The sequence of positive integers $\{a_n, n\ge 1\}$ is such that $a_n\le a_{n+1}\le a_n+5$ and $a_n$ is divisible by $n$ for all $n \ge 1$. What are the possible values of $a_1$?

Determine all positive integers $n$ such that $\frac{2^n}{n^2}$ is an integer.

Prove that for every non-negative integer $ n, $ there exists a non-negative integer $ m $ such that $$ \left( 1+\sqrt{2} \right)^n=\sqrt m +\sqrt{m+1} . $$

[url=http://www.mathlinks.ro/Forum/viewtopic.php?t=107262]IMO 2007 HKTST 1[/url] Problem 6 Determine all pairs $(x,y)$ of positive integers such that $\frac{x^{2}y+x+y}{xy^{2}+y+11}$ is an integer.

An infinite arithmetic progression whose terms are positive integers contains the square of an integer and the cube of an integer. Show that it contains the sixth power of an integer.

For all $n$, $t_{n+1} = 2(t_n)^2 - 1$. Prove that gcd $(t_n,t_m) = 1$ if $n \ne m$.

[b]p1.[/b] Compute the value of $N$, where $$N = 818^3 - 6 \cdot 818^2 \cdot 209 + 12 \cdot 818 \cdot 209^2 - 8 \cdot 209^3$$ [b]p2.[/b] Suppose $x \le 2019$ is a positive integer that is divisible by $2$ and $5$, but not $3$. If $7$ is one of the digits in $x$, how many possible values of $x$ are there? [b]p3.[/b] Find all non-negative integer solutions $(a,b)$ to the equation $$b^2 + b + 1 = a^2.$$ [b]p4.[/b] Compute the remainder when $\sum^{2019}_{n=1} n^4$ is divided by $53$. [b]p5.[/b] Let $ABC$ be an equilateral triangle and $CDEF$ a square such that $E$ lies on segment $AB$ and $F$ on segment $BC$. If the perimeter of the square is equal to $4$, what is the area of triangle $ABC$? [img]https://cdn.artofproblemsolving.com/attachments/1/6/52d9ef7032c2fadd4f97d7c0ea051b3766b584.png[/img] [b]p6.[/b] $$S = \frac{4}{1\times 2\times 3}+\frac{5}{2\times 3\times 4} +\frac{6}{3\times 4\times 5}+ ... +\frac{101}{98\times 99\times 100}$$ Let $T = \frac54 - S$. If $T = \frac{m}{n}$ , where $m$ and $n$ are relatively prime integers, find the value of $m + n$. [b]p7.[/b] Find the sum of $$\sum^{2019}_{i=0}\frac{2^i}{2^i + 2^{2019-i}}$$ [b]p8.[/b] Let $A$ and $B$ be two points in the Cartesian plane such that $A$ lies on the line $y = 12$, and $B$ lies on the line $y = 3$. Let $C_1$, $C_2$ be two distinct circles that intersect both $A$ and $B$ and are tangent to the $x$-axis at $P$ and $Q$, respectively. If $PQ = 420$, determine the length of $AB$. [b]p9.[/b] Zion has an average $2$ out of $3$ hit rate for $2$-pointers and $1$ out of $3$ hit rate for $3$-pointers. In a recent basketball match, Zion scored $18$ points without missing a shot, and all the points came from $2$ or $3$-pointers. What is the probability that all his shots were $3$-pointers? [b]p10.[/b] Let $S = \{1,2, 3,..., 2019\}$. Find the number of non-constant functions $f : S \to S$ such that $$f(k) = f(f(k + 1)) \le f(k + 1) \,\,\,\, for \,\,\,\, all \,\,\,\, 1 \le k \le 2018.$$ Express your answer in the form ${m \choose n}$, where $m$ and $n$ are integers. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $ a, b, c, d,m, n \in \mathbb{Z}^\plus{}$ such that \[ a^2\plus{}b^2\plus{}c^2\plus{}d^2 \equal{} 1989,\] \[ a\plus{}b\plus{}c\plus{}d \equal{} m^2,\] and the largest of $ a, b, c, d$ is $ n^2.$ Determine, with proof, the values of $m$ and $ n.$

