A positive real $M$ is $strong$ if for any positive reals $a$, $b$, $c$ satisfying $$ \text{max}\left\{ \frac{a}{b+c} , \frac{b}{c+a} , \frac{c}{a+b} \right\} \geqslant M $$ then the following inequality holds: $$\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b} > 20.$$ (a) Prove that $M=20-\frac{1}{20}$ is not $strong$. (b) Prove that $M=20-\frac{1}{21}$ is $strong$.

Given is an integer $k\geq 2$. Determine the smallest positive integer $n$, such that, among any $n$ points in the plane, there exist $k$ points among them, such that all distances between them are less than or equal to $2$, or all distances are strictly greater than $1$.

A box contains $100$ tickets. Each ticket has a real number written on it. There are no restrictions on the type of number except that they are all different (they can be integers, rational, positive, negative, irrational, large or small). Of course there is one ticket that has the highest number and that is the winner. The game consists of drawing a ticket at random, looking at it and deciding whether to keep it or not. If we choose to keep him, it is verified if he was the oldest, in which case we win a million pesos (if we don't win, the game is over). If we don't think it's the biggest, we can discard it and draw another one, repeating the process until we like one or we run out of tickets. Going back to choose a previously discarded ticket is prohibited. Find a game strategy that gives at least a $25\%$ chance of winning.

How many integers $n$, satisfy $|n|<2020$ and the equation $11^3|n^3+3n^2-107n+1$ (A)$0$ (B)$101$ (C)$367$ (D)$368$

There are $3$ unit squares in a row as shown in the figure below. Each side of this figure is painted by one of the three colors: Blue, Green or Red. It is known that for any square, all the three colors are used and no two adjacent sides have the same color. Find the number of possible colorings. [img]https://cdn.artofproblemsolving.com/attachments/e/c/8963e6716b7d9b23479dd7e106b4bd9a3267c1.png[/img] A. $48$ B. $96$ C. $108$ D. $192$ E. $216$

A [i]zigzag [/i] is a polyline on a plane formed from two parallel rays and a segment connecting the origins of these rays. What is the maximum number of parts a plane can be split into using $ n $ zigzags?

[i]Palindromic partitioning [/i] of the natural number $ A $ is called, when $ A $ is written as the sum of natural the terms $ A = a_1 + a_2 + \ ldots + a_ {n-1} + a_n $ ($ n \geq 1 $), in which $ a_1 = a_n , a_2 = a_ {n-1} $ and in general, $ a_i = a_ {n + 1 - i} $ with $ 1 \leq i \leq n $. For example, $ 16 = 16 $, $ 16 = 2 + 12 + 2 $ and $ 16 = 7 + 1 + 1 + 7 $ are [i]palindromic partitions[/i] of the number $16$. Find the number of all [i]palindromic partitions[/i] of the number $2006$.

Given $f_1(x)=2x-2$ and, for $k\ge2$, defined $f_k(x)=f(f_{k-1}(x))$ to be a real-valued function of $x$. Find the remainder when $f_{2013}(2012)$ is divided by the prime $2011$.

In how many ways can one select eight integers $a_1,a_2, ... ,a_8$, not necesarily distinct, such that $1 \le a_1 \le ... \le a_8 \le 8$?

We are given an equilateral triangle $ABC$, of side $a$, inside its circumscribed circle. We consider the smallest of the two portions of circle limited between $AB$ and the circumference. If we consider parallel lines to $BC$, some of them cut the portion of circle in a segment. Which is the maximum possible length for one of the segments?

Byan Rai is currently standing on the origin of a $2$D plane. In each second: [list] [*] he jumps one unit up with probability $\frac{6}{11}$, [*] he jumps three units down with probability $\frac{2}{11}$, [*] he jumps four units right with probability $\frac{3}{22}$, [*] he jumps four units left with probability $\frac{3}{22}$. [/list] Suppose Byan ends up at $(x, y)$ after $2024$ seconds. The expected value of $x^2 + y^2$ can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$. [i]Lightning 4.3[/i]

Nebraska, the home of the AMC, changed its license plate scheme. Each old license plate consisted of a letter followed by four digits. Each new license plate consists of three letters followed by three digits. By how many times is the number of possible license plates increased? $ \textbf{(A)}\ \frac{26}{10} \qquad \textbf{(B)}\ \frac{26^2}{10^2} \qquad \textbf{(C)}\ \frac{26^2}{10} \qquad \textbf{(D)}\ \frac{26^3}{10^3} \qquad \textbf{(E)}\ \frac{26^3}{10^2}$

Let $ABC$ be a triangle with integral side lengths. The angle bisector drawn from $B$ and the altitude drawn from $C$ meet at point $P$ inside the triangle. Prove that the ratio of areas of triangles $APB$ and $APC$ is a rational number.

