Let ABCD be a square and K be a circle with diameter AB. For an arbitrary point P on side CD, segments AP and BP meet K again at points M and N, respectively, and lines DM and CN meet at point Q. Prove that Q lies on the circle K and that AQ : QB = DP : PC.

Point $O$ is the origin of a space. Points $A_1, A_2,\dots, A_n$ have nonnegative coordinates. Prove the following inequality: $$|\overrightarrow{OA_1}|+|\overrightarrow {OA_2}|+\dots+|\overrightarrow {OA_n}|\leq \sqrt{3}|\overrightarrow {OA_1}+\overrightarrow{OA_2}+\dots+\overrightarrow{OA_n}|$$ [I]Proposed by A. Khrabrov[/i]

The point $D$ at the side $AB$ of triangle $ABC$ is given. Construct points $E,F$ at sides $BC, AC$ respectively such that the midpoints of $DE$ and $DF$ are collinear with $B$ and the midpoints of $DE$ and $EF$ are collinear with $C.$

The two squares shown share the same center $O$ and have sides of length 1. The length of $\overline{AB}$ is $43/99$ and the area of octagon $ABCDEFGH$ is $m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$ [asy] real alpha = 25; pair W=dir(225), X=dir(315), Y=dir(45), Z=dir(135), O=origin; pair w=dir(alpha)*W, x=dir(alpha)*X, y=dir(alpha)*Y, z=dir(alpha)*Z; draw(W--X--Y--Z--cycle^^w--x--y--z--cycle); pair A=intersectionpoint(Y--Z, y--z), C=intersectionpoint(Y--X, y--x), E=intersectionpoint(W--X, w--x), G=intersectionpoint(W--Z, w--z), B=intersectionpoint(Y--Z, y--x), D=intersectionpoint(Y--X, w--x), F=intersectionpoint(W--X, w--z), H=intersectionpoint(W--Z, y--z); dot(O); label("$O$", O, SE); label("$A$", A, dir(O--A)); label("$B$", B, dir(O--B)); label("$C$", C, dir(O--C)); label("$D$", D, dir(O--D)); label("$E$", E, dir(O--E)); label("$F$", F, dir(O--F)); label("$G$", G, dir(O--G)); label("$H$", H, dir(O--H));[/asy]

A rectangle has perimeter $10$ and diagonal $\sqrt{15}$. What is its area?

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.

