Determine the least real number $M$ such that the inequality \[|ab(a^{2}-b^{2})+bc(b^{2}-c^{2})+ca(c^{2}-a^{2})| \leq M(a^{2}+b^{2}+c^{2})^{2}\] holds for all real numbers $a$, $b$ and $c$.

Find the limit, when $n$ tends to the infinity, of $$\frac{\sum_{k=0}^{n} {{2n} \choose {2k}} 3^k} {\sum_{k=0}^{n-1} {{2n} \choose {2k+1}} 3^k}$$

The ratio of the median $AM$ of a triangle $ABC$ to the side $BC$ equals $\sqrt{3}:2$. The points on the sides of $ABC$ dividing these side into $3$ equal parts are marked. Prove that some $4$ of these $6$ points are concyclic.

Chandler the Octopus is making a concoction to create the perfect ink. He adds $1.2$ grams of melanin, $4.2$ grams of enzymes, and $6.6$ grams of polysaccharides. But Chandler accidentally added n grams of an extra ingredient to the concoction, Chemical $X$, to create glue. Given that Chemical $X$ contains none of the three aforementioned ingredients, and the percentages of melanin, enzymes, and polysaccharides in the final concoction are all integers, find the sum of all possible positive integer values of $n$. [i]Proposed by Taiki Aiba[/i]

Consider the system \[x+y=z+u,\] \[2xy=zu.\] Find the greatest value of the real constant $m$ such that $m \le \frac{x}{y}$ for any positive integer solution $(x, y, z, u)$ of the system, with $x \ge y$.

Let $\phi (n)$ be the number of positive integers less than or equal to $n$ that are relatively prime to $n$. Evaluate $$\sum \limits _{n=1} ^{64} (-1)^{n} \left \lfloor \frac{64}{n} \right \rfloor \phi (n).$$

A trapezoid has bases with lengths equal to $5$ and $15$ and legs with lengths equal to $13$ and $13.$ Determine the area of the trapezoid.

Find all functions $f$ defined on the integers greater than or equal to $1$ that satisfy: (a) for all $n,f(n)$ is a positive integer. (b) $f(n + m) =f(n)f(m)$ for all $m$ and $n$. (c) There exists $n_0$ such that $f(f(n_0)) = [f(n_0)]^2$ .

How many triples $(a,b,c)$ with $a,b,c \in \mathbb{R}$ satisfy the following system? $$\begin{cases} a^4-b^4=c \\ b^4-c^4=a \\ c^4-a^4=b \end{cases}$$.

Let $M\subset \Bbb{N}^*$ such that $|M|=2004.$ If no element of $M$ is equal to the sum of any two elements of $M,$ find the least value that the greatest element of $M$ can take.

A $ 7 \times 7$ grid of unit-length squares is given. Twenty-four $1 \times 2$ dominoes are placed in the grid, each covering two whole squares and in total leaving one empty space. It is allowed to take a domino adjacent to the empty square and slide it lengthwise to fill the whole square, leaving a new one empty and resulting in a different configuration of dominoes. Given an initial configuration of dominoes for which the maximum possible number of distinct configurations can be reached through any number of slides, compute the maximum number of distinct configurations.

If $ 4^x \minus{} 4^{x \minus{} 1} \equal{} 24$, then $ (2x)^x$ equals: $ \textbf{(A)}\ 5\sqrt{5}\qquad \textbf{(B)}\ \sqrt{5}\qquad \textbf{(C)}\ 25\sqrt{5}\qquad \textbf{(D)}\ 125\qquad \textbf{(E)}\ 25$

Find the number of nondegenerate triangles whose vertices lie in the set of points $(s,t)$ in the plane such that $0 \leq s \leq 4$, $0 \leq t \leq 4$, $s$ and $t$ are integers.

