Find all the pairs of integers $ (m, n)$ such that $ \sqrt {n +\sqrt {2016}} +\sqrt {m-\sqrt {2016}} \in \mathbb {Q}.$

Find the set of all possible values of the expression $\lfloor m^2+\sqrt{2} n \rfloor$, where $m$ and $n$ are positive integers. Note: The symbol $\lfloor x\rfloor$ denotes the largest integer less than or equal to $x$.

The sequence $ u_n$, $ n\equal{}0,1,2,...$ is defined by $ u_0\equal{}0, u_1\equal{}1$ and for each $ n \ge 1$, $ u_{n\plus{}1}$ is the smallest positive integer greater than $ u_n$ such that $ \{ u_0,u_1,...,u_{n\plus{}1} \}$ contains no three elements in arithmetic progression. Find $ u_{100}$.

Let $\sigma(\cdot)$ denote the divisor sum function and $d(\cdot)$ denote the divisor counting function. Find all positve integers $n$ such that $\sigma(d(n))=n.$ [i]Andrei Bâra[/i]

[b]p1.[/b] How many ordered triples of integers $(a, b, c)$ where $1 \le a, b, c \le 10$ are such that for every natural number, the equation $(a + n)x^2 + (b + 2n)x + c + n = 0$ has at least one real root? [b]p2.[/b] Find the smallest integer $n$ such that we can cut a $n \times n$ grid into $5$ rectangles with distinct side lengths in $\{1, 2, 3..., 10\}$. Every value is used exactly once. [b]p3.[/b] A plane is flying at constant altitude along a circle of radius $12$ miles with center at a point $A$.The speed of the aircraft is v. At some moment in time, a missile is fired at the aircraft from the point $A$, which has speed v and is guided so that its velocity vector always points towards the aircraft. How far does the missile travel before colliding with the aircraft? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Find all integers $n$ for which there exists an equiangular $n$-gon whose side lengths are distinct rational numbers.

Does there exist a $4$-digit integer which cannot be changed into a multiple of $1992$ by changing $3$ of its digits?

For which $m > 1$ is $(m -1)!$ divisible by $m$?

Let $\mathbb{P}$ be the set of all primes and let $M$ be a subset of $\mathbb{P}$ with at least three elements. Suppose that for all $k \geq 1$ and for all subsets $A=\{p_1,p_2,\dots ,p_k \}$ of $M$ ,$A\neq M$ , all prime factors of $p_1p_2\dots p_k-1$ are in $M$ . Prove that $M=\mathbb{P}$.

Find all $a,n\in\mathbb{Z}^+$ ($a>2$) such that each prime divisor of $a^n-1$ is also prime divisor of $a^{3^{2016}}-1$

