A point $D$ lies inside a triangle $ABC$ on the bisector of angle $B$. Let $\omega_{1}$ and $\omega_{2}$ be the circles touching $AD$ and $CD$ at $D$ and passing through $B$; $P$ and $Q$ be the common points of $\omega_{1}$ and $\omega_{2}$ with the circumcircle of $ABC$ distinct from $B$. Prove that the circumcircles of the triangles $PQD$ and $ACD$ are tangent. Proposed by: L Shatunov

Given two parallel lines $r$ and $r'$ and a point $P$ on the plane that contains them and that is not on them, determine an equilateral triangle whose vertex is point $P$ , and the other two, one on each of the two lines. [img]https://cdn.artofproblemsolving.com/attachments/9/3/1d475eb3e9a8a48f4a85a2a311e1bda978e740.png[/img]

Let $ABCD$ be a cyclic quadrilateral with $AB = 1$, $BC = 2$, $CD = 3$, $DA = 4$. Find the square of the area of quadrilateral $ABCD$.

Let $O,H$ be the circumcenter and the orthocenter of triangle $ABC$. Let $F$ be the foot of the perpendicular from C onto AB, and $M$ the midpoint of $CH$. Let N be the foot of the perpendicular from C onto the parallel through H at $OM$. Let $D$ be on $AB$ such that $CA=CD$. Let $BN$ intersect $CD$ at $P$. Let $PH$ intersect $CA$ at $Q$. Prove that $QF\perp OF$.

Let $ABC$ be a triangle and let $O$ be its circumcentre. The internal and external bisectrices of the angle $BAC$ meet the line $BC$ at points $D$ and $E$, respectively. Let further $M$ and $L$ respectively denote the midpoints of the segments $BC$ and $DE$. The circles $ABC$ and $ALO$ meet again at point $N$. Show that the angles $BAN$ and $CAM$ are equal.

The diagonals $ AC$ and $ BD$ of a convex quadrilateral $ ABCD$ intersect at point $ M$. The bisector of $ \angle ACD$ meets the ray $ BA$ at $ K$. Given that $ MA \cdot MC \plus{}MA \cdot CD \equal{} MB \cdot MD$, prove that $ \angle BKC \equal{} \angle CDB$.

Let $ABC$ be a triangle and let $D$ be a point on the segment $BC$ such that $AD=BC$. \\ Suppose $\angle{CAD}=x^{\circ}, \angle{ABC}=y^{\circ}$ and $\angle{ACB}=z^{\circ}$ and $x,y,z$ are in an arithmetic progression in that order where the first term and the common difference are positive integers. Find the largest possible value of $\angle{ABC}$ in degrees.

There are several discs whose radii are no more that $1$, and whose centers all lie on a segment with length ${l}$. Prove that the union of all the discs has a perimeter not exceeding $4l+8$. [i]Proposed by Morteza Saghafian - Iran[/i]

Let $P$ be a plane and two points $A \in (P),O \notin (P)$. For each line in $(P)$ through $A$, let $H$ be the foot of the perpendicular from $O$ to the line. Find the locus $(c)$ of $H$. Denote by $(C)$ the oblique cone with peak $O$ and base $(c)$. Prove that all planes, either parallel to $(P)$ or perpendicular to $OA$, intersect $(C)$ by circles. Consider the two symmetric faces of $(C)$ that intersect $(C)$ by the angles $\alpha$ and $\beta$ respectively. Find a relation between $\alpha$ and $\beta$.

[b]p1.[/b] Len's Spanish class has four tests in the first term. Len scores $72$, $81$, and $78$ on the first three tests. If Len wants to have an 80 average for the term, what is the minimum score he needs on the last test? [b]p2.[/b] In $1824$, the Electoral College had $261$ members. Andrew Jackson won $99$ Electoral College votes and John Quincy Adams won $84$ votes. A plurality occurs when no candidate has more than $50\%$ of the votes. Should a plurality occur, the vote goes to the House of Representatives to break the tie. How many more votes would Jackson have needed so that a plurality would not have occurred? [b]p3.[/b] $\frac12 + \frac16 + \frac{1}{12} + \frac{1}{20} + \frac{1}{30}= 1 - \frac{1}{n}$. Find $n$. [b]p4.[/b] How many ways are there to sit Samuel, Esun, Johnny, and Prat in a row of $4$ chairs if Prat and Johnny refuse to sit on an end? [b]p5.[/b] Find an ordered quadruple $(w, x, y, z)$ that satisfies the following: $$3^w + 3^x + 3^y = 3^z$$ where $w + x + y + z = 2017$. [b]p6.[/b] In rectangle $ABCD$, $E$ is the midpoint of $CD$. If $AB = 6$ inches and $AE = 6$ inches, what is the length of $AC$? [b]p7.[/b] Call an integer interesting if the integer is divisible by the sum of its digits. For example, $27$ is divisible by $2 + 7 = 9$, so $27$ is interesting. How many $2$-digit interesting integers are there? [b]p8.[/b] Let $a\#b = \frac{a^3-b^3}{a-b}$ . If $a, b, c$ are the roots of the polynomial $x^3 + 2x^2 + 3x + 4$, what is the value of $a\#b + b\#c + c\#a$? [b]p9.[/b] Akshay and Gowri are examining a strange chessboard. Suppose $3$ distinct rooks are placed into the following chessboard. Find the number of ways that one can place these rooks so that they don't attack each other. Note that two rooks are considered attacking each other if they are in the same row or the same column. [img]https://cdn.artofproblemsolving.com/attachments/f/1/70f7d68c44a7a69eb13ce12291c0600d11027c.png[/img] [b]p10.[/b] The Earth is a very large sphere. Richard and Allen have a large spherical model of Earth, and they would like to (for some strange reason) cut the sphere up with planar cuts. If each cut intersects the sphere, and Allen holds the sphere together so it does not fall apart after each cut, what is the maximum number of pieces the sphere can be cut into after $6$ cuts? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $ABC$ be a triangle with circumcircle $\omega$ and let $\Omega_A$ be the $A$-excircle. Let $X$ and $Y$ be the intersection points of $\omega$ and $\Omega_A$. Let $P$ and $Q$ be the projections of $A$ onto the tangent lines to $\Omega_A$ at $X$ and $Y$ respectively. The tangent line at $P$ to the circumcircle of the triangle $APX$ intersects the tangent line at $Q$ to the circumcircle of the triangle $AQY$ at a point $R$. Prove that $\overline{AR} \perp \overline{BC}$.

