Two circles have radius $9$, and one circle has radius $7$. Each circle is externally tangent to the other two circles, and each circle is internally tangent to two sides of an isosceles triangle, as shown. The sine of the base angle of the triangle is $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m + n$. [img]https://cdn.artofproblemsolving.com/attachments/7/f/c34ff6bcaf6f07e6ba81a7d256e15a61f0e1fa.png[/img]

The pages of a notebook are numbered consecutively so that the numbers $1$ and $2$ are on the second sheet, numbers $3$ and $4$, and so on. A sheet is torn out of this notebook. All of the remaining page numbers are addedand have sum $2021$. (a) How many pages could the notebook originally have been? (b) What page numbers can be on the torn sheet? (Walther Janous)

Let $G$ be a finite group of order $n$. Define the set \[H=\{x:x\in G\text{ and }x^2=e\},\] where $e$ is the neutral element of $G$. Let $p=|H|$ be the cardinality of $H$. Prove that a) $|H\cap xH|\ge 2p-n$, for any $x\in G$, where $xH=\{xh:h\in H\}$. b) If $p>\frac{3n}{4}$, then $G$ is commutative. c) If $\frac{n}{2}<p\le\frac{3n}{4}$, then $G$ is non-commutative. [i]Marian Andronache[/i]

Let $\mathbb{Z}^+$ be the set of positive integers. Determine all functions $f : \mathbb{Z}^+\to\mathbb{Z}^+$ such that $a^2+f(a)f(b)$ is divisible by $f(a)+b$ for all positive integers $a,b$.

A list of $11$ positive integers has a mean of $10$, a median of $9$, and a unique mode of $8$. What is the largest possible value of an integer in the list? ${ \textbf{(A)}\ 24\qquad\textbf{(B)}\ 30\qquad\textbf{(C)}\ 31\qquad\textbf{(D)}}\ 33\qquad\textbf{(E)}\ 35 $

The function $g$ is defined about the natural numbers and satisfies the following conditions: $g(2) = 1$ $g(2n) = g(n)$ $g(2n+1) = g(2n) +1.$ Where $n$ is a natural number such that $1 \leq n \leq 2002$. Find the maximum value $M$ of $g(n).$ Also, calculate how many values of $n$ satisfy the condition of $g(n) = M.$

Suppose $a$, $b$, $c$, and $d$ are positive real numbers that satisfy the system of equations \begin{align*}(a+b)(c+d)&=143,\\(a+c)(b+d)&=150,\\(a+d)(b+c)&=169.\end{align*} Compute the smallest possible value of $a^2+b^2+c^2+d^2$.

$2014$ points are placed on a circumference. On each of the segments with end points on two of the $2014$ points is written a non-negative real number. For any convex polygon with vertices on some of the $2014$ points, the sum of the numbers written on their sides is less or equal than $1$. Find the maximum possible value for the sum of all the written numbers.

Find all function $f,g: \mathbb{Q} \to \mathbb{Q}$ such that \[\begin{array}{l} f\left( {g\left( x \right) - g\left( y \right)} \right) = f\left( {g\left( x \right)} \right) - y \\ g\left( {f\left( x \right) - f\left( y \right)} \right) = g\left( {f\left( x \right)} \right) - y \\ \end{array}\] for all $x,y \in \mathbb{Q}$.

For $n>2$, an $n\times n$ grid of squares is coloured black and white like a chessboard, with its upper left corner coloured black. Then we can recolour some of the white squares black in the following way: choose a $2\times 3$ (or $3\times 2$) rectangle which has exactly $3$ white squares and then colour all these $3$ white squares black. Find all $n$ such that after a series of such operations all squares will be black.

