Given a natural number $n \geq 2$, consider all the fractions of the form $\frac{1}{ab}$, where $a$ and $b$ are natural numbers, relative primes and such that: $a < b \leq n$, $a+b>n$. Show that for each $n$, the sum of all this fractions are $\frac12$.

Find all the pairs of prime numbers $ (p,q)$ such that $ pq|5^p\plus{}5^q.$

Let $n$ be a given positive integer. Determine all positive divisors $d$ of $3n^2$ such that $n^2 + d$ is the square of an integer.

Given $n$ integers $a_1 = 1, a_2,..., a_n$ such that $a_i \le a_{i+1} \le 2a_i$ ($i = 1, 2, 3,..., n - 1$) and whose sum is even. Find whether it is possible to divide them into two groups so that the sum of numbers in one group is equal to the sum of numbers in the other group.

For a given prime $ p > 2$ and positive integer $ k$ let \[ S_k \equal{} 1^k \plus{} 2^k \plus{} \ldots \plus{} (p \minus{} 1)^k\] Find those values of $ k$ for which $ p \, |\, S_k$.

Prove that there is no polynomial $P$ with integer coefficients such that $P\left(\sqrt[3]5+\sqrt[3]{25}\right)=5+\sqrt[3]5$.

[b]p1.[/b] A rectangular pool has diagonal $17$ units and area $120$ units$^2$. Joey and Rachel start on opposite sides of the pool when Rachel starts chasing Joey. If Rachel runs $5$ units/sec faster than Joey, how long does it take for her to catch him? [b]p2. [/b] Alice plays a game with her standard deck of $52$ cards. She gives all of the cards number values where Aces are $1$’s, royal cards are $10$’s and all other cards are assigned their face value. Every turn she flips over the top card from her deck and creates a new pile. If the flipped card has value $v$, she places $12 - v$ cards on top of the flipped card. For example: if she flips the $3$ of diamonds then she places $9$ cards on top. Alice continues creating piles until she can no longer create a new pile. If the number of leftover cards is $4$ and there are $5$ piles, what is the sum of the flipped over cards? [b]p3.[/b] There are $5$ people standing at $(0, 0)$, $(3, 0)$, $(0, 3)$, $(-3, 0)$, and $(-3, 0)$ on a coordinate grid at a time $t = 0$ seconds. Each second, every person on the grid moves exactly $1$ unit up, down, left, or right. The person at the origin is infected with covid-$19$, and if someone who is not infected is at the same lattice point as a person who is infected, at any point in time, they will be infected from that point in time onwards. (Note that this means that if two people run into each other at a non-lattice point, such as $(0, 1.5)$, they will not infect each other.) What is the maximum possible number of infected people after $t = 7$ seconds? [b]p4.[/b] Kara gives Kaylie a ring with a circular diamond inscribed in a gold hexagon. The diameter of the diamond is $2$ mm. If diamonds cost $\$100/ mm ^2$ and gold costs $\$50 /mm ^2$ , what is the cost of the ring? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

[url=https://artofproblemsolving.com/community/c677808][b]Cyprus IMO TST 2018[/b][/url] [url=https://artofproblemsolving.com/community/c6h1666662p10591751][b]Problem 1.[/b][/url] Determine all integers $n \geq 2$ for which the number $11111$ in base $n$ is a perfect square. [url=https://artofproblemsolving.com/community/c6h1666663p10591753][b]Problem 2.[/b][/url] Consider a trapezium $AB \Gamma \Delta$, where $A\Delta \parallel B\Gamma$ and $\measuredangle A = 120^{\circ}$. Let $E$ be the midpoint of $AB$ and let $O_1$ and $O_2$ be the circumcenters of triangles $AE \Delta$ and $BE\Gamma$, respectively. Prove that the area of the trapezium is equal to six time the area of the triangle $O_1 E O_2$. [url=https://artofproblemsolving.com/community/c6h1666660p10591747][b]Problem 3.[/b][/url] Find all triples $(\alpha, \beta, \gamma)$ of positive real numbers for which the expression $$K = \frac{\alpha+3 \gamma}{\alpha + 2\beta + \gamma} + \frac{4\beta}{\alpha+\beta+2\gamma} - \frac{8 \gamma}{\alpha+ \beta + 3\gamma}$$obtains its minimum value. [url=https://artofproblemsolving.com/community/c6h1666661p10591749][b]Problem 4.[/b][/url] Let $\Lambda= \{1, 2, \ldots, 2v-1,2v\}$ and $P=\{\alpha_1, \alpha_2, \ldots, \alpha_{2v-1}, \alpha_{2v}\}$ be a permutation of the elements of $\Lambda$. (a) Prove that $$\sum_{i=1}^v \alpha_{2i-1}\alpha_{2i} \leq \sum_{i=1}^v (2i-1)2i.$$(b) Determine the largest positive integer $m$ such that we can partition the $m\times m$ square into $7$ rectangles for which every pair of them has no common interior points and their lengths and widths form the following sequence: $$1,2,3,4,5,6,7,8,9,10,11,12,13,14.$$

