Find all positive integers $n$ such that $n^{2}+2^{n}$ is square of an integer.

Let $k$ be an integer $\geq 3$. Sequence $\{a_n\}$ satisfies that $a_k = 2k$ and for all $n > k$, we have \[a_n = \begin{cases} a_{n-1}+1 & \text{if } (a_{n-1},n) = 1 \\ 2n & \text{if } (a_{n-1},n) > 1 \end{cases} \] Prove that there are infinitely many primes in the sequence $\{a_n - a_{n-1}\}$.

Find all triples $ (x,y,z)$ of natural numbers satisfying $ 2xz\equal{}y^2$ and $ x\plus{}z\equal{}1987$.

Find all composite positive integers \(m\) such that, whenever the product of two positive integers \(a\) and \(b\) is \(m\), their sum is a power of $2$. [i]Proposed by Harun Khan[/i]

The graphs of $y=3(x-h)^2+j$ and $y=2(x-h)^2+k$ have $y$-intercepts of $2013$ and $2014$, respectively, and each graph has two positive integer $x$-intercepts. Find $h$.

Show that there exists a set of infinite positive integers such that the sum of an arbitrary finite subset of these is never a perfect square. What happens if we change the condition from not being a perfect square to not being a perfect power?

The sequence of positive integers $a_0, a_1, a_2, . . .$ is defined by $a_0 = 3$ and $$a_{n+1} - a_n = n(a_n - 1)$$ for all $n \ge 0$. Determine all integers $m \ge 2$ for which $gcd (m, a_n) = 1$ for all $n \ge 0$.

The sequences $(a_n)$ and $(b_n)$ are defined by $a_1 = b_1 = 1$ and $a_{n+1} = a_n +b_n, b_{n+1} = a_nb_n$ for $n = 1,2,...$ Show that every two distinct terms of the sequence $(a_n)$ are coprime

Let $p$ be a prime number. Show that $7^p+3p-4$ is not a perfect square.

$2020$ positive integers are written in one line. Each of them starting with the third is divisible by previous and by the sum of two previous numbers. What is the smallest value the last number can take? A. Gribalko

[b]p1.[/b] Nicky is studying biology and has a tank of $17$ lizards. In one day, he can either remove $5$ lizards or add $2$ lizards to his tank. What is the minimum number of days necessary for Nicky to get rid of all of the lizards from his tank? [b]p2.[/b] What is the maximum number of spheres with radius $1$ that can fit into a sphere with radius $2$? [b]p3.[/b] A positive integer $x$ is sunny if $3x$ has more digits than $x$. If all sunny numbers are written in increasing order, what is the $50$th number written? [b]p4.[/b] Quadrilateral $ABCD$ satisfies $AB = 4$, $BC = 5$, $DA = 4$, $\angle DAB = 60^o$, and $\angle ABC = 150^o$. Find the area of $ABCD$. [b]p5. [/b]Totoro wants to cut a $3$ meter long bar of mixed metals into two parts with equal monetary value. The left meter is bronze, worth $10$ zoty per meter, the middle meter is silver, worth $25$ zoty per meter, and the right meter is gold, worth $40$ zoty per meter. How far, in meters, from the left should Totoro make the cut? [b]p6.[/b] If the numbers $x_1, x_2, x_3, x_4$, and $x5$ are a permutation of the numbers $1, 2, 3, 4$, and $5$, compute the maximum possible value of $$|x_1 - x_2| + |x_2 - x_3| + |x_3 - x_4| + |x_4 - x_5|.$$ [b]p7.[/b] In a $3 \times 4$ grid of $12$ squares, find the number of paths from the top left corner to the bottom right corner that satisfy the following two properties: $\bullet$ The path passes through each square exactly once. $\bullet$ Consecutive squares share a side. Two paths are considered distinct if and only if the order in which the twelve squares are visited is different. For instance, in the diagram below, the two paths drawn are considered the same. [img]https://cdn.artofproblemsolving.com/attachments/7/a/bb3471bbde1a8f58a61d9dd69c8527cfece05a.png[/img] [b]p8.[/b] Scott, Demi, and Alex are writing a computer program that is $25$ ines long. Since they are working together on one computer, only one person may type at a time. To encourage collaboration, no person can type two lines in a row, and everyone must type something. If Scott takes $10$ seconds to type one line, Demi takes $15$ seconds, and Alex takes $20$ seconds, at least how long, in seconds, will it take them to finish the program? [b]p9.[/b] A hand of four cards of the form $(c, c, c + 1, c + 1)$ is called a tractor. Vinjai has a deck consisting of four of each of the numbers $7$, $8$, $9$ and $10$. If Vinjai shuffles and draws four cards from his deck, compute the probability that they form a tractor. [b]p10. [/b]The parabola $y = 2x^2$ is the wall of a fortress. Totoro is located at $(0, 4)$ and fires a cannonball in a straight line at the closest point on the wall. Compute the y-coordinate of the point on the wall that the cannonball hits. [b]p11. [/b]How many ways are there to color the squares of a $10$ by $10$ grid with black and white such that in each row and each column there are exactly two black squares and between the two black squares in a given row or column there are exactly [b]4[/b] white squares? Two configurations that are the same under rotations or reflections are considered different. [b]p12.[/b] In rectangle $ABCD$, points $E$ and $F$ are on sides $AB$ and $CD$, respectively, such that $AE = CF > AD$ and $\angle CED = 90^o$. Lines $AF, BF, CE$ and $DE$ enclose a rectangle whose area is $24\%$ of the area of $ABCD$. Compute $\frac{BF}{CE}$ . [b]p13.[/b] Link cuts trees in order to complete a quest. He must cut $3$ Fenwick trees, $3$ Splay trees and $3$ KD trees. If he must also cut 3 trees of the same type in a row at some point during his quest, in how many ways can he cut the trees and complete the quest? (Trees of the same type are indistinguishable.) [b]p14.[/b] Find all ordered pairs (a, b) of positive integers such that $\sqrt{64a + b^2} + 8 = 8\sqrt{a} + b$. [b]p15.[/b] Let $ABCDE$ be a convex pentagon such that $\angle ABC = \angle BCD = 108^o$, $\angle CDE = 168^o$ and $AB =BC = CD = DE$. Find the measure of $\angle AEB$ PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Find all integer sequences of the form $ x_i, 1 \le i \le 1997$, that satisfy $ \sum_{k\equal{}1}^{1997} 2^{k\minus{}1} x_{k}^{1997}\equal{}1996\prod_{k\equal{}1}^{1997}x_k$.