[b]p1.[/b] Consider a cube with side length $2$. Take any one of its vertices and consider the three midpoints of the three edges emanating from that vertex. What is the distance from that vertex to the plane formed by those three midpoints? [b]p2.[/b] Digits $H$, $M$, and $C$ satisfy the following relations where $\overline{ABC}$ denotes the number whose digits in base $10$ are $A$, $B$, and $C$. $$\overline{H}\times \overline{H} = \overline{M}\times \overline{C} + 1$$ $$\overline{HH}\times \overline{H} = \overline{MC}\times \overline{C} + 1$$ $$\overline{HHH}\times \overline{H} = \overline{MCC}\times \overline{C} + 1$$ Find $\overline{HMC}$. [b]p3.[/b] Two players play the following game on a table with fair two-sided coins. The first player starts with one, two, or three coins on the table, each with equal probability. On each turn, the player flips all the coins on the table and counts how many coins land heads up. If this number is odd, a coin is removed from the table. If this number is even, a coin is added to the table. A player wins when he/she removes the last coin on the table. Suppose the game ends. What is the probability that the first player wins? [b]p4.[/b] Cyclic quadrilateral $[BLUE]$ has right $\angle E$. Let $R$ be a point not in $[BLUE]$. If $[BLUR] =[BLUE]$, $\angle ELB = 45^o$, and $\overline{EU} = \overline{UR}$, find $\angle RUE$. [b]p5.[/b] There are two tracks in the $x, y$ plane, defined by the equations $$y =\sqrt{3 - x^2}\,\,\, \text{and} \,\,\,y =\sqrt{4- x^2}$$ A baton of length $1$ has one end attached to each track and is allowed to move freely, but no end may be picked up or go past the end of either track. What is the maximum area the baton can sweep out? [b]p6.[/b] For integers $1 \le a \le 2$, $1 \le b \le 10$,$ 1 \le c \le 12$, $1 \le d \le 18$, let $f(a, b, c, d)$ be the unique integer between $0$ and $8150$ inclusive that leaves a remainder of a when divided by $3$, a remainder of $b$ when divided by $11$, a remainder of $c$ when divided by $13$, and a remainder of $d$ when divided by $19$. Compute $$\sum_{a+b+c+d=23}f(a, b, c, d).$$ [b]p7.[/b] Compute $\cos ( \theta)$ if $$\sum^{\infty}_{n=0} \frac{ \cos (n\theta)}{3^n} = 1.$$ [b]p8.[/b] How many solutions does this equation $$\left(\frac{a+b}{2}\right)^2=\left(\frac{b+c}{2019}\right)^2$$ have in positive integers $a, b, c$ that are all less than $2019^2$? [b]p9.[/b] Consider a square grid with vertices labeled $1, 2, 3, 4$ clockwise in that order. Fred the frog is jumping between vertices, with the following rules: he starts at the vertex label $1$, and at any given vertex he jumps to the vertex diagonally across from him with probability $\frac12$ and the vertices adjacent to him each with probability $\frac14$ . After $2019$ jumps, suppose the probability that the sum of the labels on the last two vertices he has visited is $3$ can be written as $2^{-m} -2^{-n}$ for positive integers $m,n$. Find $m + n$. [b]p10.[/b] The base ten numeral system uses digits $0-9$ and each place value corresponds to a power of $10$. For example, $$2019 = 2 \cdot 10^3 + 0 \cdot 10^2 + 1 \cdot 10^1 + 9 \cdot 10^0.