In the convex quadrilateral $ABCD$, point $K$ is the midpoint of $AB$, point $L$ is the midpoint of $BC$, point $M$ is the midpoint of CD, and point $N$ is the midpoint of $DA$. Let $S$ be a point lying inside the quadrilateral $ABCD$ such that $KS = LS$ and $NS = MS$ .Prove that $\angle KSN = \angle MSL$.

The altitude $AH, BT$, and $CR$ are drawn in the non isosceles triangle $ABC$. On the side $BC$ mark the point $P$; points $X$ and $Y$ are projections of $P$ on $AB$ and $AC$. Two common external tangents to the circumscribed circles of triangles $XBH$ and $HCY$ intersect at point $Q$. The lines $RT$ and $BC$ intersect at point $K$. a). Prove that the point $Q$ lies on a fixed line independent of choice$ P$. b). Prove that $KQ = QH$.

Let $ABC$ be a triangle with $|AC| \ne |BC|$. Let $P$ and $Q$ be the intersection points of the line $AB$ with the internal and external angle bisectors at $C$, so that $P$ is between $A$ and $B$. Prove that if $M$ is any point on the circle with diameter $PQ$, then $\angle AMP = \angle BMP$.

Let $a$ and $b$ be two positive real numbers, and $AB$ a segment of length $a$ on a plane. Let $C,D$ be two variable points on the plane such that $ABCD$ is a non-degenerate convex quadrilateral with $BC=CD=b$ and $DA=a$. It is easy to see that there is a circle tangent to all four sides of the quadrilateral $ABCD$. Find the precise locus of the point $I$.

The diagonals of a convex quadrilateral $ABCD$ intersect at a point $O$. Find all angles of this quadrilateral if $\measuredangle OBA=30^{\circ},\measuredangle OCB=45^{\circ},\measuredangle ODC=45^{\circ}$, and $\measuredangle OAD=30^{\circ}$.

The incircle of triangle $ABC$ with $AB\neq AC$ has center $I$ and is tangent to $BC, CA,$ and $AB$ at $D, E,$ and $F$ respectively. The circumcircle of triangle $ADI$ intersects $AB$ and $AC$ again at $X$ and $Y.$ Prove that $EF$ bisects $XY.$

Two circles intersect at points $A$ and $B$. The minor arcs $AB$ measure $30^\circ$ on one circle and $60^\circ$ on the other circle. What is the ratio of the area of the larger circle to the area of the smaller circle? $\textbf{(A) }2\qquad \textbf{(B) }1+\sqrt3\qquad \textbf{(C) }3\qquad \textbf{(D) }2+\sqrt3\qquad \textbf{(E) }4\qquad$

Let $I$ and $H$ be the incenter and orthocenter of an acute triangle $ABC$. $M$ is the midpoint of arc $AC$ of circumcircle of triangle $ABC$ which does not contain point $B$. If $MI=MH$, find the measure of angle $\angle ABC$. [I]Proposed by F. Bakharev[/i]

We took a non-equilateral acute-angled triangle and marked $4$ wonderful points in it: the centers of the inscribed and circumscribed circles, the center of gravity (the point of intersection of the medians) and the intersection point of altitudes. Then the triangle itself was erased. It turned out that it was impossible to establish which of the centers corresponds to each of the marked points. Find the angles of the triangle

Georg and his mother love pizza. They buy a pizza shaped as an equilateral triangle. Georg demands to be allowed to divide the pizza by a straight cut and then make the first choice. The mother accepts this reluctantly, but she wants to choose a point of the pizza through which the cut must pass. Determine the largest fraction of the pizza which the mother is certain to get by this procedure.

We have a convex quadrilateral $ABCD$ and a point $M$ on its side $AD$ such that $CM$ and $BM$ are parallel to $AB$ and $CD$ respectively. Prove that $S_{ABCD} \geq 3 S_{BCM}.$ [i]Remark.[/i] $S$ denotes the area function.

A paper rectangle $ABCD $ of area $1$ is folded along a straight line so that $C$ coincides with $A$. Prove that the area of the pentagon thus obtained is less than $3/4$.

Let $ ABCD $ be a convex quadrilateral. Let angle bisectors of $ \angle B $ and $ \angle C $ intersect at $ E $. Let $ AB $ intersect $ CD $ at $ F $. Prove that if $ AB+CD=BC $, then $A,D,E,F$ is cyclic.

$O$ is the intersection point of the diagonals of the trapezoid $ABCD$. A line passing through $C$ and a point symmetric to $B$ with respect to $O$, intersects the base $AD$ at the point $K$. Prove that $S_{AOK} = S_{AOB} + S_{DOK}$.