Find all ordered triples $(a,b, c)$ of real numbers such that $$(a-b)(b-c) + (b-c)(c-a) + (c-a)(a-b) = 0.$$

Let $ABC$ be a fixed triangle. Point $D$ is an arbitrary point on the side $BC$. Point $P$ is fixed on $AD$. The circumcircle of triangle $BPD$ meets $AB$ at $E$ distinct from $B$. Point $Q$ varies on $AP$. Let $BQ$ and $CQ$ meet the circumcircles of triangles $BPD, CPD$ respectively at $F,Z$ distinct from $B,C$. Prove that the circumcircle $EFZ$ is through a fixed point distinct from $E$ and this fixed point is on the circumcircle of triangle $CPD$. Kostas Vittas

In an $n\times{n}$ grid, there are some cats living in each cell (the number of cats in a cell must be a non-negative integer). Every midnight, the manager chooses one cell: (a) The number of cats living in the chosen cell must be greater than or equal to the number of neighboring cells of the chosen cell. (b) For every neighboring cell of the chosen cell, the manager moves one cat from the chosen cell to the neighboring cell. (Two cells are called "neighboring" if they share a common side, e.g. there are only $2$ neighboring cells for a cell in the corner of the grid) Find the minimum number of cats living in the whole grid, such that the manager is able to do infinitely many times of this process.

Let $m$ be a line in the plane and $M$ a point not in $m$. Find the locus of the focus of the parabolas with vertex $M$ that are tangent to $m$.

Given triangle $ABC$ such that $AB- BC = \frac{AC}{\sqrt2}$ . Let $M$ be the midpoint of $AC$, and $N$ be the foot of the angle bisector from $B$. Prove that $\angle BMC + \angle BNC = 90^o$. (A.Akopjan)

Let $f(x)$ be the polynomial $\prod_{k=1}^{50} \bigl( x - (2k-1) \bigr)$. Let $c$ be the coefficient of $x^{48}$ in $f(x)$. When $c$ is divided by 101, what is the remainder? (The remainder is an integer between 0 and 100.)

In the triangle $ABC$ the incircle is tangent to the sides $AB$ and $AC$ at $D$ and $E$ respectively. The line $DE$ intersects the circumcircle at $P$ and $Q$, with $P$ in the small arc $AB$ and $Q$ in the small arc $AC$. If $P$ is the midpoint of the arc $AB$, find the angle A and the ratio $\frac{PQ}{BC}$.

Jackson's paintbrush makes a narrow strip that is $6.5$ mm wide. Jackson has enough paint to make a strip of 25 meters. How much can he paint, in $\text{cm}^2$? $\textbf{(A) }162{,}500\qquad\textbf{(B) }162.5\qquad\textbf{(C) }1{,}625\qquad\textbf{(D) }1{,}625{,}000\qquad\textbf{(E) }16{,}250$

For a given number $ n $, let us denote by $ p_n $ the probability that when randomly selecting a pair of integers $ k, m $ satisfying the conditions $ 0 \leq k \leq m \leq 2^n $ (the selection of each pair is equally probable) the number $\binom{m}{k}$ will be even. Calculate $ \lim_{n\to \infty} p_n $.

[b]p1.[/b] For what integers $x$ is it possible to find an integer $y$ such that $$x(x + 1) (x + 2) (x + 3) + 1 = y^2 ?$$ [b]p2.[/b] Two tangents to a circle are parallel and touch the circle at points $A$ and $B$, respectively. A tangent to the circle at any point $X$, other than $A$ or $B$, meets the first tangent at $Y$ and the second tangent at $Z$. Prove $AY \cdot BZ$ is independent of the position of $X$. [b]p3.[/b] If $a, b, c$ are positive real numbers, prove that $$8abc \le (b + c) (c + a) (a + b)$$ by first verifying the relation in the special case when $c = b$. [b]p4.[/b] Solve the equation $$\frac{x^2}{3}+\frac{48}{x^2}=10 \left( \frac{x}{3}-\frac{4}{x}\right)$$ [b]p5.[/b] Tom and Bill live on the same street. Each boy has a package to deliver to the other boy’s house. The two boys start simultaneously from their own homes and meet $600$ yards from Bill's house. The boys continue on their errand and they meet again $700$ yards from Tom's house. How far apart do the boy's live? [b]p6.[/b] A standard set of dominoes consists of $28$ blocks of size $1$ by $2$. Each block contains two numbers from the set $0,1,2,...,6$. We can denote the block containing $2$ and $3$ by $[2, 3]$, which is the same block as $[3, 2]$. The blocks $[0, 0]$, $[1, 1]$,..., $[6, 6]$ are in the set but there are no duplicate blocks. a) Show that it is possible to arrange the twenty-eight dominoes in a line, end-to-end, with adjacent ends matching, e. g., $... [3, 1]$ $[1, 1]$ $[1, 0]$ $[0, 6] ...$ . b) Consider the set of dominoes which do not contain $0$. Show that it is impossible to arrange this set in such a line. c) Generalize the problem and prove your generalization. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $N$ be a ten digit positive integer divisible by $7$. Suppose the first and the last digit of $N$ are interchanged and the resulting number (not necessarily ten digit) is also divisible by $7$ then we say that $N$ is a good integer. How many ten digit good integers are there?