Let $n$ be a nonnegative integer. Determine the number of ways that one can choose $(n+1)^2$ sets $S_{i,j}\subseteq\{1,2,\ldots,2n\}$, for integers $i,j$ with $0\leq i,j\leq n$, such that: [list] [*] for all $0\leq i,j\leq n$, the set $S_{i,j}$ has $i+j$ elements; and [*] $S_{i,j}\subseteq S_{k,l}$ whenever $0\leq i\leq k\leq n$ and $0\leq j\leq l\leq n$. [/list] [i]Proposed by Ricky Liu[/i]

Given a quadrilateral $ABCD$ inscribed in circle $\Gamma$.From a point P outside $\Gamma$, draw tangents $PA$ and $PB$ with $A$ and $B$ as touspoints. The line $PC$ intersects $\Gamma$ at point $D$. Draw a line through $B$ parallel to $PA$, this line intersects $AC$ and $AD$ at points $E$ and $F$ respectively. Prove that $BE = BF$.

[u]Round 5[/u] [b]p13.[/b] Points $A, B, C$, and $D$ lie in a plane with $AB = 6$, $BC = 5$, and $CD = 5$, and $AB$ is perpendicular to $BC$. Point E lies on line $AD$ such that $D \ne E$, $AE = 3$ and $CE = 5$. Find $DE$. [b]p14.[/b] How many ordered pairs of integers $(x,y)$ are solutions to $x^2y = 36 + y$? [b]p15.[/b] Chicken nuggets come in boxes of two sizes, $a$ nuggets per box and $b$ nuggets per box. We know that $899$ nuggets is the largest number of nuggets we cannot obtain with some combination of $a$-sized boxes and $b$-sized boxes. How many different pairs $(a, b)$ are there with $a < b$? [u]Round 6[/u] [b]p16.[/b] You are playing a game with coins with your friends Alice and Bob. When all three of you flip your respective coins, the majority side wins. For example, if Alice, Bob, and you flip Heads, Tails, Heads in that order, then you win. If Alice, Bob, and you flip Heads, Heads, Tails in that order, then you lose. Notice that more than one person will “win.” Alice and Bob design their coins as follows: a value $p$ is chosen randomly and uniformly between $0$ and $1$. Alice then makes a biased coin that lands on heads with probability $p$, and Bob makes a biased coin that lands on heads with probability $1 -p$. You design your own biased coin to maximize your chance of winning without knowing $p$. What is the probability that you win? [b]p17.[/b] There are $N$ distinct students, numbered from $1$ to $N$. Each student has exactly one hat: $y$ students have yellow hats, $b$ have blue hats, and $r$ have red hats, where $y + b + r = N$ and $y, b, r > 0$. The students stand in a line such that all the $r$ people with red hats stand in front of all the $b$ people with blue hats. Anyone wearing red is standing in front of everyone wearing blue. The $y$ people with yellow hats can stand anywhere in the line. The number of ways for the students to stand in a line is $2016$. What is $100y + 10b + r$? [b]p18.[/b] Let P be a point in rectangle $ABCD$ such that $\angle APC = 135^o$ and $\angle BPD = 150^o$. Suppose furthermore that the distance from P to $AC$ is $18$. Find the distance from $P$ to $BD$. [u]Round 7 [/u] [b]p19.[/b] Let triangle $ABC$ be an isosceles triangle with $|AB| = |AC|$. Let $D$ and $E$ lie on $AB$ and $AC$, respectively. Suppose $|AD| = |BC| = |EC|$ and triangle $ADE$ is isosceles. Find the sum of all possible values of $\angle BAC$ in radians. Write your answer in the form $2 arcsin \left( \frac{a}{b}\right) + \frac{c}{d} \pi$, where $\frac{a}{b}$ and $\frac{c}{d}$ are in lowest terms, $-1 \le \frac{a}{b} \le 1$, and $-1 \le \frac{c}{d} \le 1$. [b]p20.[/b] Kevin is playing a game in which he aims to maximize his score. In the $n^{th}$ round, for $n \ge 1$, a real number between $0$ and $\frac{1}{3^n}$ is randomly generated. At each round, Kevin can either choose to have the randomly generated number from that round as his score and end the game, or he can choose to pass on the number and continue to the next round. Once Kevin passes on a number, he CANNOT claim that number as his score. Kevin may continue playing for as many rounds as he wishes. If Kevin plays optimally, the expected value of his score is $a + b\sqrt{c}$ where $a, b$, and $c$ are integers and $c$ is positive and not divisible by any positive perfect square other than $1$. What is $100a + 10b + c$? [b]p21.[/b] Lisa the ladybug (a dimensionless ladybug) lives on the coordinate plane. She begins at the origin and walks along the grid, at each step moving either right or up one unit. The path she takes ends up at $(2016, 2017)$. Define the “area” of a path as the area below the path and above the $x$-axis. The sum of areas over all paths that Lisa can take can be represented as as $a \cdot {{4033} \choose {2016}}$ . What is the remainder when $a$ is divided by $1000$? PS. You should use hide for answers. Rounds 1-4 have been posted [url=https://artofproblemsolving.com/community/c4h2782871p24446475]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Quadrilateral $ABCD$ has both an inscribed and a circumscribed circle and sidelengths $BC = 4, CD = 5, DA = 6$. Find the area of $ABCD$.