Consider a positive integer $n=\overline{a_1a_2...a_k},k\ge 2$.A [i]trunk[/i] of $n$ is a number of the form $\overline{a_1a_2...a_t},1\le t\le k-1$.(For example,the number $23$ is a [i]trunk[/i] of $2351$.) By $T(n)$ we denote the sum of all [i]trunk[/i] of $n$ and let $S(n)=a_1+a_2+...+a_k$.Prove that $n=S(n)+9\cdot T(n)$.

We say $m \circ n$ for natural m,n $\Longleftrightarrow$ nth number of binary representation of m is 1 or mth number of binary representation of n is 1. and we say $m \bullet n$ if and only if $m,n$ doesn't have the relation $\circ$ We say $A \subset \mathbb{N}$ is golden $\Longleftrightarrow$ $\forall U,V \subset A$ that are finite and arenot empty and $U \cap V = \emptyset$,There exist $z \in A$ that $\forall x \in U,y \in V$ we have $z \circ x ,z \bullet y$ Suppose $\mathbb{P}$ is set of prime numbers.Prove if $\mathbb{P}=P_1 \cup ... \cup P_k$ and $P_i \cap P_j = \emptyset$ then one of $P_1,...,P_k$ is golden.

[b]p1.[/b] Find the smallest positive integer $N$ such that $2N -1$ and $2N +1$ are both composite. [b]p2.[/b] Compute the number of ordered pairs of integers $(a, b)$ with $1 \le a, b \le 5$ such that $ab - a - b$ is prime. [b]p3.[/b] Given a semicircle $\Omega$ with diameter $AB$, point $C$ is chosen on $\Omega$ such that $\angle CAB = 60^o$. Point $D$ lies on ray $BA$ such that $DC$ is tangent to $\Omega$. Find $\left(\frac{BD}{BC} \right)^2$. [b]p4.[/b] Let the roots of $x^2 + 7x + 11$ be $r$ and $s$. If $f(x)$ is the monic polynomial with roots $rs + r + s$ and $r^2 + s^2$, what is $f(3)$? [b]p5.[/b] Regular hexagon $ABCDEF$ has side length $3$. Circle $\omega$ is drawn with $AC$ as its diameter. $BC$ is extended to intersect $\omega$ at point $G$. If the area of triangle $BEG$ can be expressed as $\frac{a\sqrt{b}}{c}$ for positive integers $a, b, c$ with $b$ squarefree and $gcd(a, c) = 1$, find $a + b + c$. [b]p6.[/b] Suppose that $x$ and $y$ are positive real numbers such that $\log_2 x = \log_x y = \log_y 256$. Find $xy$. [b]p7.[/b] Call a positive three digit integer $\overline{ABC}$ fancy if $\overline{ABC} = (\overline{AB})^2 - 11 \cdot \overline{C}$. Find the sum of all fancy integers. [b]p8.[/b] Let $\vartriangle ABC$ be an equilateral triangle. Isosceles triangles $\vartriangle DBC$, $\vartriangle ECA$, and $\vartriangle FAB$, not overlapping $\vartriangle ABC$, are constructed such that each has area seven times the area of $\vartriangle ABC$. Compute the ratio of the area of $\vartriangle DEF$ to the area of $\vartriangle ABC$. [b]p9.[/b] Consider the sequence of polynomials an(x) with $a_0(x) = 0$, $a_1(x) = 1$, and $a_n(x) = a_{n-1}(x) + xa_{n-2}(x)$ for all $n \ge 2$. Suppose that $p_k = a_k(-1) \cdot a_k(1)$ for all nonnegative integers $k$. Find the number of positive integers $k$ between $10$ and $50$, inclusive, such that $p_{k-2} + p_{k-1} = p_{k+1} - p_{k+2}$. [b]p10.[/b] In triangle $ABC$, point $D$ and $E$ are on line segments $BC$ and $AC$, respectively, such that $AD$ and $BE$ intersect at $H$. Suppose that $AC = 12$, $BC = 30$, and $EC = 6$. Triangle BEC has area 45 and triangle $ADC$ has area $72$, and lines CH and AB meet at F. If $BF^2$ can be expressed as $\frac{a-b\sqrt{c}}{d}$ for positive integers $a$, $b$, $c$, $d$ with c squarefree and $gcd(a, b, d) = 1$, then find $a + b + c + d$. [b]p11.[/b] Find the minimum possible integer $y$ such that $y > 100$ and there exists a positive integer x such that $x^2 + 18x + y$ is a perfect fourth power. [b]p12.[/b] Let $ABCD$ be a quadrilateral such that $AB = 2$, $CD = 4$, $BC = AD$, and $\angle ADC + \angle BCD = 120^o$. If the sum of the maximum and minimum possible areas of quadrilateral $ABCD$ can be expressed as $a\sqrt{b}$ for positive integers $a$, $b$ with $b$ squarefree, then find $a + b$. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Consider the sequence $(a^n + 1)_{n\geq 1}$, with $a>1$ a fixed integer. i) Prove there exist infinitely many primes, each dividing some term of the sequence. ii) Prove there exist infinitely many primes, none dividing any term of the sequence. [i](Dan Schwarz)[/i]

