[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 $\angle XAY$.

Let $ \{\phi_n(x) \}$ be a sequence of functions belonging to $ L^2(0,1)$ and having norm less that $ 1$ such that for any subsequence $ \{\phi_{n_k}(x) \}$ the measure of the set \[ \{x \in (0,1) : \;|\frac{1}{\sqrt{N}} \sum _{k=1}^N \phi_{n_k}(x)| \geq y\ \}\] tends to $ 0$ as $ y$ and $ N$ tend to infinity. Prove that $ \phi_n$ tends to $ 0$ weakly in the function space $ L^2(0,1).$ [i]F. Moricz[/i]

Given that $\frac{1+3+5+\cdots+(2n-1)}{2+4+6+\cdots+(2n)}=\frac{2011}{2012}$, determine n.

A set of distinct positive integers is called [i]singular [/i] if, for each of its elements, after crossing out that element, the remaining ones can be grouped into two sets with no common elements such that the sum of the elements in the two groups is the same. Find the smallest positive integer $n>1$ such that there exists a singular set $A$ with $n$ items.

[b](a)[/b] Each of Peter and Basil thinks of three positive integers. For each pair of his numbers, Peter writes down the greatest common divisor of the two numbers. For each pair of his numbers, Basil writes down the least common multiple of the two numbers. If both Peter and Basil write down the same three numbers, prove that these three numbers are equal to each other. [b](b)[/b] Can the analogous result be proved if each of Peter and Basil thinks of four positive integers instead?

Prove that for any $n \ge 1$, $LCM _{0\le k\le n} \big \{$ $n \choose k$ $\big\} = \frac{1}{n + 1} LCM \{1, 2,3,...,n + 1\}$

A real number $a \ne 0$ is given. Determine all functions $f : R \to R$ satisfying $f(x)f(y) + f(x + y) = axy$ for all real numbers $x, y$.

Let $AB$ be a diameter of a circle $\Omega$ and $P$ be any point not on the line through $AB$. Suppose that the line through $PA$ cuts $\Omega$ again at $U$, and the line through $PB$ cuts $\Omega$ at $V$. Note that in case of tangency, $U$ may coincide with $A$ or $V$ might coincide with $B$. Also, if $P$ is on $\Omega$ then $P=U=V$. Suppose that $|PU|=s|PA|$ and $|PV|=t|PB|$ for some $0\le s,t\in \mathbb{R}$. Determine $\cos \angle APB$ in terms of $s,t$.

Let the function $f:R \to R$ satisfies the following conditions: 1) for all $x, y\in R$, $ f(x +y) = f(x) +f(y)$ 2)$ f(1)=1$ 3) for all $x \ne 0$ , $ f(1/x) =\frac{f(x)}{x^2}$ Prove that for all $x \in R$, $f(x) = x$.

Let $f:[0,1]\to\mathbb R$ be a continuous function. Define a sequence of functions $f_n:[0,1]\to\mathbb R$ in the following way: $$f_0(x)=f(x),\qquad f_{n+1}(x)=\int^x_0f_n(t)\text dt,\qquad n=0,1,2,\ldots.$$Prove that if $f_n(1)=0$ for all $n$, then $f(x)\equiv0$.