Prove that there are different points $A_0 \,\, ,A_1 \,\, , \cdots A_{2550}$ on the $XY$ plane corresponding to the following properties simultaneously. (i) Any three points are not on the same line. (ii) If $ d(A_i,A_j)$ represents the distance between $A_i\,\, , A_j $ then $$ \sum_{0 \leq i < j \leq 2550}\{d(A_i,A_j)\} < 10^{-2008}$$ Note : $ \{x \}$ represents the decimal part of x e.g. $ \{ 3.16\} = 0.16$. [i] (passer-by)[/i]

Consider a tetrahedron$ A_1A_2A_3A_4$. A point $N$ is said to be a Servais point if its projections on the six edges of the tetrahedron lie in a plane $\alpha(N)$ (called Servais plane). Prove that if all the six points $Nij$ symmetric to a point $M$ with respect to the midpoints $Bij$ of the edges $A_iA_j$ are Servais points, then $M$ is contained in all Servais planes $\alpha(Nij )$

Given a cyclic quadrilateral $ABCD$ with circumcenter $O$, let the circle $(AOD)$ intersect the segments $AB$, $AC$, $DB$, $DC$ at $P$, $Q$, $R$, $S$ respectively. Suppose $X$ is the reflection of $D$ about $PQ$ and $Y$ is the reflection of $A$ about $RS$. Prove that the circles $(AOD)$, $(BPX)$, $(CSY)$ meet at a common point. [i]Proposed by Leia Mayssa & Ivan Chan Kai Chin[/i]

Let $ABC$ be a triangle, $P$ and $Q$ the intersections of the parallel line to $BC$ that passes through $A$ with the external angle bisectors of angles $B$ and $C$, respectively. The perpendicular to $BP$ at $P$ and the perpendicular to $CQ$ at $Q$ meet at $R$. Let $I$ be the incenter of $ABC$. Show that $AI = AR$.

The angles subtended by a tower at distances $100$, $200$ and $300$ from its foot sum to $90^o$. What is its height?

(F.Nilov) Given isosceles triangle $ ABC$ with base $ AC$ and $ \angle B \equal{} \alpha$. The arc $ AC$ constructed outside the triangle has angular measure equal to $ \beta$. Two lines passing through $ B$ divide the segment and the arc $ AC$ into three equal parts. Find the ratio $ \alpha / \beta$.

Let $A$ be a set consisting of 6 points in the plane. denoted $n(A)$ as the number of the unit circles which meet at least three points of $A$. Find the maximum of $n(A)$

We are given the square $ABCD$. On sides $AB$ and $CD$ we are given points $ K$ and $L$ respectively, and on segment $KL$ we are given point $M$ . Prove that the second intersection point (i.e. the one other than $M$) of the intersection points of circles circumscribed around triangles $AKM$ and $MLC$ lies on the diagonal $AC$. (V . N . Dubrovskiy)

Let $\omega$ be a circle with centre $O$. Let $\gamma$ be another circle passing through $O$ and intersecting $\omega$ at points $A$ and $B$. $A$ diameter $CD$ of $\omega$ intersects $\gamma$ at a point $P$ different from $O$. Prove that $\angle APC= \angle BPD$

Let $\vartriangle ABC$ be a triangle with $\angle BAC = 90^o$, $\angle ABC = 60^o$, and $\angle BCA = 30^o$ and $BC = 4$. Let the incircle of $\vartriangle ABC$ meet sides $BC$, $CA$, $AB$ at points $A_0$, $B_0$, $C_0$, respectively. Let $\omega_A$, $\omega_B$, $\omega_C$ denote the circumcircles of triangles $\vartriangle B_0IC_0$ , $\vartriangle C_0IA_0$ , $\vartriangle A_0IB_0$, respectively. We construct triangle $T_A$ as follows: let $A_0B_0$ meet $\omega_B$ for the second time at $A_1\ne A_0$, let $A_0C_0$ meet $\omega_C$ for the second time at $A_2\ne A_0$, and let $T_A$ denote the triangle $\vartriangle A_0A_1A_2$. Construct triangles $T_B$, $T_C$ similarly. If the sum of the areas of triangles $T_A$, $T_B$, $T_C$ equals $\sqrt{m} - n$ for positive integers $m$, $n$, find $m + n$.

