Pentagon $ABCDE$ is inscribed in circle $\Omega$. Line $AD$ meets $CE$ at $F$, and $P (\neq E, F)$ is a point on segment $EF$. The circumcircle of triangle $AFP$ meets $\Omega$ at $Q(\neq A)$ and $AC$ at $R(\neq A)$. Line $AD$ meets $BQ$ at $S$, and the circumcircle of triangle $DES$ meets line $BQ, BD$ at $T(\neq S), U(\neq D)$, respectively. Prove that if $F, P, T, S$ are concyclic, then $P, T, R, U$ are concyclic.

Let $ABCD$ be a convex quadrilateral whose sides $AD$ and $BC$ are not parallel. Suppose that the circles with diameters $AB$ and $CD$ meet at points $E$ and $F$ inside the quadrilateral. Let $\omega_E$ be the circle through the feet of the perpendiculars from $E$ to the lines $AB,BC$ and $CD$. Let $\omega_F$ be the circle through the feet of the perpendiculars from $F$ to the lines $CD,DA$ and $AB$. Prove that the midpoint of the segment $EF$ lies on the line through the two intersections of $\omega_E$ and $\omega_F$. [i]Proposed by Carlos Yuzo Shine, Brazil[/i]

Aidan owns a plot of land that is in the shape of a triangle with side lengths $5$,$10$, and $5\sqrt3$ feet. Aidan wants to plant radishes such that there are no two radishes that are less than $1$ foot apart. Determine the maximum number of radishes Aidan can plant

Let there be a square $ABCD$. Points $E$ and $F$ are on sides $(BC)$ and $(AB)$ such that $BF=CE$. LInes $AE$ and $CF$ intersect in point $G$. Prove that $EF$ and $DG$ are perpendicular.

Consider all possible rectangles that can be drawn on a $8 \times 8$ chessboard, covering only whole cells. Calculate the sum of their areas. What formula is obtained if “$8 \times 8$” is replaced with “$a \times b$”, where $a, b$ are positive integers?

$ABCD$ is a trapezoid with $AB \parallel CD$. The diagonals intersect at $P$. Let $\omega _1$ be a circle passing through $B$ and tangent to $AC$ at $A$. Let $\omega _2$ be a circle passing through $C$ and tangent to $BD$ at $D$. $\omega _3$ is the circumcircle of triangle $BPC$. Prove that the common chord of circles $\omega _1,\omega _3$ and the common chord of circles $\omega _2, \omega _3$ intersect each other on $AD$. [i]Proposed by Kasra Ahmadi[/i]

A square $ ABCD$ of side length $ 2$ is given on a plane. The segment $ AB$ is moved continuously towards $ CD$ until $ A$ and $ C$ coincide with $ C$ and $ D$, respectively. Let $ S$ be the area of the region formed by the segment $ AB$ while moving. Prove that $ AB$ can be moved in such a way that $ S <\frac{5\pi}{6}$.

Point $H$ is the foot of the altitude from $A$ of triangle $ABC$. On the lines $AB$ and $AC$ points $X$ and $Y$ are marked such that the circumcircles of triangles $BXH$ and $CYH$ are tangent, call this circles $w_B$ and $w_C$ respectively. Tangent lines to circles $w_B$ and $w_C$ at $X$ and $Y$ intersect at $Z$. Prove that $ZA=ZH$. [i]Vadzim Kamianetski[/i]

Let $ABC$ be a triangle with orthocenter $H$ and circumcircle $\Gamma$. Let $D$ be the reflection of $A$ in $B$ and let $E$ the reflection of $A$ in $C$. Let $M$ be the midpoint of segment $DE$. Show that the tangent to $\Gamma$ in $A$ is perpendicular to $HM$.

