Ryan Alweiss storms into the Fine Hall common room with a gigantic eraser and erases all integers $n$ in the interval $[2, 728]$ such that $3^t$ doesn’t divide $n!$, where $t = \left\lceil \frac{n-3}{2} \right\rceil$. Find the sum of the leftover integers in that interval modulo $1000$.

At Easter Day, $4$ children and their mothers participated in a game in which they had to find hidden chocolate eggs. Augustine found $4$ eggs, Bruno found $6$, Carlos found $9$ and Daniel found $12$. Mrs. Gómez found the same number of eggs as her son, Mrs. Junco found twice as many eggs as her son, Mrs. Messi found three times as many eggs as her son, and Mrs. Núñez found five times as many eggs as her son. At the end of the day, they put all the eggs in boxes, with $18$ eggs in each box, and only one egg was left over. Determine who the mother of each child is.

Let $n$ be a positive integer. We call a $n$-tuple $(a_1, . . . , a_n)$ of positive integers [i]nice [/i] if $\bullet$ $gcd (a_1, . . . , a_n) = 1$, and $\bullet$ $a_i|a_{i-1} + a_{i+1}$, for all $i = 1, . . . , n$ (we define $a_0 = a_n$ and $a_{n+1} = a1$ here). Find the maximal possible value of the sum $a_1 +...+ a_n$ if $(a_1, . . . , a_n)$ is a nice $n$-tuple.

Solve the equation in integer numbers $$y^3-x^3=91$$

The number $1- \frac12 +\frac13-\frac14+...+\frac{1}{2n-1}-\frac{1}{2n}$ is represented as an irreducible fraction. If $3n+1$ is a prime number, prove that the numerator of this fraction is a multiple of $3n + 1$.

If $n$ is a positive integer such that $2n$ has $28$ positive divisors and $3n$ has $30$ positive divisors, then how many positive divisors does $6n$ have? $\text{(A)}\ 32 \qquad \text{(B)}\ 34 \qquad \text{(C)}\ 35 \qquad \text{(D)}\ 36\qquad \text{(E)}\ 38$

Let $a$ be a positive integer such that the last two digits of $a^2$ are both non-zero. When the last two digits of $a^2$ are deleted, the resulting number is still a perfect square. Find, with justification, all possible values of $a$.

Let $N$ be the number of integral solutions of the equation \[x^2 - y^2 = z^3 - t^3\] satisfying the condition $0 \leq x, y, z, t \leq 10^6$, and let $M$ be the number of integral solutions of the equation \[x^2 - y^2 = z^3 - t^3 + 1\] satisfying the condition $0 \leq x, y, z, t \leq 10^6$. Prove that $N >M.$

We are given that $a, b$ and $c$ are whole numbers (i.e. positive integers) . Prove that if $a = b + c$ then $a^4 + b^4 + c^4$ is double the square of a whole number. (Folklore)

Consider two positive integers $a$ and $b$ such that $a^{n+1} + b^{n+1}$ is divisible by $a^n + b^n$ for infinitely many positive integers $n$. Is it necessarily true that $a = b$? (Boris Frenkin)

Compute the remainder of $3^{2021}$ mod $15$

Do there exist three different prime numbers $p$, $q$ and $r$ such that $p^2 + d$ is divisible by $qr$, $q^2 + d$ is divisible by $rp$ and $r^2 + d$ is divisible by $pq$, if (a) $d = 10$; (b) $d = 11$? (V Senderov)

Find all positive integers $n\geq1$ such that there exists a pair $(a,b)$ of positive integers, such that $a^2+b+3$ is not divisible by the cube of any prime, and $$n=\frac{ab+3b+8}{a^2+b+3}.$$

Given an integer $a>1$. Prove that there exists a sequence of positive integers \[ n_1, n_2, n_3, \ldots \] Such that \[ \gcd(a^{n_i+1} + a^{n_i} - 1, \ a^{n_j + 1} + a^{n_j} - 1) =1 \] For every $i \neq j$.

Find a way to write all the digits of $1$ to $9$ in a sequence and without repetition, so that the numbers determined by any two consecutive digits of the sequence are divisible by $7$ or $13$.

The positive integers $ \alpha, \beta, \gamma$ are the roots of a polynomial $ f(x)$ with degree $ 4$ and the coefficient of the first term is $ 1$. If there exists an integer such that $ f(\minus{}1)\equal{}f^2(s)$. Prove that $ \alpha\beta$ is not a perfect square.