[hide=B stands for Bernoulli, G stands for Germain]they had two problem sets under those two names[/hide] [b]B1[/b] What is the sum of the first $5$ positive integers? [b]B2[/b] Bread picks a number $n$. He finds out that if he multiplies $n$ by $23$ and then subtracts $20$, he gets $46279$. What is $n$? [b]B3[/b] A [i]Harshad [/i] Number is a number that is divisible by the sum of its digits. For example, $27$ is divisible by $2 + 7 = 9$. Only one two-digit multiple of $9$ is not a [i]Harshad [/i] Number. What is this number? [b]B4 / G1[/b] There are $5$ red balls and 3 blue balls in a bag. Alice randomly picks a ball out of the bag and then puts it back in the bag. Bob then randomly picks a ball out of the bag. What is the probability that Alice gets a red ball and Bob gets a blue ball, assuming each ball is equally likely to be chosen? [b]B5[/b] Let $a$ be a $1$-digit positive integer and $b$ be a $3$-digit positive integer. If the product of $a$ and $b$ is a$ 4$-digit integer, what is the minimum possible value of the sum of $a$ and $b$? [b]B6 / G2[/b] A circle has radius $6$. A smaller circle with the same center has radius $5$. What is the probability that a dart randomly placed inside the outer circle is outside the inner circle? [b]B7[/b] Call a two-digit integer “sus” if its digits sum to $10$. How many two-digit primes are sus? [b]B8 / G3[/b] Alex and Jeff are playing against Max and Alan in a game of tractor with $2$ standard decks of $52$ cards. They take turns taking (and keeping) cards from the combined decks. At the end of the game, the $5$s are worth $5$ points, the $10$s are worth $10$ points, and the kings are worth 10 points. Given that a team needs $50$ percent more points than the other to win, what is the minimal score Alan and Max need to win? [b]B9 / G4[/b] Bob has a sandwich in the shape of a rectangular prism. It has side lengths $10$, $5$, and $5$. He cuts the sandwich along the two diagonals of a face, resulting in four pieces. What is the volume of the largest piece? [b]B10 / G5[/b] Aven makes a rectangular fence of area $96$ with side lengths $x$ and $y$. John makesva larger rectangular fence of area 186 with side lengths $x + 3$ and $y + 3$. What is the value of $x + y$? [b]B11 / G6[/b] A number is prime if it is only divisible by itself and $1$. What is the largest prime number $n$ smaller than $1000$ such that $n + 2$ and $n - 2$ are also prime? Note: $1$ is not prime. [b]B12 / G7[/b] Sally has $3$ red socks, $1$ green sock, $2$ blue socks, and $4$ purple socks. What is the probability she will choose a pair of matching socks when only choosing $2$ socks without replacement? [b]B13 / G8[/b] A triangle with vertices at $(0, 0)$,$ (3, 0)$, $(0, 6)$ is filled with as many $1 \times 1$ lattice squares as possible. How much of the triangle’s area is not filled in by the squares? [b]B14 / G10[/b] A series of concentric circles $w_1, w_2, w_3, ...$ satisfy that the radius of $w_1 = 1$ and the radius of $w_n =\frac34$ times the radius of $w_{n-1}$. The regions enclosed in $w_{2n-1}$ but not in $w_{2n}$ are shaded for all integers $n > 0$. What is the total area of the shaded regions? [b]B15 / G12[/b] $10$ cards labeled 1 through $10$ lie on a table. Kevin randomly takes $3$ cards and Patrick randomly takes 2 of the remaining $7$ cards. What is the probability that Kevin’s largest card is smaller than Patrick’s largest card, and that Kevin’s second-largest card is smaller than Patrick’s smallest card? [b]G9[/b] Let $A$ and $B$ be digits. If $125A^2 + B161^2 = 11566946$. What is $A + B$? [b]G11[/b] How many ordered pairs of integers $(x, y)$ satisfy $y^2 - xy + x = 0$? [b]G13[/b] $N$ consecutive integers add to $27$. How many possible values are there for $N$? [b]G14[/b] A circle with center O and radius $7$ is tangent to a pair of parallel lines $\ell_1$ and $\ell_2$. Let a third line tangent to circle $O$ intersect $\ell_1$ and $\ell_2$ at points $A$ and $B$. If $AB = 18$, find $OA + OB$. [b]G15[/b] Let $$ M =\prod ^{42}_{i=0}(i^2 - 5).$$ Given that $43$ doesn’t divide $M$, what is the remainder when M is divided by $43$? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Find all positive integers $n$ such that $3^n - 2^n - 1$ is a perfect square.

For any natural number $ n, $ denote $ x_n $ as being the number of natural numbers of $ n $ digits that are divisible by $ 4 $ and formed only with the digits $ 0,1,2 $ or $ 6. $ [b]a)[/b] Calculate $ x_1,x_2,x_3,x_4. $ [b]b)[/b] Find the natural number $ m $ such that $$ 1+\left\lfloor \frac{x_2}{x_1}\right\rfloor +\left\lfloor \frac{x_3}{x_2}\right\rfloor +\left\lfloor \frac{x_4}{x_3}\right\rfloor +\cdots +\left\lfloor \frac{x_{m+1}}{x_m}\right\rfloor =2016 , $$ where $ \lfloor\rfloor $ is the usual integer part.

For some positive integer $n$, Katya wrote on the board next to each other numbers $2^n$ and $14^n$ (in this order), thus forming a new number $A$. Can the number $A - 1$ be prime? [i]Proposed by Oleksii Masalitin[/i]

Let $a_1,a_2,...,a_n$ be $n$ different positive integers where $n\ge 1$. Show that $$\sum_{i=1}^n a_i^3 \ge \left(\sum_{i=1}^n a_i\right)^2$$

Determine all strictly increasing functions $f:\mathbb{N}_0\to\mathbb{N}_0$ which satisfy \[f(x)\cdot f(y)\mid (1+2x)\cdot f(y)+(1+2y)\cdot f(x)\]for all non-negative integers $x{}$ and $y{}$.

