Determine, with proof, all integers $ x$ for which $ x(x\plus{}1)(x\plus{}7)(x\plus{}8)$ is a perfect square.

Prove the value of the expression $$\displaystyle \dfrac {\sqrt{n + \sqrt{0}} + \sqrt{n + \sqrt{1}} + \sqrt{n + \sqrt{2}} + \cdots + \sqrt{n + \sqrt{n^2-1}} + \sqrt{n + \sqrt{n^2}}} {\sqrt{n - \sqrt{0}} + \sqrt{n - \sqrt{1}} + \sqrt{n - \sqrt{2}} + \cdots + \sqrt{n - \sqrt{n^2-1}} + \sqrt{n - \sqrt{n^2}}}$$ is constant over all positive integers $n$. ([i]Folklore (also Philippines Olympiad)[/i])

Find all prime $p$ numbers such that $p^3-4p+9$ is perfect square.

Consider a positive integer $a > 1$. If $a$ is not a perfect square then at the next move we add $3$ to it and if it is a perfect square we take the square root of it. Define the trajectory of a number $a$ as the set obtained by performing this operation on $a$. For example the cardinality of $3$ is $\{3, 6, 9\}$. Find all $n$ such that the cardinality of $n$ is finite. The following part problems may attract partial credit. $\textbf{(a)}$Show that the cardinality of the trajectory of a number cannot be $1$ or $2$. $\textbf{(b)}$Show that $\{3, 6, 9\}$ is the only trajectory with cardinality $3$. $\textbf{(c)}$ Show that there for all $k \geq 3$, there exists a number such that the cardinality of its trajectory is $k$. $\textbf{(d)}$ Give an example of a number with cardinality of trajectory as infinity.

Let $a_n = 2^{3n-1} + 3^{6n-2} + 5^{6n-3}$. Compute gcd$(a_1, a_2, ... , a_{25})$

Suppose $a$, $b$, $c$, and $d$ are non-negative integers such that \[(a+b+c+d)(a^2+b^2+c^2+d^2)^2=2023.\] Find $a^3+b^3+c^3+d^3$. [i]Proposed by Connor Gordon[/i]

Show that the only integral solution to \[\left\{ \begin{array}{l} xy + yz + zx = 3n^2 - 1\\ x + y + z = 3n \\ \end{array} \right. \] with $x \geq y \geq z$ is $x=n+1$, $y=n$, $z=n-1$.

Assume that $a$ is a given irrational number. (a) Prove that for each positive real number $\epsilon$ there exists at least one integer $q\ge0$ such that $aq-\lfloor aq\rfloor<\epsilon$. (b) Prove that for given $\epsilon>0$ there exist infinitely many rational numbers $\frac pq$ such that $q>0$ and $\left|a-\frac pq\right|<\frac\epsilon q$.

For any integer $n+1,\ldots, 2n$ ($n$ is a natural number) consider its greatest odd divisor. Prove that the sum of all these divisors equals $n^2.$

The product $a_1 \cdot a_2 \cdot ... \cdot a_{100}$ is written on the board , where $a_1$, $a_2$, $ ... $, $a_{100}$, are natural numbers. Let's consider $99$ expressions, each of which is obtained by replacing one of the multiplication signs with an addition sign. It is known that the values of exactly $32$ of these expressions are even. What is the largest number of even numbers among $a_1$, $a_2$, $ ... $, $a_{100}$ could it be?

Let $n$ be a natural number with $120$ positive divisors (including $1$ and $n$). For each divisor $d$ of $n$, let $q$ be the quotient and $r$ the remainder when dividing $4n - 3$ by $d$. Let $Q$ be the sum of all the quotients $q$, and $R$ the sum of all the remainders $r$ for the $120$ divisions of $4n - 3$ by $d$. Determine all posible values of $Q - 4R$

For all $n\in\mathbb{N}$, we denote by $s(n)$ the sum of its digits. Find all integers $k\geq 2$ such that there exist $a,b\in\mathbb{N}$ with $$s(n^3+an+b)\equiv s(n)\pmod k,$$ for all $n\in\mathbb{N}^*$.

Find all integer solutions $ (a_1,a_2,\dots,a_{2008})$ of the following equation: $ (2008\minus{}a_1)^2\plus{}(a_1\minus{}a_2)^2\plus{}\dots\plus{}(a_{2007}\minus{}a_{2008})^2\plus{}a_{2008}^2\equal{}2008$

The number $N$ is the product of $200$ distinct positive integers. Prove that it has at least 19901 distinct divisors (including 0 and itself). [I]Proposed by A. Golovanov[/i]

Let $ a$ and $ b$ be two positive integers such that $ a \cdot b \plus{} 1$ divides $ a^{2} \plus{} b^{2}$. Show that $ \frac {a^{2} \plus{} b^{2}}{a \cdot b \plus{} 1}$ is a perfect square.