[b]p1.[/b] A $10 \times 10$ array of squares is given. In each square, a student writes the product of the row number and the column number of the square (the upper left hand corner of this array is shown below). Determine the sum of the $100$ integers written in the array. [img]https://cdn.artofproblemsolving.com/attachments/5/9/527fdf90529221f6d06af169de1728da296538.png[/img] [b]p2.[/b] The equilateral triangle $DEF$ is inscribed in the equilateral triangle $ABC$ so that $ED$ is perpendicular to $BC$. If the area of $ABC$ equals one square unit, determine the area of $DEF$. [img]https://cdn.artofproblemsolving.com/attachments/c/0/6e1a303a45fa89576e26bc8fd30ce6564aaad1.png[/img] [b]p3.[/b] Consider a symmetric triangular set of points as shown (every point lies a distance of one unit from each of its neighbors). A collection of $m$ lines has the property that for every point in the arrangement, there is at least one line in the collection that passes through that point. Prove or disprove that $m \ge 10$. [img]https://cdn.artofproblemsolving.com/attachments/0/9/540f2781312f86672df1578bfe4f68b51d3b2c.png[/img] [b]p4.[/b] Let $P$ be a convex polygon drawn on graph paper (defined as the grid of all lines with equations $x = a$ and $y = b$, with $a$ and $b$ integers). We know that all the vertices of $P$ are at the intersections of grid lines and none of its sides is parallel to a grid line. Let $H$ be the sum of the lengths of the horizontal segments of the grid which are contained in the interior of $P$, and let $V$ be the sum of the lengths of the vertical segments of the grid in the interior of $P$. Prove that $H = V$ . [b]p5.[/b] Peter, Paul, and Mary play the following game. Given a fixed positive integer $k$ which is at most $2013$, they randomly choose a subset $A$ of $\{1, 2,..., 2013\}$ with $k$ elements. The winner is Peter, Paul, or Mary, respectively, if the sum of the numbers in $A$ leaves a remainder of $0$, $1$, or $2$ when divided by $3$. Determine the values of $k$ for which this game is fair (i.e., such that the three possible outcomes are equally likely). PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Given $n$ points $A_1, A_2, \ldots, A_n,$ no three collinear, show that the $n$- gon $A_1 A_2 \ldots A_n,$ is inscribed in a circle if and only if $A_1 A_2 \cdot A_3 A_n \cdot \ldots \cdot A_{n-1} A_n + A_2 A_3 \cdot A_4 A_n \cdot \ldots A_{n-1} A_n \cdot A_1 A_n + \ldots$ $+ A_{n-1} A_{n-2} \cdot A_1 A_n \cdot \ldots \cdot A_{n-3} A_n$ $= A_1 A_{n-1} \cdot A_2 A_n \cdot \ldots \cdot A_{n-2} A_n$, where $XY$ denotes the length of the segment $XY.$

The triangle $ABC$ satisfies $0\le AB\le 1\le BC\le 2\le CA\le 3$. What is the maximum area it can have?

Square $ABCD$ has side length $5$. Draw E on $BC$ and $F$ on $AD$ such that $BE < AF$. Next, flip $ABCD$ across $EF$ to a square $A'B'C'D'$ such that $C'$ lies in the interior of $ABCD$ and $C$ lies in the interior of $A'B'C'D'$. Suppose that $CC' = 4$ and $DD' = 2$. Compute $AA'$.

In a square of area $1$ there are situated $2024$ polygons whose total area is greater than $2023$. Prove that they have a point in common.

In $\triangle{PQR}$, $PQ=4$, $PR=5$, and $QR=6$. Assume that an equilateral hexagon $ABCDEF$ is able to be drawn inside $\triangle{PQR}$ so that $\overline{AB}$ is parallel to $\overline{QR}$, $\overline{CD}$ is parallel to $\overline{PQ}$, $\overline{EF}$ is parallel to $\overline{RP}$, $\overline{BC}$ lies on $\overline{RP}$, $\overline{DE}$ lies on $\overline{QR}$, and $\overline{AF}$ lies on $\overline{PQ}$. Find the area of hexagon $ABCDEF$.

