Prove that there exists two strictly increasing sequences $(a_{n})$ and $(b_{n})$ such that $a_{n}(a_{n}+1)$ divides $b^{2}_{n}+1$ for every natural n.

Find all pairs $(k,n)$ of positive integers such that \[ k!=(2^n-1)(2^n-2)(2^n-4)\cdots(2^n-2^{n-1}). \] [i]Proposed by Gabriel Chicas Reyes, El Salvador[/i]

Let $m$ and $n$ be positive integers such that $m>n$ and $m \equiv n \pmod{2}$. If $(m^2-n^2+1) \mid n^2-1$, then prove that $m^2-n^2+1$ is a perfect square. (proposed by G. Batzaya, folklore)

Prove that for any polynomial $P$ with integer coefficients and any natural number $k$ there exists a natural number $n$ such that $P(1) + P(2) + ...+ P(n)$ is divisible by $k$.

Prove that if $p$ is an odd prime, then $p^2(p^2 -1999)$ is divisible by $6$ but not by $12$.

Find all pairs of integers $x,y$ for which \[x^3+x^2+x=y^2+y.\]

Let $r$ and $s$ be positive integers. Define $a_0 = 0$, $a_1 = 1$, and $a_n = ra_{n-1} + sa_{n-2}$ for $n \geq 2$. Let $f_n = a_1a_2\cdots a_n$. Prove that $\displaystyle\frac{f_n}{f_kf_{n-k}}$ is an integer for all integers $n$ and $k$ such that $0 < k < n$. [i]Evan O' Dorney.[/i]

For which $ p$ prime numbers, there is an integer root of the polynominal $ 1 \plus{} p \plus{} Q(x^1)\cdot\ Q(x^2)\ldots\ Q(x^{2p \minus{} 2})$ such that $ Q(x)$ is a polynominal with integer coefficients?

Let $a,b$ and $c$ be positive integers, no two of which have a common divisor greater than $1$. Show that $2abc-ab-bc-ca$ is the largest integer which cannot be expressed in the form $xbc+yca+zab$, where $x,y,z$ are non-negative integers.

A four - digit natural number which is divisible by $7$ is given. The number obtained by writing the digits in reverse order is also divisible by $7$. Furthermore, both the numbers leave the same remainder when divided by $37$. Find the 4-digit number.

Find all pairs of positive integers $x,y$ such that $$x^3+y^3=4(x^2y+xy^2-5).$$

Given $ m$ and $ n$ as relatively prime positive integers greater than one, show that \[ \frac{\log_{10} m}{\log_{10} n}\] is not a rational number.

Let $F$ be the set of all fractions $m/n$ where $m$ and $n$ are positive integers with $m+n\le 2005$. Find the largest number $a$ in $F$ such that $a < 16/23$.

In natural numbers $m,n$ Solve : $n(n+1)(n+2)(n+3)=m(m+1)^2(m+2)^3(m+3)^4$

Consider a polynomial $P(x) = \prod^9_{j=1}(x+d_j),$ where $d_1, d_2, \ldots d_9$ are nine distinct integers. Prove that there exists an integer $N,$ such that for all integers $x \geq N$ the number $P(x)$ is divisible by a prime number greater than 20. [i]Proposed by Luxembourg[/i]

Let $a$ and $b$ be two positive integers. Prove that the integer \[a^2+\left\lceil\frac{4a^2}b\right\rceil\] is not a square. (Here $\lceil z\rceil$ denotes the least integer greater than or equal to $z$.) [i]Russia[/i]

Find eight positive integers $n_1, n_2, \dots , n_8$ with the following property: For every integer $k$, $-1985 \leq k \leq 1985$, there are eight integers $a_1, a_2, \dots, a_8$, each belonging to the set $\{-1, 0, 1\}$, such that $k=\sum_{i=1}^{8} a_i n_i .$

Find the value of $$(2^{40}+12^{41}+23^{42}+67^{43}+87^{44})^{45!+46}\mod11$$ (variation but same answer) [hide=Answer]3[/hide]

Show that the fraction $ \frac{n^2+n-1}{n^2+2n}$ is irreducible for every positive integer n.

