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 $c$.

A finite set $S$ of unit squares is chosen out of a large grid of unit squares. The squares of $S$ are tiled with isosceles right triangles of hypotenuse $2$ so that the triangles do not overlap each other, do not extend past $S$, and all of $S$ is fully covered by the triangles. Additionally, the hypotenuse of each triangle lies along a grid line, and the vertices of the triangles lie at the corners of the squares. Show that the number of triangles must be a multiple of $4$.

A rectangular floor measures $ a$ by $ b$ feet, where $ a$ and $ b$ are positive integers with $ b > a$. An artist paints a rectangle on the floor with the sides of the rectangle parallel to the sides of the floor. The unpainted part of the floor forms a border of width $ 1$ foot around the painted rectangle and occupies half of the area of the entire floor. How many possibilities are there for the ordered pair $ (a,b)$? $ \textbf{(A)}\ 1\qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D)}\ 4\qquad\textbf{(E)}\ 5$

Let $\mathcal{S}$ be a set of $10$ points in a plane that lie within a disk of radius $1$ billion. Define a $move$ as picking a point $P \in \mathcal{S}$ and reflecting it across $\mathcal{S}$'s centroid. Does there always exist a sequence of at most $1500$ moves after which all points of $\mathcal{S}$ are contained in a disk of radius $10$? [i]Advaith Avadhanam[/i]

In a quadrilateral $ABCD$ we have $\angle A = \angle C = 90^o$. Let $E$ be a point in the interior of $ABCD$. Let $M$ be the midpoint of $BE$. Prove that $\angle ADB = \angle EDC$ if and only if $|MA| = |MC|$.

Let $A, B, C$, and $D$ be points on a circle, in that order, such that $\overline{AD}$ is a diameter of the circle. Let $E$ be the intersection of $\overleftrightarrow{AB}$ and $\overleftrightarrow{DC}$, let $F$ be the intersection of $\overleftrightarrow{AC}$ and $\overleftrightarrow{BD}$, and let $G$ be the intersection of $\overleftrightarrow{EF}$ and $\overleftrightarrow{AD}$. If $AD = 8$, $AE = 9$, and $DE = 7$, compute $EG$.

Find the number of rectangles that can be formed inside a fixed regular dodecagon ($12$-gon) where each side of the rectangle lies on either a side or a diagonal of the dodecagon. The diagram below shows three of those rectangles. [asy] unitsize(40); real r = pi/6; pair A1 = (cos(r),sin(r)); pair A2 = (cos(2r),sin(2r)); pair A3 = (cos(3r),sin(3r)); pair A4 = (cos(4r),sin(4r)); pair A5 = (cos(5r),sin(5r)); pair A6 = (cos(6r),sin(6r)); pair A7 = (cos(7r),sin(7r)); pair A8 = (cos(8r),sin(8r)); pair A9 = (cos(9r),sin(9r)); pair A10 = (cos(10r),sin(10r)); pair A11 = (cos(11r),sin(11r)); pair A12 = (cos(12r),sin(12r)); draw(A1--A2--A3--A4--A5--A6--A7--A8--A9--A10--A11--A12--cycle); filldraw(A2--A1--A8--A7--cycle, mediumgray, linewidth(1.2)); draw(A4--A11); draw(0.365*A3--0.365*A12, linewidth(1.2)); dot(A1); dot(A2); dot(A3); dot(A4); dot(A5); dot(A6); dot(A7); dot(A8); dot(A9); dot(A10); dot(A11); dot(A12); [/asy]

Let $n\ge 3$ be an integer. In a country there are $n$ airports and $n$ airlines operating two-way flights. For each airline, there is an odd integer $m\ge 3$, and $m$ distinct airports $c_1, \dots, c_m$, where the flights offered by the airline are exactly those between the following pairs of airports: $c_1$ and $c_2$; $c_2$ and $c_3$; $\dots$ ; $c_{m-1}$ and $c_m$; $c_m$ and $c_1$. Prove that there is a closed route consisting of an odd number of flights where no two flights are operated by the same airline.

Let $p$ be a prime number, and let $a_1, a_2, \dots , a_p$ and $b_1, b_2, \dots , b_p$ be $2p$ (not necessarily distinct) integers chosen from the set $\{1, 2, \dots , p - 1\}$. Prove that there exist integers $i$ and $j$ such that $1 \le i < j \le p$ and $p$ divides $a_ib_j-a_jb_i$.

Let $P,Q,R$ be the polynomials with real or complex coefficients such that at least one of them is not constant. If $P^n+Q^n+R^n = 0$, prove that $n < 3.$

A chord $AB = a$ is drawn in a circle of radius $B$. A circle with center on line $AB$ passes through $A$ and intersects this circle a second time at point $C$. Let $M$ be an arbitrary point of the second circle. Straight lines $MA$ and $MC$ intersect the first circle a second time at points $P$ and $Q$. Find $PQ$.

Suppose three direction on the plane . We draw $ 11$ lines in each direction . Find maximum number of the points on the plane which are on three lines .

In triangle $ABC, O$ is the circumcenter, $H$ is the orthocenter. Construct the circumcircles of triangles $CHB, CHA$ and $AHB$, and let their centers be $A_1, B_1, C_1$, respectively. Prove that triangles $ABC$ and $A_1B_1C_1$ are congruent, and their nine-point circles coincide.

