Two vertices $A,B$ of a triangle $ABC$ are located on a parabola $y=ax^2 + bx + c$ with $a>0$ in such a way that the sides $AC,BC$ are tangent to the parabola. Let $m_c$ be the length of the median $CC_1$ of triangle $ABC$ and $S$ be the area of triangle $ABC$. Find \[\frac{S^2}{m_c^3}\]

Consider a unit square $ABCD$ and a (variable) equilateral triangle $XYZ$ such that $X, Z$ lie on rays $AB, DC,$ respectively, and $Y$ lies on segment $AD$. Compute the area of triangle $XYZ$ in terms of $x=AX$ and determine its maximum and minimum.

Let $ABCD$ be a convex quadrilateral. The perpendicular bisectors of the sides $[AD]$ and $[BC]$ intersect at a point $P$ inside the quadrilateral and the perpendicular bisectors of the sides $[AB]$ and $[CD]$ also intersect at a point $Q$ inside the quadrilateral. Show that, if $\angle APD = \angle BPC$ then $\angle AQB = \angle CQD$

In the parallelogram $ABCD$, $\angle BAD =60 ^ \circ$. Let $E $ be the intersection point of the diagonals. The circle circumscribed to the triangle $ACD$ intersects the line $AB$ at the point $K$ (different from $A$), the line $BD$ at the point $P$ (different from $D$), and to the line $BC$ in $L$ (different from $C$). The line $EP$ intersects the circumscribed circle of the triangle $CEL$ at the points $E$ and $M$. Show that the triangles $KLM$ and $CAP$ are congruent.

Points $A, B, C, D$ and $E$ are located on the plane. It is known that $CA = 12$, $AB = 8$, $BC = 4$, $CD = 5$, $DB = 3$, $BE = 6$ and $ED = 3$. Find the length of $AE$.

In the diagram below, all circles are tangent to each other as shown. The six outer circles are all congruent to each other, and the six inner circles are all congruent to each other. Compute the ratio of the area of one of the outer circles to the area of one of the inner circles. [img]https://cdn.artofproblemsolving.com/attachments/b/6/4cfbc1df86b8d38e082b7ad0a71b9e366548b3.png[/img]

A parallelopiped has vertices $A_1, A_2, ... , A_8$ and center $O$. Show that: \[ 4 \sum_{i=1}^8 OA_i ^2 \leq \left(\sum_{i=1}^8 OA_i \right) ^2 \]

Let $D$ be an interior point of the acute triangle $ABC$ with $AB > AC$ so that $\angle DAB = \angle CAD.$ The point $E$ on the segment $AC$ satisfies $\angle ADE =\angle BCD,$ the point $F$ on the segment $AB$ satisfies $\angle FDA =\angle DBC,$ and the point $X$ on the line $AC$ satisfies $CX = BX.$ Let $O_1$ and $O_2$ be the circumcenters of the triangles $ADC$ and $EXD,$ respectively. Prove that the lines $BC, EF,$ and $O_1O_2$ are concurrent.

Let $ABC$ be a triangle, $O$ its circumcenter, $S$ its centroid, and $H$ its orthocenter. Denote by $A_1, B_1$, and $C_1$ the centers of the circles circumscribed about the triangles $CHB, CHA$, and $AHB$, respectively. Prove that the triangle $ABC$ is congruent to the triangle $A_1B_1C_1$ and that the nine-point circle of $\triangle ABC$ is also the nine-point circle of $\triangle A_1B_1C_1$.

A unit equilateral triangle is given. Divide each side into three equal parts. Remove the equilateral triangles whose bases are middle one-third segments. Now we have a new polygon. Remove the equilateral triangles whose bases are middle one-third segments of the sides of the polygon. After repeating these steps for infinite times, what is the area of the new shape? $ \textbf{(A)}\ \dfrac {1}{2\sqrt 3} \qquad\textbf{(B)}\ \dfrac {\sqrt 3}{8} \qquad\textbf{(C)}\ \dfrac {\sqrt 3}{10} \qquad\textbf{(D)}\ \dfrac {1}{4\sqrt 3} \qquad\textbf{(E)}\ \text{None of the above} $

Point $ O $ is the center of the circumscribed circle of an acute triangle $ Abc $. A certain circle passes through the points $ B $ and $ C $ and intersects sides $ AB $ and $ AC $ of a triangle. On its arc lying inside the triangle, points $ D $ and $ E $ are chosen so that the segments $ BD $ and $ CE $ pass through the point $ O $. Perpendicular $ DD_1 $ to $ AB $ side and perpendicular $ EE_1 $ to $ AC $ side intersect at $ M $. Prove that the points $ A $, $ M $ and $ O $ lie on the same straight line.

In $\triangle ABC$, $AB<AC$. $D$ and $P$ are the feet of the internal and external angle bisectors of $\angle BAC$, respectively. $M$ is the midpoint of segment $BC$, and $\omega$ is the circumcircle of $\triangle APD$. Suppose $Q$ is on the minor arc $AD$ of $\omega$ such that $MQ$ is tangent to $\omega$. $QB$ meets $\omega$ again at $R$, and the line through $R$ perpendicular to $BC$ meets $PQ$ at $S$. Prove $SD$ is tangent to the circumcircle of $\triangle QDM$. [i]Proposed by Ray Li[/i]

Find the greatest natural number $N$ such that, for any arrangement of the numbers $1, 2, \ldots, 400$ in a chessboard $20 \times 20$, there exist two numbers in the same row or column, which differ by at least $N.$

a) What is the least number of points that can be chosen on a circle of length $1956$, so that for each of these points there is exactly one chosen point at distance $1$, and exactly one chosen point at distance $2$ (distances are measured along the circle)? b) On a circle of length $15$ there are selected $n$ points such that for each of them there is exactly one selected point at distance $1$ from it, and exactly one is selected point at distance $2$ from it. (All distances are measured along the circle.) Prove that $n$ is divisible by $10$.

