A “large” circular disk is attached to a vertical wall. It rotates clockwise with one revolution per minute. An insect lands on the disk and immediately starts to climb vertically upward with constant speed $\frac{\pi}{3}$ cm per second (relative to the disk). Describe the path of the insect [i](a)[/i] relative to the disk; [i](b)[/i] relative to the wall.

Given some monic polynomials $P_1, \ldots, P_n$ with real coefficients, for any real number $y$, let $S_y$ be the set of real number $x$ such that $y = P_i(x)$ for some $i = 1, 2, ..., n$. If the sets $S_{y_1}, S_{y_2}$ have the same size for any two real numbers $y_1, y_2$, show that $P_1, \ldots, P_n$ have the same degree. [i] Proposed by usjl[/i]

In a football tournament $20$ teams participated, each pair of teams played exactly one game. For the win the team is awarded $3$ points, for the draw -- $1$ point, for the lose no points awarded. The total number of points of all teams in the tournament is $554$. Prove that there exist $7$ teams each having at least one draw.

Let $a_0 = 2$, $a_1 = 5$, and $a_2 = 8$, and for $n>2$ define $a_n$ recursively to be the remainder when $4(a_{n-1} + a_{n-2} + a_{n-3})$ is divided by $11$. Find $a_{2018}\cdot a_{2020}\cdot a_{2022}$.

$T$ is a set of all the positive integers of the form $2^k 3^l$, where $k, l$ are some non-negetive integers. Show that there exists $1998$ different elements of $T$ that satisfy the following condition. [b]Condition[/b] The sum of the $1998$ elements is again an element of $T$.

$ABC$ is a right triangle with $\angle C=90$,$CD$ is perpendicular to $AB$,and $D$ is the foot,$\omega$ is the circumcircle of triangle $BCD$,$\omega_{1}$ is a circle inside triangle $ACD$,tangent to $AD$ and $AC$ at $M$ and $N$ respectively,and $\omega_{1}$ is also tangent to $\omega$.prove that: (1)$BD*CN+BC*DM=CD*BM$ (2)$BM=BC$

For $a, b \ge 0$ we define $a * b = \frac{a+b+1}{ab+12}$ . Compute $0*(1*(2*(... (2003*(2004*2005))...)))$.

Triangle $\triangle{ABC}$ is isosceles with $AB = AC$. Let the incircle of $\triangle{ABC}$ intersect $BC$ and $AC$ at $D$ and $E$ respectively. Let $F \neq A$ be the point such that $DF = DA$ and $EF = EA$. If $AF = 8$ and the circumradius of $\triangle{AED}$ is $5$, find the area of $\triangle{ABC}$. [i]Proposed by Anthony Yang and Andy Xu[/i]

Let a tree on $mn + 1$ vertices be $(m,n)$-nice if the following conditions hold: [list] [*] $m + 1$ colors are assigned to the nodes of the tree [*] for the first $m$ colors, there will be exactly $n$ nodes of color $i$ $(1\le i \le m)$ [*] the root node of the tree will be the unique node of color $m+1$. \item the $(m,n)$-nice trees must also satisfy the condition that for any two non-root nodes $i, j$, if the color of $i$ equals the color of $j$, then the color of the parent of $i$ equals the color of the parent of $j$. [*] Nodes of the same color are considered indistinguishable (swapping any two of them results in the same tree). [/list] Let $N(u,v,l)$ denote the number of $(u,v)$-nice trees with $l$ leaves. Note that $N(2,2,2) = 2, N(2,2,3) = 4, N(2,2,4) = 6$. Compute the remainder when $\sum_{l = 123}^{789} N(8,101,l)$ is divided by $101$. Definition: Any rooted, ordered tree consists of some set of nodes, each of which has a (possibly empty) ordered list of children. Each node is the child of exactly one other node, with the exception of the root, which has not parent. There also cannot be any cycles of nodes which are all linearly children of each other. [i]Proposed by Advait Nene[/i]

Does there exist a natural number $N$ which is a power of$2$, such that one can permute its decimal digits to obtain a different power of $2$?

In the triangle $ABC$, $X$ lies on the side $BC$, $Y$ on the side $CA$, and $Z$ on the side $AB$ with $YX \| AB, ZY \| BC$, and $XZ \| CA$. Show that $X,Y$, and $Z$ are the midpoints of the respective sides of $ABC$.

Show that $abc \le (ab + bc + ca)(a^2 + b^2 + c^2)^2$ for all positive real numbers $a, b$ and $c$ such that $a + b + c = 1$.

Prove that in a triangle one of the sides is twice as large as the other if and only if a median and an angle bisector of this triangle are perpendicular

An arrow is assigned to each edge of a polyhedron such that for each vertex, there is an arrow pointing towards that vertex and an arrow pointing away from that vertex. Prove that there exist at least two faces such that the arrows on their boundaries form a cycle.

