Let $n$ be a natural number greater than 2. $l$ is a line on a plane. There are $n$ distinct points $P_1$, $P_2$, …, $P_n$ on $l$. Let the product of distances between $P_i$ and the other $n-1$ points be $d_i$ ($i = 1, 2,$ …, $n$). There exists a point $Q$, which does not lie on $l$, on the plane. Let the distance from $Q$ to $P_i$ be $C_i$ ($i = 1, 2,$ …, $n$). Find $S_n = \sum_{i = 1}^{n} (-1)^{n-i} \frac{c_i^2}{d_i}$.

Let $d \ge 2$ be an integer. Prove that there exists a constant $C(d)$ such that the following holds: For any convex polytope $K\subset \mathbb{R}^d$, which is symmetric about the origin, and any $\varepsilon \in (0, 1)$, there exists a convex polytope $L \subset \mathbb{R}^d$ with at most $C(d) \varepsilon^{1-d}$ vertices such that \[(1-\varepsilon)K \subseteq L \subseteq K.\] Official definitions: For a real $\alpha,$ a set $T \in \mathbb{R}^d$ is a [i]convex polytope with at most $\alpha$ vertices[/i], if $T$ is a convex hull of a set $X \in \mathbb{R}^d$ of at most $\alpha$ points, i.e. $T = \{\sum\limits_{x\in X} t_x x | t_x \ge 0, \sum\limits_{x \in X} t_x = 1\}.$ Define $\alpha K = \{\alpha x | x \in K\}.$ A set $T \in \mathbb{R}^d$ is [i]symmetric about the origin[/i] if $(-1)T = T.$

Let $s$ be an arbitrary straight line passing through the incenter $I$ of the triangle $ABC$ . Line $s$ intersects lines $AB$ and $BC$ at points $D$ and $E$, respectively. Points $P$ and $Q$ are the centers of the circumscribed circles of triangles $DAI$ and $CEI$, respectively, and point $F$ is the second intersection point of these circles. Prove that the circumcircle of the triangle $PQF$ is always passes through a fixed point on the plane regardless of the position of the straight line $s$. (Matvii Kurskyi)

We have a rectangle with integer sides $m, n$ that is subdivided into $mn$ squares of side $1$. Find the number of little squares that are crossed by the diagonal (without counting those that are touched only in one vertex)

Let $A$ be one of the two points where the circles whose centers are the points $M$ and $N$ intersect. The tangents in $A$ to such circles intersect them again in $B$ and $C$, respectively. Let $P$ be a point such that the quadrilateral $AMPN$ is a parallelogram. Show that $P$ is the circumcenter of triangle $ABC$.

Show that for every odd integer $N>5$ there exist vectors $\bf u,v,w$ in (three-dimensional) space which are pairwise perpendicular, not parallel with any of the coordinate axes, have integer coordinates, and satisfy $N\bf =|u|=|v|=|w|.$ [i]Based on problem 2 of the 2018 Kürschák contest[/i]

A point $M$ is found inside a convex quadrilateral $ABCD$ such that triangles $AMB$ and $CMD$ are isoceles ($AM = MB, CM = MD$) and $\angle AMB= \angle CMD = 120^o$ . Prove that there exists a point N such that triangles$ BNC$ and $DNA$ are equilateral. (I.Sharygin)

$ABCD$ is a quadrilateral with right angles at $A$ and $C$. Points $E$ and $F$ are on $AC$, and $DE$ and $BF$ are perpendicular to $AC$. If $AE=3$, $DE=5$, and $CE=7$, then $BF=$ [asy] draw((0,0)--(10,0)--(3,-5)--(0,0)--(6.5,3)--(10,0)); draw((6.5,0)--(6.5,3)); draw((3,0)--(3,-5)); dot((0,0)); dot((10,0)); dot((3,0)); dot((3,-5)); dot((6.5,0)); dot((6.5,3)); label("A", (0,0), W); label("B", (6.5,3), N); label("C", (10,0), E); label("D", (3,-5), S); label("E", (3,0), N); label("F", (6.5,0), S);[/asy] $\text{(A)} \ 3.6 \qquad \text{(B)} \ 4 \qquad \text{(C)} \ 4.2 \qquad \text{(D)} \ 4.5 \qquad \text{(E)} \ 5$

Let $ABCD$ be a convex quadrilateral, where $R$ and $S$ are points in $DC$ and $AB$, respectively, such that $AD=RC$ and $BC=SA$. Let $P$, $Q$ and $M$ be the midpoints of $RD$, $BS$ and $CA$, respectively. If $\angle MPC + \angle MQA = 90$, prove that $ABCD$ is cyclic.

