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]

Given a circumscribed quadrilateral $ABCD$. Prove that its inradius is smaller than the sum of the inradii of triangles $ABC$ and $ACD$.

Point $B$ is due east of point $A$. Point $C$ is due north of point $B$. The distance between points $A$ and $C$ is $10\sqrt{2}$ meters, and $\angle BAC=45^{\circ}$. Point $D$ is $20$ meters due north of point $C$. The distance $AD$ is between which two integers? $ \textbf{(A)}\ 30\text{ and }31\qquad\textbf{(B)}\ 31\text{ and }32\qquad\textbf{(C)}\ 32\text{ and }33\qquad\textbf{(D)}\ 33\text{ and }34\qquad\textbf{(E)}\ 34\text{ and }35$

To each side of the regular $p$-gon of side length $1$ there is attached a $1 \times k$ rectangle, partitioned into $k$ unit cells, where $k$ and $p$ are given positive integers and p an odd prime. Let $P$ be the resulting nonconvex star-like polygonal figure consisting of $kp + 1$ regions ($kp$ unit cells and the $p$-gon). Each region is to be colored in one of three colors, adjacent regions having different colors. Furthermore, it is required that the colored figure should not have a symmetry axis. In how many ways can this be done?

Let $n$ be a positive integer. A 4-by-$n$ rectangle is divided into $4n$ unit squares in the usual way. Each unit square is colored black or white. Suppose that every white unit square shares an edge with at least one black unit square. Prove that there are at least $n$ black unit squares.

Let $AC$ be the hypothenuse of a right-angled triangle $ABC$. The bisector $BD$ is given, and the midpoints $E$ and $F$ of the arcs $BD$ of the circumcircles of triangles $ADB$ and $CDB$ respectively are marked (the circles are erased). Construct the centers of these circles using only a ruler.

Let $ABC$ be a triangle and let $A' \in BC$, $B' \in CA$ and $C' \in AB$ be three collinear points. a) Prove that each pair of circles of diameters $AA'$, $BB'$ and $CC'$ has the same radical axis; b) Prove that the circumcenter of the triangle formed by the intersections of the lines $AA' , BB'$ and $CC'$ lies on the common radical axis found above.

Let $ABC$ be acute-angled triangle with circumcircle $\Gamma$. Points $H$ and $M$ are the orthocenter and the midpoint of $BC$ respectively. The line $HM$ meets the circumcircle $\omega$ of triangle $BHC$ at point $N\not= H$. Point $P$ lies on the arc $BC$ of $\omega$ not containing $H$ in such a way that $\angle HMP = 90^\circ$. The segment $PM$ meets $\Gamma$ at point $Q$. Points $B'$ and $C'$ are the reflections of $A$ about $B$ and $C$ respectively. Prove that the circumcircles of triangles $AB'C'$ and $PQN$ are tangent.

Let $AD, BE$, and $CF$ denote the altitudes of triangle $\vartriangle ABC$. Points $E'$ and $F'$ are the reflections of $E$ and $F$ over $AD$, respectively. The lines $BF'$ and $CE'$ intersect at $X$, while the lines $BE'$ and $CF'$ intersect at the point $Y$. Prove that if $H$ is the orthocenter of $\vartriangle ABC$, then the lines $AX, YH$, and $BC$ are concurrent.

A convex polygon $P$ is placed inside a unit square $Q$. Prove that the perimeter of $P$ does not exceed $4$.

Let \( ABC \) and \( A_1B_1C_1 \) be two triangles such that the segments \( AA_1 \) and \( BC \) intersect, the segments \( BB_1 \) and \( AC \) intersect, and the segments \( CC_1 \) and \( AB \) intersect. If it is known that there exists a point \( X \) inside both triangles such that \[ \begin{aligned} \angle XAB &= \angle XA_1B_1, &\angle XBC &= \angle XC_1A_1, &\angle XCA &= \angle XB_1C_1,\\ \angle XAC &= \angle XB_1A_1, &\angle XBA &= \angle XA_1C_1, &\angle XCB &= \angle XC_1B_1. \end{aligned} \] Prove that the lines \( AC_1 \), \( BB_1 \), and \( CA_1 \) are concurrent or parallel.

Let $\triangle ABC$ be an acute triangle, and let $I_B, I_C,$ and $O$ denote its $B$-excenter, $C$-excenter, and circumcenter, respectively. Points $E$ and $Y$ are selected on $\overline{AC}$ such that $\angle ABY=\angle CBY$ and $\overline{BE}\perp\overline{AC}$. Similarly, points $F$ and $Z$ are selected on $\overline{AB}$ such that $\angle ACZ=\angle BCZ$ and $\overline{CF}\perp\overline{AB}$. Lines $\overleftrightarrow{I_BF}$ and $\overleftrightarrow{I_CE}$ meet at $P$. Prove that $\overline{PO}$ and $\overline{YZ}$ are perpendicular. [i]Proposed by Evan Chen and Telv Cohl[/i]

On a straight line $\ell$ there are four points, $A$, $B$, $C$ and $D$ in that order, such that $AB=BC=CD$. A point $E$ is chosen outside the straight line so that when drawing the segments $EB$ and $EC$, an equilateral triangle $EBC$ is formed . Segments $EA$ and $ED$ are drawn, and a point $F$ is chosen so that when drawing the segments $FA$ and $FE$, an equilateral triangle $FAE$ is formed outside the triangle $EAD$. Finally, the lines $EB$ and $FA$ are drawn , which intersect at the point $G$. If the area of triangle $EBD$ is $10$, calculate the area of triangle $EFG$.