On a circle we consider a set $M$ consisting of $n$ ($n \ge 3$) points, of which only one is colored red. Determine of which polygons inscribed in a circle having the vertices in the set $M$ are more: those that contain the red dot or those that do not contain those points? How many more are there than others?

It is given that $n$ and $\sqrt{12n^2+1}$ are both positive integers. Prove that: $$ \sqrt{ \frac{\sqrt{12n^2+1}+1}{2}} $$ is also positive integer.

In Oddland there are stamps with values of $1$ cent, $3$ cents, $5$ cents, etc., each for odd number there is exactly one stamp type. Oddland Post dictates: For two different values on a letter must be the number of stamps of the lower one value must be at least as large as the number of tokens of the higher value. In Squareland, on the other hand, there are stamps with values of $1$ cent, $4$ cents, $9$ cents, etc. there is exactly one stamp type for each square number. Brands can be found in Squareland can be combined as required without further regulations. Prove for every positive integer $n$: there are the same number in the two countries possibilities to send a letter with stamps worth a total of $n$ cents. It makes no difference if you have the same stamps on arrange a letter differently. (Stephan Wagner)

Prove that if for a real number $a $ , $a+\frac {1}{a} $is integer then $a^n+\frac {1}{a^n} $ is also integer where $n$ is positive integer.

Let $a,b,c>0$ so that $a+b+c=1$. Then prove that $$(a+2ab+2ac+bc)^a(b+2bc+2ba+ca)^b(c+2ca+2cb+ab)^c\le1.$$ [i]Proposed by Marius Drăgan[/i]

We call the arrangement of $n$ ones and $m$ zeros around the circle as good, if we can swap neighboring zero and one in such a way that we get an arrangement, that differs from the original by rotation. For what natural $m$ and $n$ does a good arrangement exist?

The set of $ x$-values satisfying the equation $ x^{\log_{10} x} \equal{} \frac{x^3}{100}$ consists of: $ \textbf{(A)}\ \frac{1}{10} \qquad \textbf{(B)}\ \text{10, only} \qquad \textbf{(C)}\ \text{100, only} \qquad \textbf{(D)}\ \text{10 or 100, only} \qquad \textbf{(E)}\ \text{more than two real numbers.}$

(a) Decide whether the fields of the $8 \times 8$ chessboard can be numbered by the numbers $1, 2, \dots , 64$ in such a way that the sum of the four numbers in each of its parts of one of the forms [list][img]http://www.artofproblemsolving.com/Forum/download/file.php?id=28446[/img][/list] is divisible by four. (b) Solve the analogous problem for [list][img]http://www.artofproblemsolving.com/Forum/download/file.php?id=28447[/img][/list]