Prove that an integer $n \ge 2$ is a prime if and only if $\phi (n)$ divides $(n - 1)$ and $(n + 1)$ divides $\sigma (n)$. [Here $\phi$ is the Totient function and $\sigma $ is the divisor - sum function.] [hide=Hint]$n$ is squarefree[/hide]

In quadrilateral $ ABCD$, sides $ \overline{AB}$ and $ \overline{BC}$ both have length 10, sides $ \overline{CD}$ and $ \overline{DA}$ both have length 17, and the measure of angle $ ADC$ is $ 60^\circ$. What is the length of diagonal $ \overline{AC}$? [asy]draw((0,0)--(17,0)); draw(rotate(301, (17,0))*(0,0)--(17,0)); picture p; draw(p, (0,0)--(0,10)); draw(p, rotate(115, (0,10))*(0,0)--(0,10)); add(rotate(3)*p); draw((0,0)--(8.25,14.5), linetype("8 8")); label("$A$", (8.25, 14.5), N); label("$B$", (-0.25, 10), W); label("$C$", (0,0), SW); label("$D$", (17, 0), E);[/asy] $ \textbf{(A)}\ 13.5\qquad\textbf{(B)}\ 14\qquad\textbf{(C)}\ 15.5\qquad\textbf{(D)}\ 17\qquad\textbf{(E)}\ 18.5 $

Let $a,b,c$ be real numbers with $a$ non-zero. It is known that the real numbers $x_1,x_2,\ldots,x_n$ satisfy the $n$ equations: \[ ax_1^2+bx_1+c = x_{2} \]\[ ax_2^2+bx_2 +c = x_3\]\[ \ldots \quad \ldots \quad \ldots \quad \ldots\]\[ ax_n^2+bx_n+c = x_1 \] Prove that the system has [b]zero[/b], [u]one[/u] or [i]more than one[/i] real solutions if $(b-1)^2-4ac$ is [b]negative[/b], equal to [u]zero[/u] or [i]positive[/i] respectively.

Let $n=2^{2018}$ and let $S=\{1,2,\ldots,n\}$. For subsets $S_1,S_2,\ldots,S_n\subseteq S$, we call an ordered pair $(i,j)$ [i]murine[/i] if and only if $\{i,j\}$ is a subset of at least one of $S_i, S_j$. Then, a sequence of subsets $(S_1,\ldots, S_n)$ of $S$ is called [i]tasty[/i] if and only if: 1) For all $i$, $i\in S_i$. 2) For all $i$, $\displaystyle\bigcup_{j\in S_i} S_j=S_i$. 3) There do not exist pairwise distinct integers $a_1,a_2,\ldots,a_k$ with $k\ge 3$ such that for each $i$, $(a_i, a_{i+1})$ is murine, where indices are taken modulo $k$. 4) $n$ divides $1+|S_1|+|S_2|+\ldots+|S_n|$. Find the largest integer $x$ such that $2^x$ divides the number of tasty sequences $(S_1,\ldots, S_n)$. [i]Proposed by Vincent Huang and Brandon Wang

A student on vacation for $d$ days observed that $(1)$ it rained $7$ times, morning or afternoon $(2)$ when it rained in the afternoon, it was clear in the morning $(3)$ there were five clear afternoons $(4)$ there were six clear mornings. Then $d$ equals: $ \textbf{(A)}\ 7\qquad\textbf{(B)}\ 9\qquad\textbf{(C)}\ 10\qquad\textbf{(D)}\ 11\qquad\textbf{(E)}\ 12 $

Let $a,b,c$ be real numbers, $z_{1},z_{2},z_{3}$ be complex numbers such that $\begin{cases} \displaystyle|z_1|=|z_2|=|z_3|=1\\ \displaystyle\frac{z_{1}}{z_{2}}+\frac{z_{2}}{z_{3}}+\frac{z_{3}}{z_{1}}=1\\ \end{cases}$ Find $|az_{1}+bz_{2}+cz_{3}|$.