[u]Round 9[/u] [b]p25.[/b] Maisy the Bear is at the origin of the Cartesian Plane. WhenMaisy is on the point $(m,n)$ then it can jump to either $(m,n +1)$ or $(m+1,n)$. Let $L(x, y)$ be the number of pathsMaisy can take to reach the point $(x, y)$. The sum of $L(x, y)$ over all lattice points $(x, y)$ with both coordinates between $0$ and $2020$, inclusive, can be written as ${2k \choose k} - j$ for a minimum positive integer k and corresponding positive integer $j$ . Find $k + j$ . [b]p26.[/b] A circle with center $O$ and radius $2$ and a circle with center $P$ and radius $3$ are externally tangent at $A$. Points $B$ and $C$ are on the circle with center $O$ such that $\vartriangle ABC$ is equilateral. Segment $AB$ extends past B to point $D$ and $AC$ extends past $C$ to point $E$ such that $BD = CE =\sqrt3$. A line through $D$ is tangent to circle $P$ at $F$. The value of $EF^2$ can be expressed as $\frac{a+b\sqrt{c}}{d}$ where $a$, $b$, $c$, and $d$ are integers, c is squarefree, and $gcd(a,b,d) = 1$. Find $a +b +c +d$. [b]p27.[/b] Find the number of trailing zeroes at the end of $$\sum^{2021}_{i=1}(2021^i -1) = (2021^1 -1)...(2021^{2021}-1).$$ [u]Round 10[/u] [b]p28.[/b] Points $A, B, C, P$, and $D$ lie on circle ω in that order. Let $AC$ and $BD$ intersect at $I$ . Given that $PI = PC = PD$, $\angle DAB = 137^o$, and $\angle ABC = 109^o$, find the measure of $\angle BIC$ in degrees. [b]p29.[/b] Find the sum of all positive integers $n < 2021$ such that when ${d_1,d_2,... ,d_k}$ are the positive integer factors of $n$, then $$\left( \sum^{k}_{i=1}d_i \right) \left( \sum^{k}_{i=1} \frac{1}{d_i} \right)= r^2$$ for some rational number $r$ . [b]p30.[/b] Let $a, b, c, d$ and $e$ be positive real numbers. Define the function $f (x, y) = \frac{x}{y}+\frac{y}{x}$ for all positive real numbers. Given that $f (a,b) = 7$, $f (b,c) = 5$, $f (c,d) = 3$, and $f (d,e) = 2$, find the sum of all possible values of $f (e,a)$. [u]Round 11[/u] [b]p31.[/b] There exist $100$ (not necessarily distinct) complex numbers $r_1, r_2,..., r_{100}$ such that for any positive integer $1 \le k \le 100$, we have that $P(r_k ) = 0$ where the polynomial $P$ is defined as $$P(x) = \sum^{101}_{i=1}i \cdot x^{101-i} = x^{100} +2x^{99} +3x^{98} +...+99x^2 +100x +101.$$ Find the value of $$\prod^{100}_{j=1} (r^2_j+1) = (r^2_1 +1)(r^2_2 +1)...(r^2_{100} +1).$$ [b]p32.[/b] Let $BT$ be the diameter of a circle $\omega_1$, and $AT$ be a tangent of $\omega_1$. Line $AB$ intersects $\omega_1$ at $C$, and $\vartriangle ACT$ has circumcircle $\omega_2$. Points $P$ and $S$ exist such that $PA$ and $PC$ are tangent to $\omega_2$ and $SB = BT = 20$. Given that $AT = 15$, the length of $PS$ can be written as $\frac{a\sqrt{b}}{c}$ , where $a$, $b$, and $c$ are positive integers, $b$ is squarefree, and $gcd(a,b) = 1$. Find $a +b +c$. [b]p33.[/b] There are a hundred students in math team. Each pair of students are either mutually friends or mutually enemies. It is given that if any three students are chosen, then they are not all mutually friends. The maximum possible number of ways to choose four students such that it is possible to label them $A, B, C$, and $D$ such that $A$ and $B$ are friends, $B$ and $C$ are friends, $C$ and $D$ are friends, and D and A are friends can be expressed as $n^4$. Find $n$. [u]Round 12[/u] [b]p34.[/b] Let $\{p_i\}$ be the prime numbers, such that $p_1 = 2, p_2 = 3, p_3 = 5, ...$ For each $i$ , let $q_i$ be the nearest perfect square to $p_i$ . Estimate $\sum^{2021}_{i=1}|p_i=q_i |$. If the correct answer is $A$ and your answer is $E$, your score will be $\left \lfloor 30 \cdot \max - \left(0,1-5 \cdot \left| \log_{10} \frac{A}{E} \right| \right)\right \rfloor.$ [b]p35.[/b] Estimate the number of digits of $(2021!)^{2021}$. If the correct answer is $A$ and your answer is $E$, your score will be $\left \lfloor 15 \cdot \max \left(0,2- \cdot \left| \log_{10} \frac{A}{E} \right| \right) \right \rfloor.$ [b]p36.[/b] Pick a positive integer between$ 1$ and $1000$, inclusive. If your answer is $E$ and a quarter of the mean of all the responses to this problem is $A$, your score will be $$ \lfloor \max \left(0,30- |A-E|, 2-|E-1000| \right) \rfloor.$$ Note that if you pick $1000$, you will automatically get $2$ points. PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c3h3166489p28814241]here [/url] and 5-8 [url=https://artofproblemsolving.com/community/c3h3166494p28814284]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Determine the smallest positive real number $ k$ with the following property. Let $ ABCD$ be a convex quadrilateral, and let points $ A_1$, $ B_1$, $ C_1$, and $ D_1$ lie on sides $ AB$, $ BC$, $ CD$, and $ DA$, respectively. Consider the areas of triangles $ AA_1D_1$, $ BB_1A_1$, $ CC_1B_1$ and $ DD_1C_1$; let $ S$ be the sum of the two smallest ones, and let $ S_1$ be the area of quadrilateral $ A_1B_1C_1D_1$. Then we always have $ kS_1\ge S$. [i]Author: Zuming Feng and Oleg Golberg, USA[/i]

Let $\vartriangle ABC$ be a right triangle with $\angle B = 90^o$. Let $P$ be a point on side $BC$, and let $\omega$ be the circle with diameter $CP$. Suppose that $\omega$ intersects $AC $and $AP$ again at $Q$ and $R$, respectively. Show that $CP^2 = AC \cdot CQ - AP \cdot P R$.