Let $f : \mathbb{Z} \rightarrow \mathbb{Z}^+$ be a function, and define $h : \mathbb{Z} \times \mathbb{Z} \rightarrow \mathbb{Z}^+$ by $h(x, y) = \gcd (f(x), f(y))$. If $h(x, y)$ is a two-variable polynomial in $x$ and $y$, prove that it must be constant.

[b]p1.[/b] Vivek has three letters to send out. Unfortunately, he forgets which letter is which after sealing the envelopes and before putting on the addresses. He puts the addresses on at random sends out the letters anyways. What are the chances that none of the three recipients get their intended letter? [b]p2.[/b] David is a horrible bowler. Luckily, Logan and Christy let him use bumpers. The bowling lane is $2$ meters wide, and David's ball travels a total distance of $24$ meters. How many times did David's bowling ball hit the bumpers, if he threw it from the middle of the lane at a $60^o$ degree angle to the horizontal? [b]p3.[/b] Find $\gcd \,(212106, 106212)$. [b]p4.[/b] Michael has two fair dice, one six-sided (with sides marked $1$ through $6$) and one eight-sided (with sides marked $1-8$). Michael play a game with Alex: Alex calls out a number, and then Michael rolls the dice. If the sum of the dice is equal to Alex's number, Michael gives Alex the amount of the sum. Otherwise Alex wins nothing. What number should Alex call to maximize his expected gain of money? [b]p5.[/b] Suppose that $x$ is a real number with $\log_5 \sin x + \log_5 \cos x = -1$. Find $$|\sin^2 x \cos x + \cos^2 x \sin x|.$$ [b]p6.[/b] What is the volume of the largest sphere that FIts inside a regular tetrahedron of side length $6$? [b]p7.[/b] An ant is wandering on the edges of a cube. At every second, the ant randomly chooses one of the three edges incident at one vertex and walks along that edge, arriving at the other vertex at the end of the second. What is the probability that the ant is at its starting vertex after exactly $6$ seconds? [b]p8.[/b] Determine the smallest positive integer $k$ such that there exist $m, n$ non-negative integers with $m > 1$ satisfying $$k = 2^{2m+1} - n^2.$$ [b]p9.[/b] For $A,B \subset Z$ with $A,B \ne \emptyset$, define $A + B = \{a + b|a \in A, b \in B\}$. Determine the least $n$ such that there exist sets $A,B$ with $|A| = |B| = n$ and $A + B = \{0, 1, 2,..., 2012\}$. [b]p10.[/b] For positive integers $n \ge 1$, let $\tau (n)$ and $\sigma (n)$ be, respectively, the number of and sum of the positive integer divisors of $n$ (including $1$ and $n$). For example, $\tau (1) = \sigma (1) = 1$ and $\tau (6) = 4$, $\sigma (6) = 12$. Find the number of positive integers $n \le 100$ such that $$\sigma (n) \le (\sqrt{n} - 1)^2 +\tau (n)\sqrt{n}.$$ PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $A_1,A_2,\cdots , A_n$ be arithmetic progressions of integers, each of $k$ terms, such that any two of these arithmetic progressions have at least two common elements. Suppose $b$ of these arithmetic progressions have common difference $d_1$ and the remaining arithmetic progressions have common difference $d_2$ where $0<b<n$. Prove that \[b \le 2\left(k-\frac{d_2}{gcd(d_1,d_2)}\right)-1.\]

Solve $ (n + 1)(n +10) = q^2$, for certain $q$ and maximum $n$.

Suppose that every integer has been given one of the colors red, blue, green, yellow. Let $x$ and $y$ be odd integers such that $|x| \ne |y|$. Show that there are two integers of the same color whose difference has one of the following values: $x,y,x+y,x-y$.

An arithmetic sequence of exactly $10$ positive integers has the property that any two elements are relatively prime. Compute the smallest possible sum of the $10$ numbers. [i]Proposed by Kyle Lee[/i]