Let $ABC$ be an acute triangle such that $\angle{B}=60$. The circle with diameter $AC$ intersects the internal angle bisectors of $A$ and $C$ at the points $M$ and $N$, respectively $(M\neq{A},$ $N\neq{C})$. The internal bisector of $\angle{B}$ intersects $MN$ and $AC$ at the points $R$ and $S$, respectively. Prove that $BR\leq{RS}$.

The circle $\omega_1$ passes through the vertex $A$ of the parallelogram $ABCD$ and touches the rays $CB, CD$. The circle $\omega_2$ touches the rays $AB, AD$ and touches $\omega_1$ externally at point $T$. Prove that $T$ lies on the diagonal $AC$

Suppose we are given a $19\times19$ chessboard (a table with $19^2$ squares) and remove the central square. Is it possible to tile the remaining $19^2-1=360$ squares with $4\times1$ and $1\times4$ rectangles? (So that each of the $360$ squares is covered by exactly one rectangle.) Justify your answer.

Given triangle $ABC$ with $|AC| > |BC|$. The point $M$ lies on the angle bisector of angle $C$, and $BM$ is perpendicular to the angle bisector. Prove that the area of triangle AMC is half of the area of triangle $ABC$. [img]https://cdn.artofproblemsolving.com/attachments/4/2/1b541b76ec4a9c052b8866acbfea9a0ce04b56.png[/img]

Prove that for each triangle, there exists a vertex, such that with the two sides starting from that vertex and each cevian starting from that vertex, is possible to construct a triangle.

Prove that for any tetrahedron the radius of the inscribed sphere $r <\frac{ ab}{ 2(a + b)}$, where $a$ and $b$ are the lengths of any pair of opposite edges.

$ABCD$ is a circumscribed and inscribed quadrilateral. $O$ is the circumcenter of the quadrilateral. $E,F$ and $S$ are the intersections of $AB,CD$ , $AD,BC$ and $AC,BD$ respectively. $E'$ and $F'$ are points on $AD$ and $AB$ such that $A\hat{E}E'=E'\hat{E}D$ and $A\hat{F}F'=F'\hat{F}B$. $X$ and $Y$ are points on $OE'$ and $OF'$ such that $\frac{XA}{XD}=\frac{EA}{ED}$ and $\frac{YA}{YB}=\frac{FA}{FB}$. $M$ is the midpoint of arc $BD$ of $(O)$ which contains $A$. Prove that the circumcircles of triangles $OXY$ and $OAM$ are coaxal with the circle with diameter $OS$.

Given a billiard in the form of a regular $1998$-gon $A_1A_2...A_{1998}$. A ball was released from the midpoint of side $A_1A_2$, which, reflected therefore from sides $A_2A_3$, $A_3A_4$, . . . , $A_{1998}A_1$ (according to the law, the angle of incidence is equal to the angle of reflection), returned to the starting point. Prove that the trajectory of the ball is a regular $1998$-gon.

Triangle $\triangle ABC$ in the figure has area $10$. Points $D$, $E$ and $F$, all distinct from $A$, $B$ and $C$, are on sides $AB$, $BC$ and $CA$ respectively, and $AD = 2$, $DB = 3$. If triangle $\triangle ABE$ and quadrilateral $DBEF$ have equal areas, then that area is [asy] size(200); defaultpen(linewidth(0.7)+fontsize(10)); pair A=origin, B=(10,0), C=(8,7), F=7*dir(A--C), E=(10,0)+4*dir(B--C), D=4*dir(A--B); draw(A--B--C--A--E--F--D); pair point=incenter(A,B,C); label("$A$", A, dir(point--A)); label("$B$", B, dir(point--B)); label("$C$", C, dir(point--C)); label("$D$", D, dir(point--D)); label("$E$", E, dir(point--E)); label("$F$", F, dir(point--F)); label("$2$", (2,0), S); label("$3$", (7,0), S);[/asy] $ \textbf{(A)}\ 4\qquad\textbf{(B)}\ 5\qquad\textbf{(C)}\ 6\qquad\textbf{(D)}\ \frac{5}{3}\sqrt{10}\qquad\textbf{(E)}\ \text{not uniquely determined}$