A monkey in Zoo becomes lucky if he eats three different fruits. What is the largest number of monkeys one can make lucky, by having $20$ oranges, $30$ bananas, $40$ peaches and $50$ tangerines? Justify your answer. (A): $30$ (B): $35$ (C): $40$ (D): $45$ (E): None of the above.

Find all positive integers $n$ for wich $\phi(\phi (n))$ divides $n$.

Considering the three-digit natural numbers, how many of them, when adding two of their digits, are double of their remainder? Justify your answer.

Given an integer $ m\geq$ 2, m positive integers $ a_1,a_2,...a_m$. Prove that there exist infinitely many positive integers n, such that $ a_{1}1^{n} \plus{} a_{2}2^{n} \plus{} ... \plus{} a_{m}m^{n}$ is composite.

[b]p1.[/b] Ivan Ivanovich came to the store with $20$ rubles. The store sold brooms for $1$ ruble. $17$ kopecks and basins for $1$ rub. $66$ kopecks (there are no other products left in the store). How many brooms and how many basins does he need to buy in order to spend as much money as possible? (Note: $1$ ruble = $100$ kopecks) [b]p2.[/b] On the road from city A to city B there are kilometer posts. On each pillar, on one side, the distance to city A is written, and on the other, to B. In the morning, a tourist passed by a pillar on which one number was twice the size of the other. After walking another $10$ km, the tourist saw a post on which the numbers differed exactly three times. What is the distance from A to B? List all possibilities. [b]p3.[/b] On New Year's Eve, geraniums, crocuses and cacti stood in a row (from left to right) on the windowsill. Every morning, Masha, wiping off the dust, swaps the places of the flower on the right and the flower in the center. During the day, Tanya, while watering flowers, swaps places between the one in the center and the one on the left. In what order will the flowers be in 365 days on the next New Year's Eve? [b]p4.[/b] What is the smallest number of digits that must be written in a row so that by crossing out some digits you can get any three-digit natural number from $100$ to $999$? [b]p5.[/b] An ordinary irreducible fraction was written on the board, the numerator and denominator of which were positive integers. The numerator was added to its denominator and a new fraction was obtained. The denominator was added to the numerator of the new fraction to form a third fraction. When the numerator was added to the denominator of the third fraction, the result was $13/23$. What fraction was written on the board? [b]p6.[/b] The number $x$ is such that $15\%$ of it and $33\%$ of it are positive integers. What is the smallest number $x$ (not necessarily an integer!) with this property? [b]p7.[/b] A radio-controlled toy leaves a certain point. It moves in a straight line, and on command can turn left exactly $17^o$ (relative to the previous direction of movement). What is the smallest number of commands required for the toy to pass through the starting point again? [b]p8.[/b] The square is divided by straight lines into $25$ rectangles (fig. 1). The areas of some of them are indicated in the figure (not to scale). Find the area of the rectangle marked with a question mark. [img]https://cdn.artofproblemsolving.com/attachments/0/9/591c93421067123d50382744f9d28357acf83a.png[/img] [b]p9.[/b] Petya multiplied all natural numbers from $1$ to his age inclusive. The result is a number $$8 \,\, 841 \,\,761993 \,\,739 \,\,701954 \,\,543 \,\,616 \,\,000 \,\,000.$$ How old is Petya? [b]p10.[/b] There are $100$ integers written in a line, and the sum of any three in a row is equal to $10$ or $11$. The first number is equal to one. What could the last number be? List all possibilities. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c2416727_soros_olympiad_in_mathematics]here.[/url]

Consider all points with integer coordinates in the carthesian plane. If one draws a circle with M(0,0) and a well-chose radius r, the circles goes through some of those points. (like circle with $r=2\sqrt2$ goes through 4 points) Prove that $\forall n\in \mathbb{N}, \exists r$ so that the circle with midpoint 0,0 and radius $r$ goes through at least $n$ points.

Given an integer $n\ge 2$, a function $f:\mathbb{Z}\rightarrow \{1,2,\ldots,n\}$ is called [i]good[/i], if for any integer $k,1\le k\le n-1$ there exists an integer $j(k)$ such that for every integer $m$ we have \[f(m+j(k))\equiv f(m+k)-f(m) \pmod{n+1}. \] Find the number of [i]good[/i] functions.

Find all natural numbers that can be written in the form $\frac{4ab}{ab^2+1}$ for some natural $a,b$. (nooonuii)

Some positive integer has an even number of divisors. Anya wants to split these divisors into pairs so that the products of the numbers in each pair have the same number of divisors. Prove that she can do this in exactly one way. [i]Proposed by Oleksii Masalitin[/i]

The Blablabla set contains all the different seven-digit numbers that can be formed with the digits $2, 3, 4, 5, 6, 7$ and $8$. Prove that there are not two Blablabla numbers such that one of them is divisible by the other.