\left|EBD\right|=\left|EAB\right|\cdot\left|ECD\right|+\left|EBC\right|\cdot\left|EDA\right|$. Guess what you are supposed to do and do it.

In a convex quadrilateral $ABCD$, the lines $AB$ and $CD$ meet at point $E$ and the lines $AD$ and $BC$ meet at point $F$. Let $P$ be the intersection point of diagonals $AC$ and $BD$. Suppose that $\omega_1$ is a circle passing through $D$ and tangent to $AC$ at $P$. Also suppose that $\omega_2$ is a circle passing through $C$ and tangent to $BD$ at $P$. Let $X$ be the intersection point of $\omega_1$ and $AD$, and $Y$ be the intersection point of $\omega_2$ and $BC$. Suppose that the circles $\omega_1$ and $\omega_2$ intersect each other in $Q$ for the second time. Prove that the perpendicular from $P$ to the line $EF$ passes through the circumcenter of triangle $XQY$ . Proposed by Iman Maghsoudi

Let unit blocks be unit squares in the coordinate plane with vertices at lattice points (points $(a, b)$ such that $a$ and $b$ are both integers). Prove that a circle with area $\pi$ can cover parts of no more than $9$ unit blocks. The circle below covers part of $8$ unit blocks. [img]https://cdn.artofproblemsolving.com/attachments/4/4/43da9abed06d0feba94012ba68c177e3c2835b.png[/img]

Let $ABC$ be an acute triangle with $\displaystyle{AB<AC<BC}$ inscribed in circle $ \displaystyle{c(O,R)}$.The excircle $\displaystyle{(c_A)}$ has center $\displaystyle{I}$ and touches the sides $\displaystyle{BC,AC,AB}$ of the triangle $ABC$ at $\displaystyle{D,E,Z} $ respectively.$ \displaystyle{AI}$ cuts $\displaystyle{(c)}$ at point $M$ and the circumcircle $\displaystyle{(c_1)}$ of triangle $\displaystyle{AZE}$ cuts $\displaystyle{(c)}$ at $K$.The circumcircle $\displaystyle{(c_2)}$ of the triangle $\displaystyle{OKM}$ cuts $\displaystyle{(c_1)} $ at point $N$.Prove that the point of intersection of the lines $AN,KI$ lies on $ \displaystyle{(c)}$.

Let $ABCD$ be a cyclic qudrilaterlal such that $AB.BC=2.CD.DA$ Prove that $8.BD^2 \leq 9.AC^2$