$$ Let $\phi =\frac{1 +\sqrt5}{2}$. We can define a similar numeral system, base , where we only use digits $0$ and $1$, and each place value corresponds to a power of . For example, $$11.01 = 1 \cdot \phi^1 + 1 \cdot \phi^0 + 0 \cdot \phi^{-1} + 1 \cdot \phi^{-2}$$ Note that base  representations are not unique, because, for example, $100_{\phi} = 11_{\phi}$. Compute the base $\phi$ representation of $7$ with the fewest number of $1$s. [b]p11.[/b] Let $ABC$ be a triangle with $\angle BAC = 60^o$ and with circumradius $1$. Let $G$ be its centroid and $D$ be the foot of the perpendicular from $A$ to $BC$. Suppose $AG =\frac{\sqrt6}{3}$ . Find $AD$. [b]p12.[/b] Let $f(a, b)$ be a function with the following properties for all positive integers $a \ne b$: $$f(1, 2) = f(2, 1)$$ $$f(a, b) + f(b, a) = 0$$ $$f(a + b, b) = f(b, a) + b$$ Compute: $$\sum^{2019}_{i=1} f(4^i - 1, 2^i) + f(4^i + 1, 2^i)$$ [b]p13.[/b] You and your friends have been tasked with building a cardboard castle in the two-dimensional Cartesian plane. The castle is built by the following rules: 1. There is a tower of height $2^n$ at the origin. 2. From towers of height $2^i \ge 2$, a wall of length $2^{i-1}$ can be constructed between the aforementioned tower and a new tower of height $2^{i-1}$. Walls must be parallel to a coordinate axis, and each tower must be connected to at least one other tower by a wall. If one unit of tower height costs $\$9$ and one unit of wall length costs $\$3$ and $n = 1000$, how many distinct costs are there of castles that satisfy the above constraints? Two castles are distinct if there exists a tower or wall that is in one castle but not in the other. [b]p14.[/b] For $n$ digits, $(a_1, a_2, ..., a_n)$ with $0 \le a_i < n$ for $i = 1, 2,..., n$ and $a_1 \ne 0$ define $(\overline{a_1a_2 ... a_n})_n$ to be the number with digits $a_1$, $a_2$, $...$, $a_n$ written in base $n$. Let $S_n = \{(a_1, a_2, a_3,..., a_n)| \,\,\, (n + 1)| (\overline{a_1a_2 ... a_n})_n, a_1 \ge 1\}$ be the set of $n$-tuples such that $(\overline{a_1a_2 ... a_n})_n$ is divisible by $n + 1$. Find all $n > 1$ such that $n$ divides $|S_n| + 2019$. [b]p15.[/b] Let $P$ be the set of polynomials with degree $2019$ with leading coefficient $1$ and non-leading coefficients from the set $C = \{-1, 0, 1\}$. For example, the function $f = x^{2019} - x^{42} + 1$ is in $P$, but the functions $f = x^{2020}$, $f = -x^{2019}$, and $f = x^{2019} + 2x^{21}$ are not in $P$. Define a [i]swap [/i]on a polynomial $f$ to be changing a term $ax^n$ to $bx^n$ where $b \in C$ and there are no terms with degree smaller than $n$ with coefficients equal to $a$ or $b$. For example, a swap from $x^{2019} + x^{17} - x^{15} + x^{10}$ to $x^{2019} + x^{17} - x^{15} - x^{10}$ would be valid, but the following swaps would not be valid: $$x^{2019} + x^3 \,\,\, \text{to} \,\,\, x^{2019}$$ $$x^{2019} + x^3 \,\,\, \text{to} \,\,\, x^{2019} + x^3 + x^2$$ $$x^{2019} + x^2 + x + 1 \,\,\, \text{to} \,\,\, x^{2019} - x^2 - x - 1$$ Let $B$ be the set of polynomials in $P$ where all non-leading terms have the same coefficient. There are $p$ polynomials that can be reached from each element of $B$ in exactly $s$ swaps, and there exist $0$ polynomials that can be reached from each element of $B$ in less than $s$ swaps. Compute $p \cdot s$, expressing your answer as a prime factorization. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Find all primes $q_1, q_2, q_3, q_4, q_5$ such that $q_1^4+q_2^4+q_3^4+q_4^4+q_5^4$ is the product of two consecutive even integers.

