Let $ABCD$ be a quadrilateral inscribed in a circle $\omega$, and let $P$ be a point on the extension of $AC$ such that $PB$ and $PD$ are tangent to $\omega$. The tangent at $C$ intersects $PD$ at $Q$ and the line $AD$ at $R$. Let $E$ be the second point of intersection between $AQ$ and $\omega$. Prove that $B$, $E$, $R$ are collinear.

Let $ABC$ be a triangle and $\omega$ its circumcircle. Point $D$ lies on the arc $BC$ (not containing $A$) of $\omega$ and is different from $B, C$ and the midpoint of arc $BC$ . The tangent line to $\omega$ at $D$ intersects lines $BC, CA,AB$ at $A', B',C'$ respectively. Lines $BB'$ and $CC'$ intersect at $E$. Line $AA' $ intersects again circle $\omega$ at $F$. Prove that the three points $D,E,F$ are colinear. Malik Talbi

In acute scalene triangle $ABC$ the external angle bisector of $\angle BAC$ meet $BC$ at point $X$.Lines $l_b$ and $l_c$ which tangents of $B$ and $C$ with respect to $(ABC)$.The line pass through $X$ intersects $l_b$ and $l_c$ at points $Y$ and $Z$ respectively. Suppose $(AYB)\cap(AZC)=N$ and $l_b\cap l_c=D$. Show that $ND$ is angle bisector of $\angle YNZ$. Proposed by [i]Alireza Haghi[/i]

Let $I$ be the incenter of an acute-angled triangle $ABC$. Let $P$, $Q$, $R$ be points on sides $AB$, $BC$, $CA$ respectively, such that $AP=AR$, $BP=BQ$ and $\angle PIQ = \angle BAC$. Prove that $QR \perp AC$.

In an acute triangle $ABC$ the point $O{}$ is the circumcenter, $H_1$ is the foot of the perpendicular from $A{}$ onto $BC$, and $M_H$ and $N_H$ are the projections of $H_1$ on $AC$ and $AB{}$, respectively. Prove that the polyline $M_HON_H$ divides the triangle $ABC$ in two figures of equal area. [i]Proposed by I. A. Kushner[/i]

A sphere with center $O$ has radius $6$. A triangle with sides of length $15, 15,$ and $24$ is situated in space so that each of its sides is tangent to the sphere. What is the distance between $O$ and the plane determined by the triangle? $ \textbf{(A) }2\sqrt{3}\qquad \textbf{(B) }4\qquad \textbf{(C) }3\sqrt{2}\qquad \textbf{(D) }2\sqrt{5}\qquad \textbf{(E) }5\qquad $

At the bottom-left corner of a $2014\times 2014$ chessboard, there are some green worms and at the top-left corner of the same chessboard, there are some brown worms. Green worms can move only to right and up, and brown worms can move only to right and down. After a while, the worms make some moves and all of the unit squares of the chessboard become occupied at least once throughout this process. Find the minimum total number of the worms.

The diagonals of a quadrilateral $ABCD$ are perpendicular: $AC\perp BD$. Four squares, $ABEF,BCGH,CDIJ,DAKL$, are erected externally on its sides. The intersection points of the pairs of straight lines $CL,DF; DF,AH; AH,BJ; BJ,CL$ are denoted by $P_1,Q_1,R_1, S_1$, respectively, and the intersection points of the pairs of straight lines $AI,BK; BK,CE;$ $ CE,DG; DG,AI$ are denoted by $P_2,Q_2,R_2, S_2$, respectively. Prove that $P_1Q_1R_1S_1 \cong P_2Q_2R_2S_2.$

Points $C_1$ and $C_2$ lie on side $AB$ of triangle $ABC$, where the point $C_1$ belongs to the segment $AC_2$ and $\angle ACC_1= \angle BCC_2$. On segments $CC_1$ and $CC_2$ points $A'$ and $B'$ are taken such that $\angle CAA'= \angle CBB' = \angle C_1CC_2$. Prove that the center of the circle $(CA'B')$ lies on the perpendicular bisector of the segment $AB$.