p1. It is known that $f$ is a function such that $f(x)+2f\left(\frac{1}{x}\right)=3x$ for every $x\ne 0$. Find the value of $x$ that satisfies $f(x) = f(-x)$. p2. It is known that ABC is an acute triangle whose vertices lie at circle centered at point $O$. Point $P$ lies on side $BC$ so that $AP$ is the altitude of triangle ABC. If $\angle ABC + 30^o \le \angle ACB$, prove that $\angle COP + \angle CAB < 90^o$. p3. Find all natural numbers $a, b$, and $c$ that are greater than $1$ and different, and fulfills the property that $abc$ divides evenly $bc + ac + ab + 2$. p4. Let $A, B$, and $ P$ be the nails planted on the board $ABP$ . The length of $AP = a$ units and $BP = b$ units. The board $ABP$ is placed on the paths $x_1x_2$ and $y_1y_2$ so that $A$ only moves freely along path $x_1x_2$ and only moves freely along the path $y_1y_2$ as in following image. Let $x$ be the distance from point $P$ to the path $y_1y_2$ and y is with respect to the path $x_1x_2$ . Show that the equation for the path of the point $P$ is $\frac{x^2}{b^2}+\frac{y^2}{a^2}=1$. [img]https://cdn.artofproblemsolving.com/attachments/4/6/d88c337370e8c3bc5a1833bc9588d3fb047bd0.png[/img] p5. There are three boxes $A, B$, and $C$ each containing $3$ colored white balls and $2$ red balls. Next, take three ball with the following rules: 1. Step 1 Take one ball from box $A$. 2. Step 2 $\bullet$ If the ball drawn from box $A$ in step 1 is white, then the ball is put into box $B$. Next from box $B$ one ball is drawn, if it is a white ball, then the ball is put into box $C$, whereas if the one drawn is red ball, then the ball is put in box $A$. $\bullet$ If the ball drawn from box $A$ in step 1 is red, then the ball is put into box $C$. Next from box $C$ one ball is taken. If what is drawn is a white ball then the ball is put into box $A$, whereas if the ball drawn is red, the ball is placed in box $B$. 3. Step 3 Take one ball each from squares $A, B$, and $C$. What is the probability that all the balls drawn in step 3 are colored red?

Two circles with radii $71$ and $100$ are externally tangent. Compute the largest possible area of a right triangle whose vertices are each on at least one of the circles.

[u]Round 5[/u] [b]p13.[/b] Determine the number of different circular bracelets can be made with $7$ beads, all either colored red or black. [b]p14.[/b] The product of $260$ and $n$ is a perfect square. The $2020$th least possible positive integer value of $n$ can be written as$ p^{e_1}_1 \cdot p^{e_2}_2\cdot p^{e_3}_3\cdot p^{e_4}_4$ . Find the sum $p_1 +p_2 +p_3 +p_4 +e_1 +e_2 +_e3 +e_4$. [b]p15.[/b] Let $B$ and $C$ be points along the circumference of circle $\omega$. Let $A$ be the intersection of the tangents at $B$ and $C$ and let $D \ne A$ be on $\overrightarrow{AC}$ such that $AC =CD = 6$. Given $\angle BAC = 60^o$, find the distance from point $D$ to the center of $\omega$. [u]Round 6[/u] [b]p16.[/b] Evaluate $\sqrt{2+\sqrt{2+\sqrt{2+...}}}$. [b]p17.[/b] Let $n(A)$ be the number of elements of set $A$ and $||A||$ be the number of subsets of set $A$. Given that $||A||+2||B|| = 2^{2020}$, find the value of $n(B)$. [b]p18.[/b] $a$ and $b$ are positive integers and $8^a9^b$ has $578$ factors. Find $ab$. [u]Round 7[/u] [b]p19.[/b] Determine the probability that a randomly chosen positive integer is relatively prime to $2019$. [b]p20.[/b] A $3$-by-$3$ grid of squares is to be numbered with the digits $1$ through $9$ such that each number is used once and no two even-numbered squares are adjacent. Determine the number of ways to number the grid. [b]p21.[/b] In $\vartriangle ABC$, point $D$ is on $AC$ so that $\frac{AD}{DC}= \frac{1}{13}$ . Let point $E$ be on $BC$, and let $F$ be the intersection of $AE$ and $BD$. If $\frac{DF}{FB}=\frac{2}{7}$ and the area of $\vartriangle DBC$ is $26$, compute the area of $\vartriangle F AB$. [u]Round 8[/u] [b]p22.[/b] A quarter circle with radius $1$ is located on a line with its horizontal base on the line and to the left of the vertical side. It is then rolled to the right until it reaches its original orientation. Determine the distance traveled by the center of the quarter circle. [b]p23.[/b] In $1734$, mathematician Leonhard Euler proved that $\frac{\pi^2}{6}=\frac11+\frac14+\frac19+\frac{1}{16}+...$. With this in mind, calculate the value of $\frac11-\frac14+\frac19-\frac{1}{16}+...$ (the series obtained by negating every other term of the original series). [b]p24.[/b] Billy the biker is competing in a bike show where he can do a variety of tricks. He knows that one trick is worth $2$ points, $1$ trick is worth $3$ points, and 1 is worth $5$ points, but he doesn’t remember which trick is worth what amount. When it’s Billy’s turn to perform, he does $6$ tricks, randomly choosing which trick to do. Compute the sum of all the possible values of points that Billy could receive in total. PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c3h3166016p28809598]here [/url] and 9-12 [url=https://artofproblemsolving.com/community/c3h3166115p28810631]here[/url].Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $ABCD$ be a parallelogram with $AC=BC.$ A point $P$ is chosen on the extension of ray $AB$ past $B.$ The circumcircle of $ACD$ meets the segment $PD$ again at $Q.$ The circumcircle of triangle $APQ$ meets the segment $PC$ at $R.$ Prove that lines $CD,AQ,BR$ are concurrent.

