For all integers $n\geq 1$ we define $x_{n+1}=x_1^2+x_2^2+\cdots +x_n^2$, where $x_1$ is a positive integer. Find the least $x_1$ such that 2006 divides $x_{2006}$.

Determine all positive integers $n$ such that $5^n - 1$ can be written as a product of an even number of consecutive integers.

[b]p13[/b] Let $ABCD$ be a square. Points $E$ and $F$ lie outside of $ABCD$ such that $ABE$ and $CBF$ are equilateral triangles. If $G$ is the centroid of triangle $DEF$, then find $\angle AGC$, in degrees. [b]p14 [/b]The silent reading $s(n)$ of a positive integer $n$ is the number obtained by dropping the zeros not at the end of the number. For example, $s(1070030) = 1730$. Find the largest $n < 10000$ such that $s(n)$ divides $n$ and $n\ne s(n)$. [b]p15[/b] Let $ABCDEFGH$ be a regular octagon with side length $12$. There exists a region $R$ inside the octagon such that for each point $X$ in $R$, exactly three of the rays $AX$, $BX$, $CX$, $DX$, $GX$, and $HX$ intersect segment $EF$. If the area of region $R$ can be expressed as $a -b\sqrt{c}$ for positive integers $a, b, c$ with $c$ squarefree, find $a + b + c$.

Find all positive integers n with the following property: There exists a positive integer $k$ and mutually distinct integers $x_1,x_2,\ldots,x_n$ such that the set $\{x_i+x_j\mid1\le i<j\le n\}$ is a set of distinct powers of $k$.

Find all prime numbers $p$ such that $(x + y)^{19} - x^{19} - y^{19}$ is a multiple of $p$ for any positive integers $x$, $y$.

The $\textit{arithmetic derivative}$ $D(n)$ of a positive integer $n$ is defined via the following rules: [list] [*] $D(1) = 0$; [*] $D(p)=1$ for all primes $p$; [*] $D(ab)=D(a)b+aD(b)$ for all positive integers $a$ and $b$. [/list] Find the sum of all positive integers $n$ below $1000$ satisfying $D(n)=n$.

How many pairs of positive integers $(a, b)$ satisfy the equation $log_a 16 = b$?

A [i]pucelana[/i] sequence is an increasing sequence of $16$ consecutive odd numbers whose sum is a perfect cube. How many pucelana sequences are there with $3$-digit numbers only?

Prove that for any positive integer $k,$ there exist finitely many sets $T$ satisfying the following two properties: $(1)T$ consists of finitely many prime numbers; $(2)\textup{ }\prod_{p\in T} (p+k)$ is divisible by $ \prod_{p\in T} p.$

$m < n$ are positive integers. Let $p=\frac{n^2+m^2}{\sqrt{n^2-m^2}}$. [b](a)[/b] Find three pairs of positive integers $(m,n)$ that make $p$ prime. [b](b)[/b] If $p$ is prime, then show that $p \equiv 1 \pmod 8$.

A natural number $n$ is at least two digits long. If we write a certain digit between the tens digit and the units digit of this number, we obtain six times the number $n$. Find all numbers $n$ with this property.

[b]p1.[/b] If $x + z = v$, $w + z = 2v$, $z - w = 2y$, and $y \ne 0$, compute the value of $$\left(x + y +\frac{x}{y} \right)^{101}.$$ [b]p2. [/b]Every minute, a snail picks one cardinal direction (either north, south, east, or west) with equal probability and moves one inch in that direction. What is the probability that after four minutes the snail is more than three inches away from where it started? [b]p3.[/b] What is the probability that a point chosen randomly from the interior of a cube is closer to the cube’s center than it is to any of the cube’s eight vertices? [b]p4.[/b] Let $ABCD$ be a rectangle where $AB = 4$ and $BC = 3$. Inscribe circles within triangles $ABC$ and $ACD$. What is the distance between the centers of these two circles? [b]p5.[/b] $C$ is a circle centered at the origin that is tangent to the line $x - y\sqrt3 = 4$. Find the radius of $C$. [b]p6.[/b] I have a fair $100$-sided die that has the numbers $ 1$ through $100$ on its sides. What is the probability that if I roll this die three times that the number on the first roll will be greater than or equal to the sum of the two numbers on the second and third rolls? [b]p7. [/b] List all solutions $(x, y, z)$ of the following system of equations with x, y, and z positive real numbers: $$x^2 + y^2 = 16$$ $$x^2 + z^2 = 4 + xz$$ $$y^2 + z^2 = 4 + yz\sqrt3$$ [b]p8.[/b] $A_1A_2A_3A_4A_5A_6A_7$ is a regular heptagon ($7$ sided-figure) centered at the origin where $A_1 = (\sqrt[91]{6}, 0)$. $B_1B_2B_3... B_{13}$ is a regular triskaidecagon ($13$ sided-figure) centered at the origin where $B_1 =(0,\sqrt[91]{41})$. Compute the product of all lengths $A_iB_j$ , where $i$ ranges between $1$ and $7$, inclusive, and $j$ ranges between $1$ and $13$, inclusive. [b]p9.[/b] How many three-digit integers are there such that one digit of the integer is exactly two times a digit of the integer that is in a different place than the first? (For example, $100$, $122$, and $124$ should be included in the count, but $42$ and $130$ should not.) [b]p10.[/b] Let $\alpha$ and $\beta$ be the solutions of the quadratic equation $$x^2 - 1154x + 1 = 0.$$ Find $\sqrt[4]{\alpha}+\sqrt[4]{\beta}$. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

A polynomial $f(x)$ with integer coefficients is said to be [i]tri-divisible[/i] if $3$ divides $f(k)$ for any integer $k$. Determine necessary and sufficient conditions for a polynomial to be tri-divisible.