Let $ABCD$ be a convex quadrilateral with $\angle B= \angle D$. Prove that the midpoint of $BD$ lies on the common internal tangent to the incircles of triangles $ABC$ and $ACD$.

Is there a triangle with $12 \, cm^2$ area and $12$ cm perimeter?

a) What can be the number of symmetry axes of a checked polygon, that is, of a polygon whose sides lie on lines of a list of checked paper? (Indicate all possible values.) b) What can be the number of symmetry axes of a checked polyhedron, that is, of a polyhedron consisting of equal cubes which border one to another by plane facets?

Let $ A$, $ B$, $ C$, $ A^{\prime}$, $ B^{\prime}$, $ C^{\prime}$, $ X$, $ Y$, $ Z$, $ X^{\prime}$, $ Y^{\prime}$, $ Z^{\prime}$ and $ P$ be pairwise distinct points in space such that $ A^{\prime} \in BC;\ B^{\prime}\in CA;\ C^{\prime}\in AB;\ X^{\prime}\in YZ;\ Y^{\prime}\in ZX;\ Z^{\prime}\in XY;$ $ P \in AX;\ P\in BY;\ P\in CZ;\ P\in A^{\prime}X^{\prime};\ P\in B^{\prime}Y^{\prime};\ P\in C^{\prime}Z^{\prime}$. Prove that $ \frac {BA^{\prime}}{A^{\prime}C}\cdot\frac {CB^{\prime}}{B^{\prime}A}\cdot\frac {AC^{\prime}}{C^{\prime}B} \equal{} \frac {YX^{\prime}}{X^{\prime}Z}\cdot\frac {ZY^{\prime}}{Y^{\prime}X}\cdot\frac {XZ^{\prime}}{Z^{\prime}Y}$.

Let $\omega$ be a semicircle and $AB$ its diameter. $\omega_1$ and $\omega_2$ are two different circles, both tangent to $\omega$ and to $AB$, and $\omega_1$ is also tangent to $\omega_2$. Let $P,Q$ be the tangent points of $\omega_1$ and $\omega_2$ to $AB$ respectively, and $P$ is between $A$ and $Q$. Let $C$ be the tangent point of $\omega_1$ and $\omega$. Find $\tan\angle ACQ$.

Two circles $\Gamma_1$ and $\Gamma_2$ meet at two distinct points $A$ and $B$. A line passing through $A$ meets $\Gamma_1$ and $\Gamma_2$ again at $C$ and $D$ respectively, such that $A$ lies between $C$ and $D$. The tangent at $A$ to $\Gamma_2$ meets $\Gamma_1$ again at $E$. Let $F$ be a point on $\Gamma_2$ such that $F$ and $A$ lie on different sides of $BD$, and $2\angle AFC=\angle ABC$. Prove that the tangent at $F$ to $\Gamma_2$, and lines $BD$ and $CE$ are concurrent.

Cells of a $2000\times2000$ board are colored according to the following rules: 1)At any moment a cell can be colored, if none of its neighbors are colored 2)At any moment a $1\times2$ rectangle can be colored, if exactly two of its neighbors are colored. 3)At any moment a $2\times2$ squared can be colored, if 8 of its neighbors are colored (Two cells are considered to be neighboring, if they share a common side). Can the entire $2000\times2000$ board be colored? [I]Proposed by K. Kohas[/i]

Let $ABCD$ be a rectangle with diagonals of length $10.$ Let $P$ be the midpoint of $\overline{AD},$ $S$ be the midpoint of $\overline{BC},$ and $T$ be the midpoint of $\overline{CD}.$ Points $Q$ and $R$ are chosen on $\overline{AB}$ such that $AP=AQ$ and $BR=BS,$ and minor arcs $\widehat{PQ}$ and $\widehat{RS}$ centered at $A$ and $B,$ respectively, are drawn. Circle $\omega$ is tangent to $\overline{CD}$ at $T$ and externally tangent to $\widehat{PQ}$ and $\widehat{RS}.$ Suppose that the radius of $\omega$ is $\tfrac{43}{18}.$ Then the sum of all possible values of the area of $ABCD$ can be written in the form $\tfrac{a+b\sqrt{c}}{d},$ where $a,\ b,\ c,$ and $d$ are positive integers, $b$ and $d$ are relatively prime, and $c$ is prime. Find $a+b+c+d.$

Let $ABCD$ be a convex quadrilateral with $\angle{BAD} = 2\angle{BCD}$ and $AB = AD$. Let $P$ be a point such that $ABCP$ is a parallelogram. Prove that $CP = DP$.