In the $9$-gon $ABCDEFGHI$, all sides have equal lengths and all angles are equal. Prove that $|AB| + |AC| = |AE|$. [img]https://cdn.artofproblemsolving.com/attachments/6/2/8c82e8a87bf8a557baaf6ac72b3d18d2ba3965.png[/img]

In rectangle $ABCD$, $AB = 3$ and $BC = 4$. If the feet of the perpendiculars from $B$ and $D$ to $AC$ are $X$ and $Y$ , the length of $X Y$ can be expressed in the form m/n , where m and n are relatively prime positive integers. Find $m +n$.

Inside the convex pentagon $ABCDE$, a point $O$ was chosen, and it turned out that all five triangles $AOB$, $BOC$, $COD$, $DOE$ and $EOA$ are congrunet to each other. Prove that these triangles are isosceles or right-angled.

Let $ABC$ be any triangle and $D, E, F$ the midpoints of $BC, CA, AB$. The medians $AD, BE$ and $CF$ intersect at point $S$. At least two of the quadrilaterals $AF SE, BDSF, CESD$ are cyclic. Show that the triangle $ABC$ is equilateral.

$ABC$ is triangle. Point $L$ is inside $ABC$ and lies on bisector of $\angle B$. $K$ is on $BL$. $\angle KAB=\angle LCB= \alpha$. Point $P$ inside triangle is such, that $AP=PC$ and $\angle APC=2\angle AKL$. Prove that $\angle KPL=2\alpha$

A rectangle is said to be $ inscribed$ in a triangle if all its vertices lie on the sides of the triangle. Prove that the locus of the centers (the meeting points of the diagonals) of all inscribed in an acute-angled triangle rectangles are three concurrent unclosed segments.

Let $ABC$ be an equilateral triangle with side length $1$. Points $A_1$ and $A_2$ are chosen on side $BC$, points $B_1$ and $B_2$ are chosen on side $CA$, and points $C_1$ and $C_2$ are chosen on side $AB$ such that $BA_1<BA_2$, $CB_1<CB_2$, and $AC_1<AC_2$. Suppose that the three line segments $B_1C_2$, $C_1A_2$, $A_1B_2$ are concurrent, and the perimeters of triangles $AB_2C_1$, $BC_2A_1$, and $CA_2B_1$ are all equal. Find all possible values of this common perimeter. [i]Ankan Bhattacharya[/i]

What is the greatest number of parts that $5$ spheres can divide the space into?

Let $ABC$ be a triangle with $AB\neq CB$. Let $C^{\prime}$ be a point on the ray $[AB$ such that $AC^{\prime}=CB$. Let $A^{\prime}$ be a point on the ray $[CB$ such that $CA^{\prime}=AB$. Let the circumcircles of triangles $ABA^{\prime}$ and $CBC^{\prime}$ intersect at a point $Q$ (apart from $B$). Prove that the line $BQ$ bisects the segment $CA$. Darij

Inside the square $ABCD$ there is the square $A'B' C'D'$ so that the segments $AA', BB', CC'$ and $DD'$ do not intersect each other neither the sides of the smaller square (the sides of the larger and the smaller square do not need to be parallel). Prove that the sum of areas of the quadrangles $AA'B' B$ and $CC'D'D$ is equal to the sum of areas of the quadrangles $BB'C'C$ and $DD'A'A$.

Let $ABC$ be an acute triangle. Let $ H $ be the foot of perpendicular from $ A $ to $ BC $. $ D, E $ are the points on $ AB, AC $ and let $ F, G $ be the foot of perpendicular from $ D, E $ to $ BC $. Assume that $ DG \cap EF $ is on $ AH $. Let $ P $ be the foot of perpendicular from $ E $ to $ DH $. Prove that $ \angle APE = \angle CPE $.

Let $ABC$ be an acute angled triangle with orthocenter $H$ and $AB<AC$. The point $T$ lies on line $BC$ so that $AT$ is a tangent to the circumcircle of $ABC$. Let lines $AH$ and $BC$ meet at point $D$ and let $M$ be the midpoint of $HC$. Let the circumcircle of $AHT$ meets $CH$ in $P \not=H$ and the circumcircle of $PDM$ meet $BC$ in $Q \not=D$. Prove that $QT=QA$.