Find the number of permutations $(a_1, a_2, . . . , a_{2013})$ of $(1, 2, \dots , 2013)$ such that there are exactly two indices $i \in \{1, 2, \dots , 2012\}$ where $a_i < a_{i+1}$.

Suppose that a sequence $t_{0}, t_{1}, t_{2}, ...$ is defined by a formula $t_{n} = An^{2} +Bn +c$ for all integers $n \geq 0$. Here $A, B$ and $C$ are real constants with $A \neq 0$. Determine values of $A, B$ and $C$ which give the greatest possible number of successive terms of the Fibonacci sequence.[i] The Fibonacci sequence is defined by[/i] $F_{0} = 0, F_{1} = 1$ [i]and[/i] $F_{m} = F_{m-1} + F_{m-2}$ [i]for[/i] $m \geq 2$.

Prove that the number $$\sqrt{1 + \frac{1}{n^2} + \frac{1}{(n+1)^2}}$$ is rational for all natural $n$.

For $m\ge 3,$ a list of $\binom m3$ real numbers $a_{ijk}$ $(1\le i<j<k\le m)$ is said to be [i]area definite[/i] for $\mathbb{R}^n$ if the inequality \[\sum_{1\le i<j<k\le m}a_{ijk}\cdot\text{Area}(\triangle A_iA_jA_k)\ge0\] holds for every choice of $m$ points $A_1,\dots,A_m$ in $\mathbb{R}^n.$ For example, the list of four numbers $a_{123}=a_{124}=a_{134}=1, a_{234}=-1$ is area definite for $\mathbb{R}^2.$ Prove that if a list of $\binom m3$ numbers is area definite for $\mathbb{R}^2,$ then it is area definite for $\mathbb{R}^3.$

problem 1 :A sequence is defined by$ x_1 = 1, x_2 = 4$ and $ x_{n+2} = 4x_{n+1} -x_n$ for $n \geq 1$. Find all natural numbers $m$ such that the number $3x_n^2 + m$ is a perfect square for all natural numbers $n$

You are given a convex quadrilateral $ABCD$. It is known that $\angle CAD = \angle DBA = 40^o$, $\angle CAB = 60^o$, $\angle CBD = 20^o$. Find the angle $\angle CDB $.

The altitudes of a triangle $\triangle ABC$ are used to form the sides of a second triangle $\triangle A_1B_1C_1$. The altitudes of $\triangle A_1B_1C_1$ are then used to form the sides of a third triangle $\triangle A_2B_2C_2$. Prove that $\triangle A_2B_2C_2$ is similar to $\triangle ABC$.

Dividing $x^{1951} - 1$ by $P(x) = x^4 + x^3 + 2x^2 + x + 1$ one gets a quotient and a remainder. Find the coefficient of $x^{14}$ in the quotient.

Prove that for all positive integers $n$, there exists a sequence of positive integers $a_1,a_2,\dots,a_n$ and $d_1,d_2,\dots,d_n$ satisfying all of the following three conditions. [list] [*] $\binom{2a_i}{a_i}$ is divisible by $d_i$ for all $i=1,2,\dots,n$ [*] $d_{i+1}=d_i+1$ for all $i=1,2,\dots, n-1$ [*] $d_i\neq m^k$ for all $i=1,2,\dots, n$ and positive integers $m$ and $k$ such that $k\geq 2$ [/list]

Let $a,b,c$ be positive integers. Prove that it is impossible to have all of the three numbers $a^2+b+c,b^2+c+a,c^2+a+b$ to be perfect squares.

Given that $a_na_{n-2}-a_{n-1}^2+a_n-na_{n-2}=-n^2+3n-1$ and $a_0=1$, $a_1=3$, find $a_{20}$.

In triangle $ABC$ with $CA=CB$, point $E$ lies on the circumcircle of $ABC$ such that $\angle ECB=90^{\circ}$. The line through $E$ parallel to $CB$ intersects $CA$ in $F$ and $AB$ in $G$. Prove that the center of the circumcircle of triangle $EGB$ lies on the circumcircle of triangle $ECF$. Proposed by Prithwijit De

Alice has two urns. Each urn contains four balls and on each ball a natural number is written. She draws one ball from each urn at random, notes the sum of the numbers written on them, and replaces the balls in the urns from which she took them. This she repeats a large number of times. Bill, on examining the numbers recorded, notices that the frequency with which each sum occurs is the same as if it were the sum of two natural numbers drawn at random from the range 1 to 4. What can he deduce about the numbers on the balls?

Let $ABCDEF$ be a regular hexagon with side 1. Point $X, Y$ are on sides $CD$ and $DE$ respectively, such that the perimeter of $DXY$ is $2$. Determine $\angle XAY$.