Find the largest positive integer with the property that each digit apart from the first and the last one is smaller than the arithmetic mean of her neighbours.

$(FRA 1)$ Let $a$ and $b$ be two nonnegative integers. Denote by $H(a, b)$ the set of numbers $n$ of the form $n = pa + qb,$ where $p$ and $q$ are positive integers. Determine $H(a) = H(a, a)$. Prove that if $a \neq b,$ it is enough to know all the sets $H(a, b)$ for coprime numbers $a, b$ in order to know all the sets $H(a, b)$. Prove that in the case of coprime numbers $a$ and $b, H(a, b)$ contains all numbers greater than or equal to $\omega = (a - 1)(b -1)$ and also $\frac{\omega}{2}$ numbers smaller than $\omega$

Prove the among $16$ consecutive integers it is always possible to find one which is relatively prime to all the rest.

$M$ is infinite set of natural numbers. If $a,b, a\neq b$ are in $M$, then $a^b+2$ or $a^b-2$ ( or both) are in $M$. Prove that there is composite number in $M$

What is the greatest common factor of $12345678987654321$ and $12345654321$? [i]Proposed by Evan Chen[/i]

Consider $10$ distinct positive integers that are all prime to each other (that is, there is no a prime factor common to all), but such that any two of them are not prime to each other. What is the smallest number of distinct prime factors that may appear in the product of $10$ numbers?

$(GBR 5)$ Let us define $u_0 = 0, u_1 = 1$ and for $n\ge 0, u_{n+2} = au_{n+1}+bu_n, a$ and $b$ being positive integers. Express $u_n$ as a polynomial in $a$ and $b.$ Prove the result. Given that $b$ is prime, prove that $b$ divides $a(u_b -1).$

the positive divisors $d_1,d_2,\cdots,d_k$ of a positive integer $n$ are ordered \[1=d_1<d_2<\cdots<d_k=n\] Suppose $d_7^2+d_{15}^2=d_{16}^2$. Find all possible values of $d_{17}$.

Find all nonnegative integers $m$ such that \[a_m=(2^{2m+1})^2+1 \] is divisible by at most two different primes.

For each $ n \in \mathbb{N}$ we have two numbers $ p_n, q_n$ with the following property: For exactly $ n$ distinct integer numbers $ x$ the number \[ x^2 \plus{} p_n \cdot x \plus{} q_n\] is the square of a natural number. (Note the definition of natural numbers includes the zero here.)

Prove that there are don't exist integers $a,b,c$ such that for every integer $x$ the number $A=(x+a)(x+b)(x+c)-x^3-1$ is divisible by $9$. [i]I. Tonov[/i]

A solid in the shape of a right circular cone is 4 inches tall and its base has a 3-inch radius. The entire surface of the cone, including its base, is painted. A plane parallel to the base of the cone divides the cone into two solids, a smaller cone-shaped solid $C$ and a frustum-shaped solid $F$, in such a way that the ratio between the areas of the painted surfaces of $C$ and $F$ and the ratio between the volumes of $C$ and $F$ are both equal to $k$. Given that $k=m/n$, where $m$ and $n$ are relatively prime positive integers, find $m+n$.

Put $1,2,....,2018$ (2018 numbers) in a row randomly and call this number $A$. Find the remainder of $A$ divided by $3$.

Find all primes $p$ such that the number of distinct positive factors of $p^2+2543$ is less than 16.

Let $(a_n)_{n\ge0}$ be a sequence of positive integers such that $a^2_n$ divides $a_{n-1}a_{n+1}$, for all $n \ge 1$. Prove that if there exists an integer $k \ge 2$ such that $a_k$ and $a_1$ are relatively prime, then $a_1$ divides $a_0$. (Malik Talbi)