Let $p=4k+3$ be a prime number. Find the number of different residues mod p of $(x^{2}+y^{2})^{2}$ where $(x,p)=(y,p)=1.$

Natural numbers $a$ and $b$ are relatively prime. Prove that the greatest common divisor of $(a+b)$ and $(a^2+b^2)$ is either $1$ or $2$.

Let $n,k$ are integers and $p$ is a prime number. Find all $(n,k,p)$ such that $|6n^2-17n-39|=p^k$

Find all functions $f : \mathbb{N} \rightarrow \mathbb{N} $ satisfying \[ f(mn) = \operatorname{lcm} (m,n) \cdot \gcd( f(m), f(n) ) \] for all positive integer $m,n$.

Box is thinking of a number, whose digits are all “$1$”. When he squares the number, the sum of its digit is $85$. How many digits is Box’s number?

How many natural numbers $n$ are there so that $n^2 + 2014$ is a perfect square? (A): $1$, (B): $2$, (C): $3$, (D): $4$, (E) None of the above.

Prove that there exists infinitely many positive integers $n$ such that $n, n+1$, and $n+2$ can be written as the sum of two perfect squares.

Prove that in every $16$-digit number there is a chain of one or more consecutive digits such that the product of those digits is a perfect square. For example, if the original number is $7862328578632785$ we can take the digits $6$, $2$ and $3$ whose product is $6^2$ (note that these appear consecutively in the number).

Let $m$ and $n$ be positive integers such that $m>n$. Define $x_k=\frac{m+k}{n+k}$ for $k=1,2,\ldots,n+1$. Prove that if all the numbers $x_1,x_2,\ldots,x_{n+1}$ are integers, then $x_1x_2\ldots x_{n+1}-1$ is divisible by an odd prime.

Four integers are marked on a circle. On each step we simultaneously replace each number by the difference between this number and next number on the circle, moving in a clockwise direction; that is, the numbers $ a,b,c,d$ are replaced by $ a\minus{}b,b\minus{}c,c\minus{}d,d\minus{}a.$ Is it possible after 1996 such to have numbers $ a,b,c,d$ such the numbers $ |bc\minus{}ad|, |ac \minus{} bd|, |ab \minus{} cd|$ are primes?

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.

Georg has bought lots of filled chocolates for a party, and when he counts how many he has, he discovers that the number is a prime number. He distributes so many of the chocolates as possible on $60$ trays with an equal number on each. He notes then that he has more than one piece left and that the number left pieces is not a prime number. How many pieces of chocolate does Georg have left?

Find all triples $(x,y,z)$ of positive integers such that $$\begin{cases} x + y +z = 2010 \\x^2 + y^2 + z^2 - xy - yz - zx =3 \end{cases}$$