[b]p1.[/b] Prair takes some set $S$ of positive integers, and for each pair of integers she computes the positive difference between them. Listing down all the numbers she computed, she notices that every integer from $1$ to $10$ is on her list! What is the smallest possible value of $|S|$, the number of elements in her set $S$? [b]p2.[/b] Jake has $2021$ balls that he wants to separate into some number of bags, such that if he wants any number of balls, he can just pick up some bags and take all the balls out of them. What is the least number of bags Jake needs? [b]p3.[/b] Claire has stolen Cat’s scooter once again! She is currently at (0; 0) in the coordinate plane, and wants to ride to $(2, 2)$, but she doesn’t know how to get there. So each second, she rides one unit in the positive $x$ or $y$-direction, each with probability $\frac12$ . If the probability that she makes it to $(2, 2)$ during her ride can be expressed as $\frac{a}{b}$ for positive integers $a, b$ with $gcd(a, b) = 1$, then find $a + b$. [b]p4.[/b] Triangle $ABC$ with $AB = BC = 6$ and $\angle ABC = 120^o$ is rotated about $A$, and suppose that the images of points $B$ and $C$ under this rotation are $B'$ and $C'$, respectively. Suppose that $A$, $B'$ and $C$ are collinear in that order. If the area of triangle $B'CC'$ can be expressed as $a - b\sqrt{c}$ for positive integers $a, b, c$ with csquarefree, find $a + b + c$. [b]p5.[/b] Find the sum of all possible values of $a + b + c + d$ if $a, b, c, $d are positive integers satisfying $$ab + cd = 100,$$ $$ac + bd = 500.$$ [b]p6.[/b] Alex lives in Chutes and Ladders land, which is set in the coordinate plane. Each step they take brings them one unit to the right or one unit up. However, there’s a chute-ladder between points $(1, 2)$ and $(2, 0)$ and a chute-ladder between points $(1, 3)$ and $(4, 0)$, whenever Alex visits an endpoint on a chute-ladder, they immediately appear at the other endpoint of that chute-ladder! How many ways are there for Alex to go from $(0, 0)$ to $(4, 4)$? [b]p7.[/b] There are $8$ identical cubes that each belong to $8$ different people. Each person randomly picks a cube. The probability that exactly $3$ people picked their own cube can be written as $\frac{a}{b}$ , where $a$ and $b$ are positive integers with $gcd(a, b) = 1$. Find $a + b$. [b]p8.[/b] Suppose that $p(R) = Rx^2 + 4x$ for all $R$. There exist finitely many integer values of $R$ such that $p(R)$ intersects the graph of $x^3 + 2021x^2 + 2x + 1$ at some point $(j, k)$ for integers $j$ and $k$. Find the sum of all possible values of $R$. [b]p9.[/b] Let $a, b, c$ be the roots of the polynomial $x^3 - 20x^2 + 22$. Find $\frac{bc}{a^2} +\frac{ac}{b^2} +\frac{ab}{c^2}$. [b]p10.[/b] In any finite grid of squares, some shaded and some not, for each unshaded square, record the number of shaded squares horizontally or vertically adjacent to it, this grid’s score is the sum of all numbers recorded this way. Deyuan shades each square in a blank $n \times n$ grid with probability $k$; he notices that the expected value of the score of the resulting grid is equal to $k$, too! Given that $k > 0.9999$, find the minimum possible value of $n$. [b]p11.[/b] Find the sum of all $x$ from $2$ to $1000$ inclusive such that $$\prod^x_{n=2} \log_{n^n}(n + 1)^{n+2}$$ is an integer. [b]p12.[/b] Let triangle $ABC$ with incenter $I$ and circumcircle $\Gamma$ satisfy $AB = 6\sqrt3$, $BC = 14$, and $CA = 22$. Construct points $P$ and $Q$ on rays $BA$ and $CA$ such that $BP = CQ = 14$. Lines $PI$ and $QI$ meet the tangents from $B$ and $C$ to $\Gamma$, respectively, at points $X$ and $Y$ . If $XY$ can be expressed as $a\sqrt{b}-c$ for positive integers $a, b, c$ with $c$ squarefree, find $a + b + c$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $P$ be the point of intersection of the diagonals of a convex quadrilateral $ABCD$.Let $X,Y,Z$ be points on the interior of $AB,BC,CD$ respectively such that $\frac{AX}{XB}=\frac{BY}{YC}=\frac{CZ}{ZD}=2$. Suppose that $XY$ is tangent to the circumcircle of $\triangle CYZ$ and that $Y Z$ is tangent to the circumcircle of $\triangle BXY$.Show that $\angle APD=\angle XYZ$.

[b]1)[/b] Prove that there are finite sequences, of any length, of nonegative integers having the property that the arithmetic mean of any choice of its elements is natural. [b]2)[/b] Study if there is an increasing infinite sequence of nonegative integers having the property that the arithmetic mean of any finite choice of its elements is natural.