In the right triangle $ABC$, the perpendicular sides $AB$ and $BC$ have lengths $3$ cm and $4$ cm, respectively. Let $M$ be the midpoint of the side $AC$ and let $D$ be a point, distinct from $A$, such that $BM = MD$ and $AB = BD$. a) Prove that $BM$ is perpendicular to $AD$. b) Calculate the area of the quadrilateral $ABDC$.

Let ABC be an acute triangle such that $|AB|=|AC|$ . Let D be a point on AB such that $<ACD = <CBD$. Let E be midpoint of BD and S be circumcenter of BCD. Prove that A,E,S,C are cyclic

For which positive integers $n \ge 4$ one can find n points in the plane, no three collinear, such that for each triangle formed with three of the $n$ points which are on the convex hull, exactly one of the $n - 3$ remaining points belongs to its interior.

In square $ABCD,$ points $E$ and $F$ are chosen in the interior of sides $BC$ and $CD$, respectively. The line drawn from $F$ perpendicular to $AE$ passes through the intersection point $G$ of $AE$ and diagonal $BD$. A point $K$ is chosen on $FG$ such that $|AK|= |EF|$. Find $\angle EKF.$

In a regular tetrahedron the centers of the four faces are the vertices of a smaller tetrahedron. The ratio of the volume of the smaller tetrahedron to that of the larger is $m/n$, where $m$ and $n$ are relatively prime positive integers. Find $m+n.$

Quadrilateral $ABCD$ is circumscribed. Its incircle touches sides $AB, BC, CD, DA$ in points $K, L, M, N$ respectively. Points $A', B', C', D'$ are the midpoints of segments $LM, MN, NK, KL$. Prove that the quadrilateral formed by lines $AA', BB', CC', DD'$ is cyclic.

In the 27 points of a cube: 8 vertexes, 12 midpoints of edges, 6 centers of surfaces, and the center of the cube, the number of groups of three collinear points is $\text{(A)}57\qquad\text{(B)}49\qquad\text{(C)}43\qquad\text{(D)}37$

Let $D$ be a unique point on segment $BC$, in $ABC$. If $AD^2 = BD \cdot CD$, show that $AB + AC = \sqrt{2}BC$.

[b]p1.[/b] One chilly morning, $10$ penguins ate a total of $50$ fish. No fish was shared by two or more penguins. Assuming that each penguin ate at least one fish, prove that at least two penguins ate the same number of fish. [b]p2.[/b] A triangle of area $1$ has sides of lengths $a > b > c$. Prove that $b > 2^{1/2}$. [b]p3.[/b] Imagine ducks as points in a plane. Three ducks are said to be in a row if a straight line passes through all three ducks. Three ducks, Huey, Dewey, and Louie, each waddle along a different straight line in the plane, each at his own constant speed. Although their paths may cross, the ducks never bump into each other. Prove: If at three separate times the ducks are in a row, then they are always in a row. [b]p4.[/b] Two computers and a number of humans participated in a large round-robin chess tournament (i.e., every participant played every other participant exactly once). In every game, the winner of the game received one point, the loser zero. If a game ended in a draw, each player received half a point. At the end of the tournament, the sum of the two computers' scores was $38$ points, and all of the human participants finished with the same total score. Describe (with proof) ALL POSSIBLE numbers of humans that could have participated in such a tournament. [b]p5.[/b] One thousand cows labeled $000$, $001$,$...$, $998$, $999$ are requested to enter $100$ empty barns labeled $00$, $01$,$...$,$98$, $99$. One hundred Dalmatians - one at the door of each barn - enforce the following rule: In order for a cow to enter a barn, the label of the barn must be obtainable from the label of the cow by deleting one of the digits. For example, the cow labeled $357$ would be admitted into any of the barns labeled $35$, $37$ or $57$, but would not admitted into any other barns. a) Demonstrate that there is a way for all $1000$ cows to enter the barns so that at least $50$ of the barns remain empty. b) Prove that no matter how they distribute themselves, after all $1000$ cows enter the barns, at most $50$ of the barns will remain empty. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

In triangle $ABC$,$AB<AC$ and $\omega$ is incirle.The $A$-excircle is tangent to $BC$ at $A^\prime$.Point $X$ lies on $AA^\prime$ such that segment $A^\prime X$ doesn't intersect with $\omega$.The tangents from $X$ to $\omega$ intersect with $BC$ at $Y,Z$.Prove that the sum $XY+XZ$ not depends to point $X$.(Mitrofanov)

Given a positive integer $n \ge 2$, let $\mathcal{P}$ and $\mathcal{Q}$ be two sets, each consisting of $n$ points in three-dimensional space. Suppose that these $2n$ points are distinct. Show that it is possible to label the points of $\mathcal{P}$ as $P_1,P_2,\dots,P_n$ and the points of $\mathcal{Q}$ as $Q_1,Q_2,\dots,Q_n$ such that for any indices $i$ and $j$, the balls of diameters $P_iQ_i$ and $P_jQ_j$ have at least one common point.