Let $A,B$ be two circles in the plane with $B$ inside $A$. Assume that $A$ has radius $3$, $B$ has radius $1$, $P$ is a point on $A$, $Q$ is a point on $B$, and $A$ and $B$ touch so that $P$ and $Q$ are the same point. Suppose that $A$ is kept fixed and $B$ is rolled once round the inside of $A$ so that $Q$ traces out a curve starting and finishing at $P$. What is the area enclosed by this curve? [img]https://services.artofproblemsolving.com/download.php?id=YXR0YWNobWVudHMvOS84LzkwMDBjOTAwODk5M2QyM2IxMGUxZGE5OTI1NWU1ZDYwMDkyYTUwLnBuZw==&rn=VlRSTUMgMjAxMC5wbmc=[/img]

Given set $S = \{ xy\left( {x + y} \right)\; |\; x,y \in \mathbb{N}\}$.Let $a$ and $n$ natural numbers such that $a+2^k\in S$ for all $k=1,2,3,...,n$.Find the greatest value of $n$.

Let $A,B$ be points on a circle $k$ with center $S$ such that $\angle ASB = 90^o$ . Circles $k_1$ and $k_2$ are tangent to each other at $Z$ and touch $k$ at $A$ and $B$ respectively. Circle $k_3$ inside $\angle ASB$ is internally tangent to $k$ at $C$ and externally tangent to $k_1$ and $k_2$ at $X$ and $Y$, respectively. Prove that $\angle XCY = 45^o$

A set of $12$ tokens ---- $3$ red, $2$ white, $1$ blue, and $6$ black ---- is to be distributed at random to $3$ game players, $4$ tokens per player. The probability that some player gets all the red tokens, another gets all the white tokens, and the remaining player gets the blue token can be written as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. What is $m+n$? $ \textbf{(A) }387 \qquad \textbf{(B) }388 \qquad \textbf{(C) }389 \qquad \textbf{(D) }390 \qquad \textbf{(E) }391 \qquad $

Paul starts with the number $19$. In one step, he can add $1$ to his number, divide his number by $2$, or divide his number by $3$. What is the minimum number of steps Paul needs to get to $1$?

The simplest form of $ 1 \minus{} \frac{1}{1 \plus{} \frac{a}{1 \minus{} a}}$ is: $ \textbf{(A)}\ {a}\text{ if }{a\not\equal{} 0} \qquad \textbf{(B)}\ 1\qquad \textbf{(C)}\ {a}\text{ if }{a\not\equal{} \minus{}1}\qquad \textbf{(D)}\ {1 \minus{} a}\text{ with not restriction on }{a}\qquad \textbf{(E)}\ {a}\text{ if }{a\not\equal{} 1}$

Let the circles $\omega$ and $\omega^ \prime$ intersect in $A$ and $B$. Tangent to circle$\omega$ at $A$ intersects$\omega^ \prime$ in $C$ and tangent to circle $\omega^ \prime$ at $A$ intersects $\omega$ in $D$. Suppose that $CD$ intersects$\omega$ and $\omega^ \prime$ in $E$ and $F$, respectively (assume that $E$ is between $F$ and $C$). The perpendicular to $AC$ from $E$ intersects $\omega^ \prime$ in point $P$ and perpendicular to $AD$ from $F$ intersects$\omega$ in point $Q$ (The points $A, P$ and $Q$ lie on the same side of the line $CD$). Prove that the points $A, P$ and $Q$ are collinear. Proposed by Mahdi Etesami Fard

Let $ S$ be a set of $ 6n$ points in a line. Choose randomly $ 4n$ of these points and paint them blue; the other $ 2n$ points are painted green. Prove that there exists a line segment that contains exactly $ 3n$ points from $ S$, $ 2n$ of them blue and $ n$ of them green.

Two circles $\gamma $ and $\gamma'$ cross one another at points $A$ and $B$ . The tangent to $\gamma'$ at $A$ meets $\gamma$ again at $C$ , the tangent to $\gamma$ at $A$ meets $\gamma'$ again at $C'$ , and the line $CC'$ separates the points $A$ and $B$ . Let $\Gamma$ be the circle externally tangent to $\gamma$ , externally tangent to $\gamma'$ , tangent to the line $CC'$, and lying on the same side of $CC'$ as $B$ . Show that the circles $\gamma$ and $\gamma'$ intercept equal segments on one of the tangents to $\Gamma$ through $A$ .

The circumcenter $O$, the incenter $I$, and the midpoint $M$ of a diagonal of a bicentral quadrilateral were marked. After this the quadrilateral was erased. Restore it.