For a positive integer $n$, let $A_n$ be the set of all positive numbers greater than $1$ and less than $n$ which are coprime to $n$. Find all $n$ such that all the elements of $A_n$ are prime numbers.

Minivan chooses a prime number. Then every second, he adds either the digit $1$ or the digit $3$ to the right end of his number (after the unit digit), such that the new number is also a prime. Can he continue indefinitely? [i](Proposed by Wong Jer Ren)[/i]

A positive integer $ K$ is given. Define the sequence $ (a_n)$ by $ a_1\equal{}1$ and $ a_n$ is the $ n$-th natural number greater than $ a_{n\minus{}1}$ which is congruent to $ n$ modulo $ K$. $ (a)$ Find an explicit formula for $ a_n$. $ (b)$ What is the result if $ K\equal{}2?$

Prove that it is possible to find $2^{n+1}$ of $2^n$ digit numbers containing only "$1$" and "$2$" as digits, such that every two of them distinguish at least in $2^{n-1}$ digits.

Find the remainder when $26!^{26} + 27!^{27}$ is divided by $29$.

Given a strictly increasing infinite sequence $\{a_n\}$ of positive real numbers such that for any $n\in N$: $$a_{n+2}=(a_{n+1}-a_{n})^{\sqrt{n}}+n^{-\sqrt{n}}$$ Prove that for any $C>0$ there exist a positive integer $m(C)$ (depended on $C$) such that $a_{m(C)}>C$.

Find all pairs $(a,b)$ of positive integers such that $a^2\mid b^3+1$ and $b^2\mid a^3+1$. [i]Linus Tang[/i]

Let $ m$ and $ n$ be positive integers such that $ 1 \le m < n$. In their decimal representations, the last three digits of $ 1978^m$ are equal, respectively, to the last three digits of $ 1978^n$. Find $ m$ and $ n$ such that $ m \plus{} n$ has its least value.

For any positive integer $n{}$ define $a_n=\{n/s(n)\}$ where $s(\cdot)$ denotes the sum of the digits and $\{\cdot\}$ denotes the fractional part.[list=a] [*]Prove that there exist infinitely many positive integers $n$ such that $a_n=1/2.$ [*]Determine the smallest positive integer $n$ such that $a_n=1/6.$ [/list] [i]Marius Burtea[/i]

A shipping company charges $.30l+.40w+.50h$ dollars to process a right rectangular prism-shaped box with dimensions $l,w,h$ in inches. The customers themselves are allowed to label the three dimensions of their box with $l,w,h$ for the purpose of calculating the processing fee. A customer finds that there are two different ways to label the dimensions of their box $B$ to get a fee of $\$8.10$, and two different ways to label $B$ to get a fee of $\$8.70$. None of the faces of $B$ are squares. Find the surface area of $B$, in square inches. [i]Proposed by Justin Hsieh[/i]

On the puzzle page of a newspaper this problem is proposed: “Two children, Antonio and José, have $160$ comics. Antonio counts his by $7$ by $7$ and there are $4$ left over. José counts his $ 8$ by $8$ and he also has $4$ left over. How many comics does he have each?" In the next issue of the newspaper this solution is given: “Antonio has $60$ comics and José has $100$.” Analyze this solution and indicate what a mathematician would do with this problem.

Let $a, b, c, d$ be digits such that $d > c > b > a \geq 0$. How many numbers of the form $1a1b1c1d1$ are multiples of $33$?

Prove that $n^4 - n^2$ is divisible by $12$ for all integers $n > 1$.