Let $\mathbb{Q}$ be the set of rational numbers. Determine all functions $f : \mathbb{Q}\to\mathbb{Q}$ satisfying both of the following conditions. [list=disc] [*] $f(a)$ is not an integer for some rational number $a$. [*] For any rational numbers $x$ and $y$, both $f(x + y) - f(x) - f(y)$ and $f(xy) - f(x)f(y)$ are integers. [/list]

[u]Round 1[/u] [b]p1.[/b] Evaluate the sum $1-2+3-...-208+209-210$. [b]p2.[/b] Tony has $14$ beige socks, $15$ blue socks, $6$ brown socks, $8$ blond socks and $7$ black socks. If Tony picks socks out randomly, how many socks does he have to pick in order to guarantee a pair of blue socks? [b]p3.[/b] The price of an item is increased by $25\%$, followed by an additional increase of $20\%$. What is the overall percentage increase? [u]Round 2[/u] [b]p4.[/b] A lamp post is $20$ feet high. How many feet away from the base of the post should a person who is $5$ feet tall stand in order to cast an 8-foot shadow? [b]p5.[/b] How many positive even two-digit integers are there that do not contain the digits $0$, $1$, $2$, $3$ or $4$? [b]p6.[/b] In four years, Jack will be twice as old as Jill. Three years ago, Jack was three times as old as Jill. How old is Jack? [u]Round 3[/u] [b]p7.[/b] Let $x \Delta y = x y^2 -2y$. Compute $20\Delta 18$. [u]p8.[/u] A spider crawls $14$ feet up a wall. If Cheenu is standing $6$ feet from the wall, and is $6$ feet tall, how far must the spider jump to land on his head? [b]p9.[/b] There are fourteen dogs with long nails and twenty dogs with long fur. If there are thirty dogs in total, and three do not have long fur or long nails, how many dogs have both long hair and long nails? [u]Round 4[/u] [b]p10.[/b] Exactly $420$ non-overlapping square tiles, each $1$ inch by $1$ inch, tesselate a rectangle. What is the least possible number of inches in the perimeter of the rectangle? [b]p11.[/b] John drives $100$ miles at fifty miles per hour to see a cat. After he discovers that there was no cat, he drives back at a speed of twenty miles per hour. What was John’s average speed in the round trip? [b]p12.[/b] What percent of the numbers $1,2,3,...,1000$ are divisible by exactly one of the numbers $4$ and $5$? PS. You should use hide for answers. Rounds 5-8 have been posted [url=https://artofproblemsolving.com/community/c3h3165992p28809294]here [/url] and 9-12 [url=https://artofproblemsolving.com/community/c3h3166045p28809814]here[/url]. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

find the smallest integer $n\geq1$ such that the equation : $$a^2+b^2+c^2-nd^2=0 $$ has $(0,0,0,0)$ as unique solution .

Determine the integers $n$ where $$|2n^2+9n+4|$$ is a prime number.

For a positive integer number $r$, the sequence $a_{1},a_{2},...$ defined by $a_{1}=1$ and $a_{n+1}=\frac{na_{n}+2(n+1)^{2r}}{n+2}\forall n\geq 1$. Prove that each $a_{n}$ is positive integer number, and find $n's$ for which $a_{n}$ is even.

Suppose $a$, $b$, $c$, and $d$ are distinct positive integers such that $a^b$ divides $b^c$, $b^c$ divides $c^d$, and $c^d$ divides $d^a$. [list](a) Is it possible to determine which of the numbers $a$, $b$, $c$, $d$ is the smallest? (b) Is it possible to determine which of the numbers $a$, $b$, $c$, $d$ is the largest?[/list]

Show that for all $n\ge 3$ there are $n$ different positive integers $x_1,x_2, ...,x_n$ such that $$\frac{1}{x_1}+\frac{1}{x_2}+...+\frac{1}{x_n}= 1.$$

It is known that unequal numbers $a$,$b$ and $c$ are successive members of an arithmetic progression, all of them are greater than $1000$ and all are squares of natural numbers. Find the smallest possible value of $b$.

Let $n$ be a positive integer, let $p$ be prime and let $q$ be a divisor of $(n + 1)^p - n^p$. Show that $p$ divides $q - 1$.