Denote by $S(a,b,c)$ the area of a triangle whose lengthes of three sides are $a,b,c$ Prove that for any positive real numbers $a_{1},b_{1},c_{1}$ and $a_{2},b_{2},c_{2}$ which can serve as the lengthes of three sides of two triangles respectively ,we have $ \sqrt{S(a_{1},b_{1},c_{1})}+\sqrt{S(a_{2},b_{2},c_{2})}\le\sqrt{S(a_{1}+a_{2},b_{1}+b_{2},c_{1}+c_{2})}$

Let $k, m, n$ be integers such that $1<n\le m-1 \le k$. Determine the maximum size of a subset $S$ of the set $\{ 1,2, \cdots, k \}$ such that no $n$ distinct elements of $S$ add up to $m$.

A quadrilateral $ABCD$ is circumscribed about a circle $\omega$. The lines $AB$ and $CD$ meet at $O$. A circle $\omega_1$ is tangent to side $BC$ at $K$ and to the extensions of sides $AB$ and $CD$, and a circle $\omega_2$ is tangent to side $AD$ at $L$ and to the extensions of sides $AB$ and $CD$. Suppose that points $O$, $K$, $L$ lie on a line. Prove that the midpoints of $BC$ and $AD$ and the center of $\omega$ also lie on a line.

Let $n \geq 3$ be an integer, and consider a set of $n$ points in three-dimensional space such that: [list=i] [*] every two distinct points are connected by a string which is either red, green, blue, or yellow; [*] for every three distinct points, if the three strings between them are not all of the same colour, then they are of three different colours; [*] not all the strings have the same colour. [/list] Find the maximum possible value of $n$.

Let $ABC$ be a triangle with $AB<AC$ and with its incircle touching the sides $AB$ and $BC$ at $M$ and $J$ respectively. A point $D$ lies on the extension of $AB$ beyond $B$ such that $AD=AC$. Let $O$ be the midpoint of $CD$. Prove that the points $J$, $O$, $M$ are collinear. [i](Proposed by Tan Rui Xuen)[/i]

Circles ${{w} _ {1}}$ and ${{w} _ {2}}$ with centers ${{O} _ {1}}$ and ${{O} _ {2}}$ intersect at points $A$ and $B$, respectively. The line ${{O} _ {1}} {{O} _ {2}}$ intersects ${{w} _ {1}}$ at the point $Q$, which does not lie inside the circle ${{w} _ {2}}$, and ${{w} _ {2}}$ at the point $X$ lying inside the circle ${{w} _ {1} }$. Around the triangle ${{O} _ {1}} AX$ circumscribe a circle ${{w} _ {3}}$ intersecting the circle ${{w} _ {1}}$ for the second time in point $T$. The line $QT$ intersects the circle ${{w} _ {3}}$ at the point $K$, and the line $QB$ intersects ${{w} _ {2}}$ the second time at the point $H$. Prove that a) points $T, \, \, X, \, \, B$ lie on one line; b) points $K, \, \, X, \, \, H$ lie on one line. (Vadym Mitrofanov)

Let $P$ be a set of people. For two people $A$ and $B$, if $A$ knows $B$, $B$ also knows $A$. Each person in $P$ knows $2$ or less people in the set. $S$, a subset of $P$ with $k$ people, is called [i][b]k-independent set[/b][/i] of $P$ if any two people in $S$ don’t know each other. $X_1, X_2, …, X_{4041}$ are [i][b]2021-independent set[/b][/i]s of $P$ (not necessarily distinct). Show that there exists a [i][b]2021-independent set[/b][/i] of $P$, $\{v_1, v_2, …, v_{2021}\}$, which satisfies the following condition: [center] For some integer $1 \le i_1 < i_2 < \cdots < i_{2021} \leq 4041$, $v_1 \in X_{i_1}, v_2 \in X_{i_2}, \ldots, v_{2021} \in X_{i_{2021}}$ [/center] [hide=Graph Wording] Thanks to Evan Chen, here's a graph wording of the problem :) Let $G$ be a finite simple graph with maximum degree at most $2$. Let $X_1, X_2, \ldots, X_{4041}$ be independent sets of size $2021$ [i](not necessarily distinct)[/i]. Prove that there exists another independent set $\{v_1, v_2, \ldots, v_{2021}\}$ of size $2021$ and indices $1 \le t_1 < t_2 < \cdots < t_{2021} \le 4041$ such that $v_i \in X_{t_i}$ for all $i$. [/hide]