In the figure, points $A$, $B$, $C$ and $D$ are on the same line and are the centers of four tangent circles at the same point. Segment $AB$ measures $8$ and segment $CD$ measures $4$. The circumferences woth centers $A$ and $C$ are of equal size. We know that the sum of the areas of the two medium circles is equivalent to the sum of the areas of the small and large circles. What is the length of segment $AD$? [img]https://cdn.artofproblemsolving.com/attachments/d/4/378243b9f4203e103af266e551eadccfc96adf.png[/img]

The positive integer $n$ is given and for all positive integers $k$, $1\leq k\leq n$, denote by $a_{kn}$ the number of all ordered sequences $(i_1,i_2,\ldots,i_k)$ of positive integers which verify the following two conditions: a) $1\leq i_1<i_2< \cdots i_k \leq n$; b) $i_{r+1}-i_r \equiv 1 \pmod 2$, for all $r \in\{1,2,\ldots,k-1\}$. Compute the number $a(n) = \sum\limits_{k=1}^n a_{kn}$. [i]Ioan Tomescu[/i]

In a simple graph, there are two disjoint set of vertices $A$ and $B$ where $A$ has $k$ and $B$ has $2016$ vertices. Four numbers are written to each vertex using the colors red, green, blue and black. There is no any edge at the beginning. For each vertex in $A$, we first choose a color and then draw all edges from this vertex to the vertices in $B$ having a larger number with the chosen color. It is known that for each vertex in $B$, the set of vertices in $A$ connected to this vertex are different. Find the minimal possible value of $k$.

An $n \times n$ board is given. In the top left corner cell there is a fox, whereas in the bottom left corner cell there is a rabbit. Every minute, the fox and the rabbit jump to a neighbouring cell at the same time. The fox can jump only to neighbouring cells that are below it or on its right, whereas the rabbit can only jump to the cells above it or in its right. They continue like this until they have no possible moves. The fox catches the rabbit if at a certain moment they are in the same cell, otherwise the rabbit gets away. Find all natural numbers $n$ for which the fox has a winning strategy to catch the rabbit. [i](Note: Two squares are considered neighbours if they have a common side.)[/i]

Let $n \ge k \ge 3$ be integers. Show that for every integer sequence $1 \le a_1 < a_2 < . . . < a_k \le n$ one can choose non-negative integers $b_1, b_2, . . . , b_k$, satisfying the following conditions: [list=i] [*] $0 \le b_i \le n$ for each $1 \le i \le k$, [*] all the positive $b_i$ are distinct, [*] the sums $a_i + b_i$, $1 \le i \le k$, form a permutation of the first $k$ terms of a non-constant arithmetic progression. [/list]

A rectangle with sides $ n$ and $p$ is divided into $np$ unit squares. Initially there are m unitary squares painted black and the remaining painted white. The following processoccurs repeatedly: if a unit square painted white has at minus two sides in common with squares painted black then Its color also turns black. Find the smallest integer $m$ that satisfies the property: there exists an initial position of $m$ black unit squares such that the entire $ n \times p$ rectangle is painted black when repeat the process a finite number of times.

Compute the number of nonnegative integer triples $(x,y,z)$ which satisfy $4x+2y+z\leq36$.

Let $p>1$. Show that there exists a constant $K_{p} >0$ such that for every $x,y\in \mathbb{R}$ with $|x|^{p}+|y|^{p}=2$, we have $$(x-y)^{2} \leq K_{p}(4-(x+y)^{2}).$$

A complete graph of $n$ vertices is a set of $n$ vertices and those vertices are connected in pairs by edges. Suppose the graph has $n$ vertices $A_1, A_2, ..., A_n$, the cycle is a set of edges of the form $A_{i_1}A_{i_2}, A_{i_2}A_{i_3},..., A_{i_m}A_{i_1}$ with $i_1, i_2, ..., i_m \in {1, 2, ..., n}$ double one different. We call $m$ the length of this cycle. Find the smallest positive integer$ n$ such that for every way of coloring all edges of a complete graph of $n$ vertices, each edge filled with one of three different colors, there is always a cycle of even length with the same color. PS. The same problem with another wording [url=https://artofproblemsolving.com/community/c6h151391p852296]here [/url].

If $x_1=5, x_2=401$, and \[ x_n=x_{n-2}-\frac 1{x_{n-1}} \] for every $3\leq n \leq m$, what is the largest value of $m$? $ \textbf{(A)}\ 406 \qquad\textbf{(B)}\ 2005 \qquad\textbf{(C)}\ 2006 \qquad\textbf{(D)}\ 2007 \qquad\textbf{(E)}\ \text{None of the above} $

Let $ABC$ be a triangle with side lengths $AB = 2011$, $BC = 2012$, $AC = 2013$. Create squares $S_1 =ABB'A''$, $S_2 = ACC''A'$ , and $S_3 = CBB''C'$ using the sides $AB$, $AC$, $BC$ respectively, so that the side $B'A''$ is on the opposite side of $AB$ from $C$, and so forth. Let square $S_4$ have side length $A''A' $, square $S_5$ have side length $C''C'$, and square $S_6$ have side length $B''B'$. Let $A(S_i)$ be the area of square $S_i$ . Compute $\frac{A(S_4)+A(S_5)+A(S_6)}{A(S_1)+A(S_2)+A(S_3)}$?

An $L$ is an arrangement of $3$ adjacent unit squares formed by deleting one unit square from a $2 \times 2$ square. a) How many $L$s can be placed on an $8 \times 8$ board (with no interior points overlapping)? b) Show that if any one square is deleted from a $1987 \times 1987$ board, then the remaining squares can be covered with $L$s (with no interior points overlapping).