Let $K$ be the midpoint of a fixed line segment $AB$, two circles $(O)$ and $(O')$ with variable radius each such that the straight line $OO'$ is throughK and $K$ is inside $(O)$, the two circles meet at $A$ and $C$, center $O'$ is on the circumference of $(O)$ and $O$ is interior to $(O')$. Assume that $M$ is the midpoint of $AC, H$ the projection of $C$ onto the perpendicular bisector of segment $AB$. Let $I$ be a variable point on the arc $AC$ of circle $(O')$ that is inside $(O), I$ is not on the line $OO'$ . Let $J$ be the reflection of $I$ about $O$. The tangent of $(O')$ at $I$ meets $AC$ at $N$. Circle $(O'JN)$ meets $IJ$ at $P$, distinct from $J$, circle $(OMP)$ intersects $MI$ at $Q$ distinct from $M$. Prove that (a) the intersection of $PQ$ and $O'I$ is on the circumference of $(O)$. (b) there exist a location of $I$ such that the line segment $AI$ meets $(O)$ at $R$ and the straight line $BI$ meets $(O')$ at $S$, then the lines $AS$ and $KR$ meets at a point on the circumference of $(O)$. (c) the intersection $G$ of lines $KC$ and $HB$ moves on a fixed line. Lê Phúc Lữ

In the $xOy$ system consider the lines $d_1\ :\ 2x-y-2=0,\ d_2\ :\ x+y-4=0,\ d_3\ :\ y=2$ and $d_4\ :\ x-4y+3=0$. Find the vertices of the triangles whom medians are $d_1,d_2,d_3$ and $d_4$ is one of their altitudes. [i]Lucian Dragomir[/i]

The area of rectangle $ABCD$ is $72$. If point $A$ and the midpoints of $\overline{BC}$ and $\overline{CD}$ are joined to form a triangle, the area of that triangle is [asy] pair A,B,C,D; A = (0,8); B = (9,8); C = (9,0); D = (0,0); draw(A--B--C--D--A--(9,4)--(4.5,0)--cycle); label("$A$",A,NW); label("$B$",B,NE); label("$C$",C,SE); label("$D$",D,SW); [/asy] $\text{(A)}\ 21 \qquad \text{(B)}\ 27 \qquad \text{(C)}\ 30 \qquad \text{(D)}\ 36 \qquad \text{(E)}\ 40$

$ABC$ is an obtuse triangle. (angle $B$ is obtuse) Its circumcircle is $O$. A tangent line for $O$ passing $C$ meets with $AB$ at $B_1$. Let $O_1$ be a circumcenter of triangle $AB_1C$. $B_2$ is a point on the segment $BB_1$. Let $C_1$ be a contact point of the tangent line for $O$ passing $B_2$, which is more closer to $C$. Let $O_2$ be a circumcenter of triangle $AB_2C_1$. Prove that if $OO_2$ and $AO_1$ is perpendicular, then five points $O,O_2,O_1,C_1,C$ are concyclic.

Let $ABC$ be a triangle such that $\angle CAB = 60^o$. Consider $D, E$ points on sides $AC$ and $AB$ respectively such that $BD$ bisects angle $\angle ABC$ , $CE$ bisects angle $\angle BCA$ and let $I$ be the intersection of them. Prove that $|ID| =|IE|$.

Let $ ABC$ be a triangle, and $ I$ its incenter. Let the incircle of $ ABC$ touch side $ BC$ at $ D$, and let lines $ BI$ and $ CI$ meet the circle with diameter $ AI$ at points $ P$ and $ Q$, respectively. Given $ BI \equal{} 6, CI \equal{} 5, DI \equal{} 3$, determine the value of $ \left( DP / DQ \right)^2$.

We are given an infinite cylinder in space (i.e. the locus of points of a given distance $R>0$ from a given straight line). Can six straight lines containing the edges of a tetrahedron all have exactly one common point with this cylinder? [i]Proposed by A. Kuznetsov[/i]

Let $\triangle ABC$ be an acute triangle. Point $D$ is arbitrarily chosen on the side $BC$. Let the circumcircle of the triangle $\triangle ADB$ intersect the segment $AC$ at $M$, and the circumcircle of the triangle $\triangle ADC$ intersect the segment $AB$ at $N$. Prove that the tangents of the circumcircle of the triangle $\triangle AMN$ at $M$ and $N$ intersect at a point that lies on the line $BC$.