Let $ABC$ be a triangle and $D$ be a point such that $A$ and $D$ are on opposite sides of $BC$. Give that $\angle ACD = 75^o$, $AC = 2$, $BD =\sqrt6$, and $AD$ is an angle bisector of both $\vartriangle ABC$ and $\vartriangle BCD$, find the area of quadrilateral $ABDC$.

Let \( \mathbb{Z^{+}} \) denote the set of all positive integers. Find all functions \( f: \mathbb{Z^{+}} \rightarrow \mathbb{Z^{+}} \) such that for every pair of positive integers \( a \) and \( b \), there exists a positive integer \( c \) satisfying: $$ f(a)f(b) - ab = 2^{c-1} - 1. $$ Proposed by Matin Yousefi

Find all functions $\mathbb{R}^+\rightarrow\mathbb{R}^+$ for which we have for all $x,y\in \mathbb{R}^+$ that $f(x)f(yf(x))=f(x+y)$.

Let $ ABC$ be a triangle such that \[ \frac{BC}{AB \minus{} BC}\equal{}\frac{AB \plus{} BC}{AC}\] Determine the ratio $ \angle A : \angle C$.

Let $ABCD$ be a convex quadrilateral with $\angle ABC>90$, $CDA>90$ and $\angle DAB=\angle BCD$. Denote by $E$ and $F$ the reflections of $A$ in lines $BC$ and $CD$, respectively. Suppose that the segments $AE$ and $AF$ meet the line $BD$ at $K$ and $L$, respectively. Prove that the circumcircles of triangles $BEK$ and $DFL$ are tangent to each other. $\emph{Slovakia}$

Solve in positive real numbers the following equation $x^{-y} + y^{-x} = 4 - x - y$.

Bob plays a game on the whiteboard. Initially, the numbers $\{1, 2, ...,n\}$ are shown. On each turn, Bob takes two numbers from the board $x$, $y$, erases them both, and writes down $2x + y$ onto the board. In terms of n, what is the maximum possible value that Bob can end up with?

Show that the planes $ACG$ and $BEH$ defined by the vertices of the cube shown in Figure are parallel. What is their distance if the edge length of the cube is $1$ meter? [img]https://cdn.artofproblemsolving.com/attachments/c/9/21585f6c462e4289161b4a29f8805c3f63ff3e.png[/img]

Let $n \ge 2$ be an integer. Let $x_1 \ge x_2 \ge ... \ge x_n$ and $y_1 \ge y_2 \ge ... \ge y_n$ be $2n$ real numbers such that $$0 = x_1 + x_2 + ... + x_n = y_1 + y_2 + ... + y_n $$ $$\text{and} \hspace{2mm} 1 =x_1^2 + x_2^2 + ... + x_n^2 = y_1^2 + y_2^2 + ... + y_n^2.$$ Prove that $$\sum_{i = 1}^n (x_iy_i - x_iy_{n + 1 - i}) \ge \frac{2}{\sqrt{n-1}}.$$ [i]Proposed by David Speyer and Kiran Kedlaya[/i]

It is known that each two of the 12 competitors, that participated in the finals of the competition “Mathematical duels”, have a common friend among the other 10. Prove that there is one of them that has at least 5 friends among the group.

Let $\omega_1$, $\omega_2$, $\omega_3$ are three externally tangent circles, with $\omega_1$ and $\omega_2$ tangent at $A$. Choose points $B$ and $C$ on $\omega_1$ so that lines $AB$ and $AC$ are tangent to $\omega_3$. Suppose the line $BC$ intersect $\omega_3$ at two distinct points, and $X$ is the intersection further away to $B$ and $C$ than the other one. Prove that one of the tangent lines of $\omega_2$ passing through $X$, is also tangent to an excircle of triangle $ABC$. [i]Proposed by Ivan Chan Kai Chin[/i]

suppose that $O$ is the circumcenter of acute triangle $ABC$. we have circle with center $O$ that is tangent too $BC$ that named $w$ suppose that $X$ and $Y$ are the points of intersection of the tangent from $A$ to $w$ with line $BC$($X$ and $B$ are in the same side of $AO$) $T$ is the intersection of the line tangent to circumcirle of $ABC$ in $B$ and the line from $X$ parallel to $AC$. $S$ is the intersection of the line tangent to circumcirle of $ABC$ in $C$ and the line from $Y$ parallel to $AB$. prove that $ST$ is tangent $ABC$.