Let $ABC$ be an acute-angled triangle with circumcircle $\omega$. A circle $\Gamma$ is internally tangent to $\omega$ at $A$ and also tangent to $BC$ at $D$. Let $AB$ and $AC$ intersect $\Gamma$ at $P$ and $Q$ respectively. Let $M$ and $N$ be points on line $BC$ such that $B$ is the midpoint of $DM$ and $C$ is the midpoint of $DN$. Lines $MP$ and $NQ$ meet at $K$ and intersect $\Gamma$ again at $I$ and $J$ respectively. The ray $KA$ meets the circumcircle of triangle $IJK$ again at $X\neq K$. Prove that $\angle BXP = \angle CXQ$. [i]Kian Moshiri, United Kingdom[/i]

A unit square has a circle of radius $r$ with center at it's midpoint. The four quarter circles are centered on the vertices of the square and are tangent to the central circle (see figure). Find the maximum and minimum possible value of the area of the striped figure in the figure and the corresponding values of $r$ such these, the maximum and minimum are achieved. [img]https://2.bp.blogspot.com/-DOT4_B5Mx-8/XnmsTlWYfyI/AAAAAAAALgs/TVYkrhqHYGAeG8eFuqFxGDCTnogVbQFUwCK4BGAYYCw/s400/96%2Bestonia%2Bopen%2Bs1.4.png[/img]

A right circular cone has base radius 1 cm and slant height 3 cm is given. $P$ is a point on the circumference of the base and the shortest path from $P$ around the cone and back to $P$ is drawn (see diagram). What is the minimum distance from the vertex $V$ to this path? [asy] import graph; unitsize(1 cm); filldraw(shift(-0.15,0.37)*rotate(17)*yscale(0.3)*xscale(1.41)*(Circle((0,0),1)),gray(0.9),nullpen); draw(yscale(0.3)*(arc((0,0),1.5,0,180)),dashed); draw(yscale(0.3)*(arc((0,0),1.5,180,360))); draw((1.5,0)--(0,4)--(-1.5,0)); draw((0,0)--(1.5,0),Arrows); draw(((1.5,0) + (0.3,0.1))--((0,4) + (0.3,0.1)),Arrows); draw(shift(-0.15,0.37)*rotate(17)*yscale(0.3)*xscale(1.41)*(arc((0,0),1,0,180)),dashed); draw(shift(-0.15,0.37)*rotate(17)*yscale(0.3)*xscale(1.41)*(arc((0,0),1,180,360))); label("$V$", (0,4), N); label("1 cm", (0.75,-0.5), N); label("$P$", (-1.5,0), SW); label("3 cm", (1.7,2)); [/asy]

A point $P$ was chosen on the smaller arc $BC$ of the circumcircle of the acute-angled triangle $ABC$. Points $R$ and $S$ on the sides$ AB$ and $AC$ are respectively selected so that $CPRS$ is a parallelogram. Point $T$ on the arc $AC$ of the circumscribed circle of $\vartriangle ABC$ such that $BT \parallel CP$. Prove that $\angle TSC = \angle BAC$. (Anton Trygub)

A line parallel to the side $BC$ of a triangle $ABC$ meets $AB$ in $F$ and $AC$ in $E$. Prove that the circles on $BE$ and $CF$ as diameters intersect in a point lying on the altitude of the triangle $ABC$ dropped from $A$ to $BC.$

Let $ a,\ b$ be real numbers satisfying $ \int_0^1 (ax\plus{}b)^2dx\equal{}1$. Determine the values of $ a,\ b$ for which $ \int_0^1 3x(ax\plus{}b)\ dx$ is maximized.

Let $ABCD$ be a convex quadrilateral with perpendicular diagonals. If $AB = 20, BC = 70$ and $CD = 90$, then what is the value of $DA$?

The triangle below is divided into nine stripes of equal width each parallel to the base of the triangle. The darkened stripes have a total area of $135$. Find the total area of the light colored stripes. [img]https://cdn.artofproblemsolving.com/attachments/0/8/f34b86ccf50ef3944f5fbfd615a68607f4fadc.png[/img]

The Lattice Point Jumping Frog jumps between lattice points in a coordinate plane that are exactly $1$ unit apart. The Lattice Point Jumping Frog starts at the origin and makes $8$ jumps, ending at the origin. Additionally, it never lands on a point other than the origin more than once. How many possible paths could the frog have taken? [i]Author: Ray Li[/i] [hide="Clarifications"] [list=1][*]The Lattice Jumping Frog is allowed to visit the origin more than twice. [*]The path of the Lattice Jumping Frog is an ordered path, that is, the order in which the Lattice Jumping Frog performs its jumps matters.[/list][/hide]

Let $\vartriangle ABC$ be a triangle such that the area$ [ABC] = 10$ and $\tan (\angle ABC) = 5$. If the smallest possible value of $(\overline{AC})^2$ can be expressed as $-a + b\sqrt{c}$ for positive integers $a, b, c$, what is $a + b + c$?

Let $ABC$ be an acute scalene triangle with circumcircle $\omega$. Let $P$ and $Q$ be interior points of the sides $AB$ and $AC$, respectively, such that $PQ$ is parallel to $BC$. Let $L$ be a point on $\omega$ such that $AL$ is parallel to $BC$. The segments $BQ$ and $CP$ intersect at $S$. The line $LS$ intersects $\omega$ at $K$. Prove that $\angle BKP = \angle CKQ$. Proposed by [i]Ervin Macić, Bosnia and Herzegovina[/i]

Three chords $AB, CD$ and $EF$ of a circle intersect at the midpoint $M$ of $AB$. Show that if $CE$ produced and $DF$ produced meet the line $AB$ at the points $P$ and $Q$ respectively, then $M$ is also the midpoint of $PQ$.