[u]Round 1[/u] [b]p1.[/b] Evaluate the sum $1-2+3-...-208+209-210$. [b]p2.[/b] Tony has $14$ beige socks, $15$ blue socks, $6$ brown socks, $8$ blond socks and $7$ black socks. If Tony picks socks out randomly, how many socks does he have to pick in order to guarantee a pair of blue socks? [b]p3.[/b] The price of an item is increased by $25\%$, followed by an additional increase of $20\%$. What is the overall percentage increase? [u]Round 2[/u] [b]p4.[/b] A lamp post is $20$ feet high. How many feet away from the base of the post should a person who is $5$ feet tall stand in order to cast an 8-foot shadow? [b]p5.[/b] How many positive even two-digit integers are there that do not contain the digits $0$, $1$, $2$, $3$ or $4$? [b]p6.[/b] In four years, Jack will be twice as old as Jill. Three years ago, Jack was three times as old as Jill. How old is Jack? [u]Round 3[/u] [b]p7.[/b] Let $x \Delta y = x y^2 -2y$. Compute $20\Delta 18$. [u]p8.[/u] A spider crawls $14$ feet up a wall. If Cheenu is standing $6$ feet from the wall, and is $6$ feet tall, how far must the spider jump to land on his head? [b]p9.[/b] There are fourteen dogs with long nails and twenty dogs with long fur. If there are thirty dogs in total, and three do not have long fur or long nails, how many dogs have both long hair and long nails? [u]Round 4[/u] [b]p10.[/b] Exactly $420$ non-overlapping square tiles, each $1$ inch by $1$ inch, tesselate a rectangle. What is the least possible number of inches in the perimeter of the rectangle? [b]p11.[/b] John drives $100$ miles at fifty miles per hour to see a cat. After he discovers that there was no cat, he drives back at a speed of twenty miles per hour. What was John’s average speed in the round trip? [b]p12.[/b] What percent of the numbers $1,2,3,...,1000$ are divisible by exactly one of the numbers $4$ and $5$? PS. You should use hide for answers. Rounds 5-8 have been posted [url=https://artofproblemsolving.com/community/c3h3165992p28809294]here [/url] and 9-12 [url=https://artofproblemsolving.com/community/c3h3166045p28809814]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Prove that for a given convex polygon of area $A$ and perimeter $P$ there exists a circle of radius $\frac AP$ that is contained in the interior of the polygon.

In some squares of a $2012\times 2012$ grid there are some beetles, such that no square contain more than one beetle. At one moment, all the beetles fly off the grid and then land on the grid again, also satisfying the condition that there is at most one beetle standing in each square. The vector from the centre of the square from which a beetle $B$ flies to the centre of the square on which it lands is called the [i]translation vector[/i] of beetle $B$. For all possible starting and ending configurations, find the maximum length of the sum of the [i]translation vectors[/i] of all beetles.

Let $O$ be the circumcenter of a triangle$ ABC$. Let $M$ be the midpoint of $AO$. The $BO$ and $CO$ intersect the altitude $AD$ at points $E$ and $F$,respectively. Let $O1$ and$ O2$ be the circumcenters of the triangle ABE and $ACF$, respectively. Prove that M lies on $O1O2$.

We are given convex quadrilateral $ABCD$. Each of its sides is divided into $N$ line segments of equal length. The points of division of side $AB$ are connected with the points of division of side $CD$ by straight lines (which we call the first set of straight lines), and the points of division of side BC are connected with the points of division of side $DA$ by straight lines (which we call the second set of straight lines) as shown in the diagram, which illustrates the case $N = 4$. This forms $N^2$ smaller quadrilaterals. From these we choose $N$ quadrilaterals in such a way that any two are at least divided by one line from the first set and one line from the second set. Prove that the sum of the areas of these chosen quadrilaterals is equal to the area of $ABCD$ divided by $N$. (A Andjans, Riga) [img]http://4.bp.blogspot.com/-8Qqk4r68nhU/XVco29-HzzI/AAAAAAAAKgo/UY8mXxg7tD0OrS6bEnoAw7Vuf31BuOE8wCK4BGAYYCw/s1600/TOT%2B1980%2BSpring%2BJ4.png[/img]

The lateral sides and diagonals of a trapezoid intersect a line $l$, determining three equal segments on it. Must $l$ be parallel to the bases of the trapezoid?

The points $A, B, M$ and $N$ are on a circle with center $O$ such that the radii $OA$ and $OB$ are perpendicular to each other, and $MN$ is parallel to $AB$ and intersects the radius $OA$ at $P$. Find the radius of the circle if $|MP|= 12$ and $|P N| = 2 \sqrt{14}$

Consider triangle $ABC$ with side lengths $AB = 4$, $BC = 7$, and $AC = 8$. Let $M$ be the midpoint of segment $AB$, and let $N$ be the point on the interior of segment $AC$ that also lies on the circumcircle of triangle $MBC$. Compute $BN$.