Given points $O$ and $A$ in the plane. Every point in the plane is colored with one of a finite number of colors. Given a point $X$ in the plane, the circle $C(X)$ has center $O$ and radius $OX+{\angle AOX\over OX}$, where $\angle AOX$ is measured in radians in the range $[0,2\pi)$. Prove that we can find a point $X$, not on $OA$, such that its color appears on the circumference of the circle $C(X)$.

Let $ABCDE$ be a regular pentagon with center $M$. A point $P$ (different from $M$) is chosen on the line segment $MD$. The circumcircle of $ABP$ intersects the line segment $AE$ in $A$ and $Q$ and the line through $P$ perpendicular to $CD$ in $P$ and $R$. Prove that $AR$ and $QR$ have same length. [i]proposed by Stephan Wagner[/i]

Let $ \lambda_i \;(i=1,2,...)$ be a sequence of distinct positive numbers tending to infinity. Consider the set of all numbers representable in the form \[ \mu= \sum_{i=1}^{\infty}n_i\lambda_i ,\] where $ n_i \geq 0$ are integers and all but finitely many $ n_i$ are $ 0$. Let \[ L(x)= \sum _{\lambda_i \leq x} 1 \;\textrm{and}\ \;M(x)= \sum _{\mu \leq x} 1 \ .\] (In the latter sum, each $ \mu$ occurs as many times as its number of representations in the above form.) Prove that if \[ \lim_{x\rightarrow \infty} \frac{L(x+1)}{L(x)}=1,\] then \[ \lim_{x\rightarrow \infty} \frac{M(x+1)}{M(x)}=1.\] [i]G. Halasz[/i]

Prove that there are positive integers $a_1, a_2,\dots, a_{2020}$ such that $$\dfrac{1}{a_1}+\dfrac{1}{2a_2}+\dfrac{1}{3a_3}+\dots+\dfrac{1}{2020a_{2020}}=1.$$

21) A particle of mass $m$ moving at speed $v_0$ collides with a particle of mass $M$ which is originally at rest. The fractional momentum transfer $f$ is the absolute value of the final momentum of $M$ divided by the initial momentum of $m$. If the collision is perfectly $elastic$, what is the maximum possible fractional momentum transfer, $f_{max}$? A) $0 < f_{max} < \frac{1}{2}$ B) $f_{max} = \frac{1}{2}$ C) $\frac{1}{2} < f_{max} < \frac{3}{2}$ D) $f_{max} = 2$ E) $f_{max} \ge 3$

$ (a)$ Prove that $ \mathbb{N}$ can be partitioned into three (mutually disjoint) sets such that, if $ m,n \in \mathbb{N}$ and $ |m\minus{}n|$ is $ 2$ or $ 5$, then $ m$ and $ n$ are in different sets. $ (b)$ Prove that $ \mathbb{N}$ can be partitioned into four sets such that, if $ m,n \in \mathbb{N}$ and $ |m\minus{}n|$ is $ 2,3,$ or $ 5$, then $ m$ and $ n$ are in different sets. Show, however, that $ \mathbb{N}$ cannot be partitioned into three sets with this property.

Find all integers $x,y$ such that $x^3(y+1)+y^3(x+1)=19$. [i]Proposed by Bulgaria[/i]

Let $a$ and $b$ be positive integers such that $10< \gcd(a,b) < 20$ and $220 < \text{lcm}(a,b) < 230$. Find the difference between the smallest and largest possible values of $ab$.

Let \(ABC\) be an acute triangle with circumcircle \(\omega\). Let \(P\) be a variable point on the arc \(BC\) of \(\omega\) not containing \(A\). Squares \(BPDE\) and \(PCFG\) are constructed such that \(A\), \(D\), \(E\) lie on the same side of line \(BP\) and \(A\), \(F\), \(G\) lie on the same side of line \(CP\). Let \(H\) be the intersection of lines \(DE\) and \(FG\). Show that as \(P\) varies, \(H\) lies on a fixed circle. [i]Proposed by Karthik Vedula[/i]