Ingrid and Erik are playing a game. For a given odd prime $p$, the numbers $1, 2, 3, ..., p-1$ are written on a blackboard. The players take turns making moves with Ingrid starting. A move consists of one of the players crossing out a number on the board that has not yet been crossed out. If the product of all currently crossed out numbers is $1 \pmod p$ after the move, the player whose move it was receives one point, otherwise, zero points are awarded. The game ends after all numbers have been crossed out. The player who has received the most points by the end of the game wins. If both players have the same score, the game ends in a draw. For each $p$, determine which player (if any) has a winning strategy

In a mathematics test number of participants is $N < 40$. The passmark is fixed at $65$. The test results are the following: The average of all participants is $66$, that of the promoted $71$ and that of the repeaters $56$. However, due to an error in the wording of a question, all scores are increased by $5$. At this point the average of the promoted participants becomes $75$ and that of the non-promoted $59$. (a) Find all possible values ​​of $N$. (b) Find all possible values ​​of $N$ in the case where, after the increase, the average of the promoted had become $79$ and that of non-promoted $47$.

What percent of the numbers $1, 2, 3, ... 1000$ are divisible by exactly one of the numbers $4$ and $5?$

An arbitrary prime $ p$ is given. If an integer sequence $ (n_1 , n_2 , \cdots , n_k )$ satisfying the conditions - For all $ i\equal{} 1, 2, \cdots , k$, $ n_i \geq \frac{p\plus{}1}{2}$ - For all $ i\equal{} 1, 2, \cdots , k$, $ p^{n_i} \minus{} 1$ is divisible by $ n_{i\plus{}1}$, and $ \frac{p^{n_i} \minus{} 1}{n_{i\plus{}1}}$ is coprime to $ n_{i\plus{}1}$. Let $ n_{k\plus{}1} \equal{} n_1$. exists not for $ k\equal{}1$, but exists for some $ k \geq 2$, then call the prime a good prime. Prove that a prime is good iff it is not $ 2$.

Let $a$ and $b$ be positive integers. Show that if $a \cdot b$ is even, then there are positive integers $c$ and $d$ with $a^2 + b^2 + c^2 = d^2$; if, on the other hand, $a\cdot b$ is odd, there are no such positive integers $c$ and $d$.

a) Two positive integers are chosen. The sum is revealed to logician $A$, and the sum of squares is revealed to logician $B$. Both $A$ and $B$ are given this information and the information contained in this sentence. The conversation between $A$ and $B$ goes as follows: $B$ starts B: ` I can't tell what they are.' A: ` I can't tell what they are.' B: ` I can't tell what they are.' A: ` I can't tell what they are.' B: ` I can't tell what they are.' A: ` I can't tell what they are.' B: ` Now I can tell what they are.' What are the two numbers? b) When $B$ first says that he cannot tell what the two numbers are, $A$ receives a large amount of information. But when $A$ first says that he cannot tell what the two numbers are, $B$ already knows that $A$ cannot tell what the two numbers are. What good does it do $B$ to listen to $A$?

Show that every number of the form $2^{p}3^{q}$ , where $p,q$ are nonnegative integers, divides some number of the form $a_{2k}10^{2k}+a_{2k-2}10^{2k-2}+...+a_{2}10^{2}+a_{0}$, where $a_{2i}\in\{1,2,...,9\}$

For which integer \( h \), are there infinitely many positive integers \( n \) such that \( \lfloor \sqrt{h^2 + 1} \cdot n \rfloor \) is a perfect square? (Here \( \lfloor x \rfloor \) denotes the integer part of the real number \( x \)?

For each positive integer $\,n,\;S(n)\,$ is defined to be the greatest integer such that, for every positive integer $\,k\leq S(n),\;n^{2}\,$ can be written as the sum of $\,k\,$ positive squares. [b]a.)[/b] Prove that $\,S(n)\leq n^{2}-14\,$ for each $\,n\geq 4$. [b]b.)[/b] Find an integer $\,n\,$ such that $\,S(n)=n^{2}-14$. [b]c.)[/b] Prove that there are infintely many integers $\,n\,$ such that $S(n)=n^{2}-14.$