Let \( ABC \) be an acute-angled scalene triangle with circumcenter \( O \). Denote by \( M \), \( N \), and \( P \) the midpoints of sides \( BC \), \( CA \), and \( AB \), respectively. Let \( \omega \) be the circle passing through \( A \) and tangent to \( OM \) at \( O \). The circle \( \omega \) intersects \( AB \) and \( AC \) at points \( E \) and \( F \), respectively (where \( E \) and \( F \) are distinct from \( A \)). Let \( I \) be the midpoint of segment \( EF \), and let \( K \) be the intersection of lines \( EF \) and \( NP \). Prove that \( AO = 2IK \) and that triangle \( IMO \) is isosceles.

Let $ABCD$ be a square of side length $4$. Points $E$ and $F$ are chosen on sides $BC$ and $DA$, respectively, such that $EF = 5$. Find the sum of the minimum and maximum possible areas of trapezoid $BEDF$. [i]Proposed by Andrew Wu[/i]

In the $xOy$ system, consider the points $A_n(n,n^3)$ with $n\in \mathbb{N}^*$ and the point $B(0,1)$. Prove that a) for any positive integers $k>j>i\ge 1$, the points $A_i,A_j,A_k$ cannot be collinear. b) for any positive integers $i_k>i_{k-1}>\ldots>i_1\ge 1$, we have \[\mu(\widehat{A_{i_1}OB})+\mu(\widehat{A_{i_2}OB})+\cdots+\mu(\widehat{A_{i_k}OB})<\frac{\pi}{2}\] [i]***[/i]

In the interior of the convex polygon $A_1A_2...A_{2n}$ there is point $M$. Prove that at least one side of the polygon has not intersection points with the lines $MA_i$, $1\le i\le 2n$. (Spain)

Each pair of opposite sides of a convex hexagon has the following property: the distance between their midpoints is equal to $\dfrac{\sqrt{3}}{2}$ times the sum of their lengths. Prove that all the angles of the hexagon are equal.

Let $\triangle{PQR}$ be a right triangle with $PQ=90$, $PR=120$, and $QR=150$. Let $C_{1}$ be the inscribed circle. Construct $\overline{ST}$ with $S$ on $\overline{PR}$ and $T$ on $\overline{QR}$, such that $\overline{ST}$ is perpendicular to $\overline{PR}$ and tangent to $C_{1}$. Construct $\overline{UV}$ with $U$ on $\overline{PQ}$ and $V$ on $\overline{QR}$ such that $\overline{UV}$ is perpendicular to $\overline{PQ}$ and tangent to $C_{1}$. Let $C_{2}$ be the inscribed circle of $\triangle{RST}$ and $C_{3}$ the inscribed circle of $\triangle{QUV}$. The distance between the centers of $C_{2}$ and $C_{3}$ can be written as $\sqrt{10n}$. What is $n$?

[b]p1.[/b] Let $f(x) = \frac{3x^3+7x^2-12x+2}{x^2+2x-3}$ . Find all integers $n$ such that $f(n)$ is an integer. [b]p2.[/b] How many ways are there to arrange $10$ trees in a line where every tree is either a yew or an oak and no two oak trees are adjacent? [b]p3.[/b] $20$ students sit in a circle in a math class. The teacher randomly selects three students to give a presentation. What is the probability that none of these three students sit next to each other? [b]p4.[/b] Let $f_0(x) = x + |x - 10| - |x + 10|$, and for $n \ge 1$, let $f_n(x) = |f_{n-1}(x)| - 1$. For how many values of $x$ is $f_{10}(x) = 0$? [b]p5.[/b] $2$ red balls, $2$ blue balls, and $6$ yellow balls are in a jar. Zion picks $4$ balls from the jar at random. What is the probability that Zion picks at least $1$ red ball and$ 1$ blue ball? [b]p6.[/b] Let $\vartriangle ABC$ be a right-angled triangle with $\angle ABC = 90^o$ and $AB = 4$. Let $D$ on $AB$ such that $AD = 3DB$ and $\sin \angle ACD = \frac35$ . What is the length of $BC$? [b]p7.[/b] Find the value of of $$\dfrac{1}{1 +\dfrac{1}{2+ \dfrac{1}{1+ \dfrac{1}{2+ \dfrac{1}{1+ ...}}}}}$$ [b]p8.[/b] Consider all possible quadrilaterals $ABCD$ that have the following properties; $ABCD$ has integer side lengths with $AB\parallel CD$, the distance between $\overline{AB}$ and $\overline{CD}$ is $20$, and $AB = 18$. What is the maximum area among all these quadrilaterals, minus the minimum area? [b]p9.[/b] How many perfect cubes exist in the set $\{1^{2018},2^{2017}, 3^{2016},.., 2017^2, 2018^1\}$? [b]p10.[/b] Let $n$ be the number of ways you can fill a $2018\times 2018$ array with the digits $1$ through $9$ such that for every $11\times 3$ rectangle (not necessarily for every $3 \times 11$ rectangle), the sum of the $33$ integers in the rectangle is divisible by $9$. Compute $\log_3 n$. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