Prove that for every given positive integer $n$, there exists a prime $p$ and an integer $m$ such that $(a)$ $p \equiv 5 \pmod 6$ $(b)$ $p \nmid n$ $(c)$ $n \equiv m^3 \pmod p$

Denote the set of all positive integers by $\mathbb{N}$, and the set of all ordered positive integers by $\mathbb{N}^2$. For all non-negative integers $k$, define [i]good functions of order k[/i] recursively for all non-negative integers $k$, among all functions from $\mathbb{N}^2$ to $\mathbb{N}$ as follows: (i) The functions $f(a,b)=a$ and $f(a,b)=b$ are both good functions of order $0$. (ii) If $f(a,b)$ and $g(a,b)$ are good functions of orders $p$ and $q$, respectively, then $\gcd(f(a,b),g(a,b))$ is a good function of order $p+q$, while $f(a,b)g(a,b)$ is a good function of order $p+q+1$. Prove that, if $f(a,b)$ is a good function of order $k\leq \binom{n}{3}$ for some positive integer $n\geq 3$, then there exist a positive integer $t\leq \binom{n}{2}$ and $t$ pairs of non-negative integers $(x_1,y_1),\ldots,(x_n,y_n)$ such that $$f(a,b)=\gcd(a^{x_1}b^{y_1},\ldots,a^{x_t}b^{y_t})$$ holds for all positive integers $a$ and $b$. [i]Proposed by usjl[/i]

Let $$\frac{r}{s}= 0.k_1k_2k_3 ...$$ be the decimal expansion of a rational number (If this is a terminating decimal, all $k_i$ from a certain one on are $0$). Prove that at least two of the numbers $$\sigma_1 = 10\frac{r}{s} - k_i, \sigma_2 = 10^2- (10k_1 + k_2),$$ $$\sigma_3 = 10^2 - (10^2k_1 + 10k_2 + k_3), ...$$ are equal.

Determine all polynomials $P{}$ with integer coefficients, satisfying $0 \leqslant P (n) \leqslant n!$ for all non-negative integers $n$. [i]Andrei Chirita[/i]

Let $ A\equal{}\{n: 1 \le n \le 2009^{2009},n \in \mathbb{N} \}$ and let $ S\equal{}\{n: n \in A,\gcd \left(n,2009^{2009}\right)\equal{}1\}$. Let $ P$ be the product of all elements of $ S$. Prove that \[ P \equiv 1 \pmod{2009^{2009}}.\] [i]Nanang Susyanto, Jogjakarta[/i]

[b]p1.[/b] Ben and his dog are walking on a path around a lake. The path is a loop $500$ meters around. Suddenly the dog runs away with velocity $10$ km/hour. Ben runs after it with velocity $8$ km/hour. At the moment when the dog is $250$ meters ahead of him, Ben turns around and runs at the same speed in the opposite direction until he meets the dog. For how many minutes does Ben run? [b]p2.[/b] The six interior angles in two triangles are measured. One triangle is obtuse (i.e. has an angle larger than $90^o$) and the other is acute (all angles less than $90^o$). Four angles measure $120^o$, $80^o$, $55^o$ and $10^o$. What is the measure of the smallest angle of the acute triangle? [b]p3.[/b] The figure below shows a $ 10 \times 10$ square with small $2 \times 2$ squares removed from the corners. What is the area of the shaded region? [img]https://cdn.artofproblemsolving.com/attachments/7/5/a829487cc5d937060e8965f6da3f4744ba5588.png[/img] [b]p4.[/b] Two three-digit whole numbers are called relatives if they are not the same, but are written using the same triple of digits. For instance, $244$ and $424$ are relatives. What is the minimal number of relatives that a three-digit whole number can have if the sum of its digits is $10$? [b]p5.[/b] Three girls, Ann, Kelly, and Kathy came to a birthday party. One of the girls wore a red dress, another wore a blue dress, and the last wore a white dress. When asked the next day, one girl said that Kelly wore a red dress, another said that Ann did not wear a red dress, the last said that Kathy did not wear a blue dress. One of the girls was truthful, while the other two lied. Which statement was true? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $S(n)$ be the sum of the decimal digits of $n$. For example. $S(2012)=2+0+1+2=5$. Prove that there is no integer $n>0$ for which $n-S(n)=9990$.

Let $A$ be a finite set of positive numbers , $B=\{\frac{a+b}{c+d} |a,b,c,d \in A \}$. Show that: $\left | B \right | \ge 2\left | A \right |^2-1 $, where $|X| $ be the number of elements of the finite set $X$. (High School Affiliated to Nanjing Normal University )

Let $a$ and $n$ denote positive integers such that $n|a^n-1$. Prove that the numbers $a+1,a^2+2, \cdots a^n+n$ all leave different remainders when divided by $n$.

Let $(F_n)_{n\in\mathbb{N}}$ be the Fibonacci sequence difined as $F_0=F_1=1, F_{n+2}=F_{n+1}+F_n, \forall n\in\mathbb{N}$. Show that for every nonnegative integer $r$ there is a term in the Fibonacci sequence that is divided by $r$.

$a)$ Let $n$ a positive integer. Prove that $gcd(n, \lfloor n\sqrt{2} \rfloor)<\sqrt[4]{8}\sqrt{n}$. $b)$ Prove that there are infinitely many positive integers $n$ such that $gcd(n, \lfloor n\sqrt{2} \rfloor)>\sqrt[4]{7.99}\sqrt{n}$.

Let $n$ be a given positive integer. Prove that there exist infinitely many integer triples $(x,y,z)$ such that \[nx^2+y^3=z^4,\ \gcd (x,y)=\gcd (y,z)=\gcd (z,x)=1.\]