[b]p1.[/b] Two players play the following game. On the lowest left square of an $8\times 8$ chessboard there is a rook. The first player is allowed to move the rook up or to the right by an arbitrary number of squares. The second player is also allowed to move the rook up or to the right by an arbitrary number of squares. Then the first player is allowed to do this again, and so on. The one who moves the rook to the upper right square wins. Who has a winning strategy? [b]p2.[/b] In Crocodile Country there are banknotes of $1$ dollar, $10$ dollars, $100$ dollars, and $1,000$ dollars. Is it possible to get 1,000,000 dollars by using $250,000$ banknotes? [b]p3.[/b] Fifteen positive numbers (not necessarily whole numbers) are placed around the circle. It is known that the sum of every four consecutive numbers is $30$. Prove that each number is less than $15$. [b]p4.[/b] Donald Duck has $100$ sticks, each of which has length $1$ cm or $3$ cm. Prove that he can break into $2$ pieces no more than one stick, after which he can compose a rectangle using all sticks. [b]p5.[/b] Three consecutive $2$ digit numbers are written next to each other. It turns out that the resulting $6$ digit number is divisible by $17$. Find all such numbers. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Are the following statements true? a) if the four vertices of a rectangle lie on the four sides of a rhombus, then the sides of the rectangle are parallel to the diagonals of the rhombus; b) if the four vertices of a square lie on the four sides of a rhombus that is not a square, then the sides of the square are parallel to the diagonals of the rhombus.

Let $\vartriangle ABC$ be a right triangle with $\angle ABC = 90^o$. Let the circle with diameter $BC$ intersect $AC$ at $D$. Let the tangent to this circle at $D$ intersect $AB$ at $E$. What is the value of $\frac{AE}{BE}$ ?

Let $ABC$ be a fixed acute-angled triangle. Consider some points $E$ and $F$ lying on the sides $AC$ and $AB$, respectively, and let $M$ be the midpoint of $EF$. Let the perpendicular bisector of $EF$ intersect the line $BC$ at $K$, and let the perpendicular bisector of $MK$ intersect the lines $AC$ and $AB$ at $S$ and $T$, respectively. If the quadrilateral $KSAT$ is cycle, prove that $\angle{KEF}=\angle{KFE}=\angle{A}$.

In the plane are fixed two internally tangent circles $\omega$ and $\Omega$, so that $\omega$ is inside $\Omega$. Denote their common point by $T$. The point $A \neq T$ moves on $\Omega$ and point $B$ on $\Omega$ is such that $AB$ is tangent to $\omega$. The line through $B$, perpendicular to $AB$, meets the external angle bisector of $\angle ATB$ at $P$. Prove that, as $A$ varies on $\Omega$, the line $AP$ passes through a fixed point.

The ratio of the radii of two concentric circles is $1:3$. If $\overline{AC}$ is a diameter of the larger circle, $\overline{BC}$ is a chord of the larger circle that is tangent to the smaller circle, and $AB = 12$, then the radius of the larger circle is [asy] size(200); defaultpen(linewidth(0.7)+fontsize(10)); pair O=origin, A=3*dir(180), B=3*dir(140), C=3*dir(0); dot(O); draw(Arc(origin,1,0,360)); draw(Arc(origin,3,0,360)); draw(A--B--C--A); label("$A$", A, dir(O--A)); label("$B$", B, dir(O--B)); label("$C$", C, dir(O--C)); [/asy] $ \textbf{(A)}\ 13\qquad\textbf{(B)}\ 18\qquad\textbf{(C)}\ 21\qquad\textbf{(D)}\ 24\qquad\textbf{(E)}\ 26 $

The integers from $1$ to $9$ can be arranged into a $3\times3$ array (as shown on the right) so that the sum of the numbers in every row, column, and diago­nal is a multiple of $9$. (a.) Prove that the number in the center of the array must be a multiple of $3$. (b.) Give an example of such an array with $6$ in the center. [asy] defaultpen(linewidth(0.7)+fontsize(10));size(100); int i,j; for(i=0; i<4; i=i+1) { draw((0,2i)--(6,2i)); draw((2i,0)--(2i,6)); } string[] letters={"G", "H", "I", "D", "E", "F", "A", "B", "C"}; for(i=0; i<3; i=i+1) { for(j=0; j<3; j=j+1) { label(letters[3i+j], (2j+1, 2i+1)); }}[/asy]