In the acute triangle $ABC$ the circle through $B$ touching the line $AC$ at $A$ has centre $P$, the circle through $A$ touching the line $BC$ at $B$ has centre $Q$. Let $R$ and $O$ be the circumradius and circumcentre of triangle $ABC$, respectively. Show that $R^2 = OP \cdot OQ$.

In the triangle $ABC$ points $M$ and $N$ are the midpoints of sides $AC$ and $AB$ respectively. $I$ is the incenter of the triangle. It is known that the angle $MIC$ is a right angle. Find the angle $NIB$.

There are $n$ points in the plane such that at least $99\%$ of quadrilaterals with vertices from these points are convex. Can we find a convex polygon in the plane having at least $90\%$ of the points as vertices? [i]Proposed by Morteza Saghafian - Iran[/i]

Let $ABC$ be a triangle with circumcircle $\Omega$ and incentre $I$. Let the line passing through $I$ and perpendicular to $CI$ intersect the segment $BC$ and the arc $BC$ (not containing $A$) of $\Omega$ at points $U$ and $V$ , respectively. Let the line passing through $U$ and parallel to $AI$ intersect $AV$ at $X$, and let the line passing through $V$ and parallel to $AI$ intersect $AB$ at $Y$ . Let $W$ and $Z$ be the midpoints of $AX$ and $BC$, respectively. Prove that if the points $I, X,$ and $Y$ are collinear, then the points $I, W ,$ and $Z$ are also collinear. [i]Proposed by David B. Rush, USA[/i]

How many square inches of paint are needed to fully paint a regular $6$-sided die with side length $2$ inches, except for the $\frac13$-inch diameter circular dots marking $1$ through $6$ (a different number per side)? The paint has negligible thickness, and the circular dots are non-overlapping.

Let $D, E$, and $F$ respectively be the feet of the altitudes from $A, B$, and $C$ of acute triangle $\vartriangle ABC$ such that $AF = 28, FB = 35$ and $BD = 45$. Let $P$ be the point on segment $BE$ such that $AP = 42$. Find the length of $CP$.

In triangle $ABC$, the angle bisectors are $BE$ and $CF$ (where $E, F$ are on the sides of the triangle), and their intersection point is $I$. Point $N$ lies on the circumcircle of $AEF$, and the angle $\angle IAN$ is right. The circumcircle of $AEF$ meets the line $NI$ a second time at the point $L$. Show that the circumcenter of $AIL$ lies on line $BC$.

A wooden cube $ n$ units on a side is painted red on all six faces and then cut into $ n^3$ unit cubes. Exactly one-fourth of the total number of faces of the unit cubes are red. What is $ n$? $ \textbf{(A)}\ 3\qquad \textbf{(B)}\ 4\qquad \textbf{(C)}\ 5\qquad \textbf{(D)}\ 6\qquad \textbf{(E)}\ 7$

Problem 19 The diagram below shows a 24×24 square ABCD. Points E and F lie on sides AD and CD, respectively, so that DE = DF = 8. Set X consists of the shaded triangle ABC with its interior, while set Y consists of the shaded triangle DEF with its interior. Set Z consists of all the points that are midpoints of segments connecting a point in set X with a point in set Y . That is, Z = {z | z is the midpoint of xy for x ∈ X and y ∈ Y}. Find the area of the set Z. For diagram to http://www.purplecomet.org/welcome/practice