Let the vertex set \( V \) of a graph be partitioned into \( h \) parts \( (V = V_1 \cup V_2 \cup \cdots \cup V_h) \), with \(|V_1| = n_1, |V_2| = n_2, \ldots, |V_h| = n_h \). If there is an edge between any two vertices only when they belong to different parts, the graph is called a complete \( h \)-partite graph, denoted as \( k(n_1, n_2, \ldots, n_h) \). Let \( n \) and \( r \) be positive integers, \( n \geq 6 \), \( r \leq \frac{2}{3}n \). Consider the complete \( r + 1 \)-partite graph \( k\left(\underbrace{1, 1, \ldots, 1}_{r}, n - r\right) \). Answer the following questions: 1. Find the maximum number of disjoint circles (i.e., circles with no common vertices) in this complete \( r + 1 \)-partite graph. 2. Given \( n \), for all \( r \leq \frac{2}{3}n \), find the maximum number of edges in a complete \( r + 1 \)-partite graph \( k(1, 1, \ldots, 1, n - r) \) where no more than one circle is disjoint.

$ABC$ is acute-angled triangle, $AA_1,CC_1$ are altitudes. $M$ is centroid. $M$ lies on circumcircle of $A_1BC_1$. Find all values of $\angle B$

Let $ P$ be any point in the interior of a $ \triangle ABC$.Prove That $ \frac{PA}{a}\plus{}\frac{PB}{b}\plus{}\frac{PC}{c}\ge \sqrt{3}$.

Find all functions $f:\mathbb{N}\to\mathbb{N}$ such that \[f(f(n))=n+2015\] where $n\in \mathbb{N}.$

Harold, Tanya, and Ulysses paint a very long picket fence. Harold starts with the first picket and paints every $h$th picket; Tanya starts with the second picket and paints everth $t$th picket; and Ulysses starts with the third picket and paints every $u$th picket. Call the positive integer $100h+10t+u$ $\textit{paintable}$ when the triple $(h,t,u)$ of positive integers results in every picket being painted exaclty once. Find the sum of all the paintable integers.

Hi guys, I was just reading over old posts that I made last year ( :P ) and saw how much the level of Getting Started became harder. To encourage more people from posting, I decided to start a Problem of the Day. This is how I'll conduct this: 1. In each post (not including this one since it has rules, etc) everyday, I'll post the problem. I may post another thread after it to give hints though. 2. Level of problem.. This is VERY important. All problems in this thread will be all AHSME or problems similar to this level. No AIME. Some AHSME problems, however, that involve tough insight or skills will not be posted. The chosen problems will be usually ones that everyone can solve after working. Calculators are allowed when you solve problems but it is NOT necessary. 3. Response.. All you have to do is simply solve the problem and post the solution. There is no credit given or taken away if you get the problem wrong. This isn't like other threads where the number of problems you get right or not matters. As for posting, post your solutions here in this thread. Do NOT PM me. Also, here are some more restrictions when posting solutions: A. No single answer post. It doesn't matter if you put hide and say "Answer is ###..." If you don't put explanation, it simply means you cheated off from some other people. I've seen several posts that went like "I know the answer" and simply post the letter. What is the purpose of even posting then? Huh? B. Do NOT go back to the previous problem(s). This causes too much confusion. C. You're FREE to give hints and post different idea, way or answer in some cases in problems. If you see someone did wrong or you don't understand what they did, post here. That's what this thread is for. 4. Main purpose.. This is for anyone who visits this forum to enjoy math. I rememeber when I first came into this forum, I was poor at math compared to other people. But I kindly got help from many people such as JBL, joml88, tokenadult, and many other people that would take too much time to type. Perhaps without them, I wouldn't be even a moderator in this forum now. This site clearly made me to enjoy math more and more and I'd like to do the same thing. That's about the rule.. Have fun problem solving! Next post will contain the Day 1 Problem. You can post the solutions until I post one. :D

Tireless Thomas and Jeremy construct a sequence. At the beginning there is one positive integer in the sequence. Then they successively write new numbers in the sequence in the following way: Thomas obtains the next number by adding to the previous number one of its (decimal) digits, while Jeremy obtains the next number by subtracting from the previous number one of its digits. Prove that there is a number in this sequence which will be repeated at least $100$ times. (A Shapovalov)

The points $A$ and $B$ lie on a horizontal line over a vertical plane. We consider the semicircumference passing through $A$ and $B$ that lies under the horizontal line. A segment of length $a$, with the same diameter that the semicircumference, moves in a way that always contains the point $A$ and one of its extremes lies always on the semicircumference. Determine the value of the cosine of the angle between this segment and the horizontal line that makes the medium point of the segment to be as down as possible.