Determine all sets of natural numbers $ A $ that have at least two elements, and satisfying the following proposition: $$ \forall x,y\in A\quad x>y\implies \frac{x-y}{\text{gcd} (x,y)} \in A. $$ [i]Marius Perianu[/i]

Let $n$ be a positive integer. Show that there exists a one-to-one function $\sigma : \{1,2,...,n\} \to \{1,2,...,n\}$ such that $$\sum_{k=1}^{n} \frac{k}{(k+\sigma(k))^2} < \frac{1}{2}.$$

Prove that for any prime number $p$ the equation $2^p+3^p=a^n$ has no solution $(a,n)$ in integers greater than $1$.

Let set $S$ be the smallest set of positive integers satisfying the following properties: [list] [*] $2$ is in set $S$. [*] If $n^2$ is in set $S$, then $n$ is also in set $S$. [*] If $n$ is in set $S$, then $(n+5)^2$ is also in set $S$. [/list] Determine which positive integers are not in set $S$.

Let $a,b, c$ be $3$ distinct numbers from $\{1, 2,3, 4, 5, 6\}$ Show that $7$ divides $abc + (7 - a)(7 - b)(7 - c)$

Suppose $S$ is the set of all positive integers. For $a,b \in S$, define \[a * b=\frac{\text{lcm}[a,b]}{\text{gcd}(a,b)}\] For example $8*12=6$. Show that [b]exactly two[/b] of the following three properties are satisfied: (i) If $a,b \in S$, then $a*b \in S$. (ii) $(a*b)*c=a*(b*c)$ for all $a,b,c \in S$. (iii) There exists an element $i \in S$ such that $a *i =a$ for all $a \in S$.

Let $n\ge2$ be an integer. Prove that if $k^2+k+n$ is prime for all integers $k$ such that $0\le k\le\sqrt{n\over3}$, then $k^2+k+n$ is prime for all integers $k$ such that $0\le k\le n-2$.[i](IMO Problem 6)[/i] [b][i]Original Formulation[/i][/b] Let $f(x) = x^2 + x + p$, $p \in \mathbb N.$ Prove that if the numbers $f(0), f(1), \cdots , f(\sqrt{p\over 3} )$ are primes, then all the numbers $f(0), f(1), \cdots , f(p - 2)$ are primes. [i]Proposed by Soviet Union. [/i]

Find the greatest natural number $n$ such there exist natural numbers $x_{1}, x_{2}, \ldots, x_{n}$ and natural $a_{1}< a_{2}< \ldots < a_{n-1}$ satisfying the following equations for $i =1,2,\ldots,n-1$: \[x_{1}x_{2}\ldots x_{n}= 1980 \quad \text{and}\quad x_{i}+\frac{1980}{x_{i}}= a_{i}.\]

Prove that, among the positive numbers $$a,2a, ...,(n - 1)a.$$ there is one that differs from an integer by at most $1/n$.

Compute the smallest positive integer $n$ for which there are at least two odd primes $p$ such that $\sum_{k=1}^{n} (-1)^{v_p(k!)} < 0$. Note: for a prime $p$ and a positive integer $m$, $v_p(m)$ is the exponent of the largest power of $p$ that divides $m$; for example, $v_3(18) = 2$.

Determine the smallest prime $p$ such that $2018!$ is divisible by $p^{3}$ , but not divisible by $p^{4}$.

If positive integers, $a,b,c$ are pair-wise co-prime, and, \[\ a^2|(b^3+c^3), b^2|(a^3+c^3), c^2|(a^3+b^3) \] find $a,b,$ and $c$

Is there a number in the decimal notation of the square which has a sequence of digits "$2018$"?

How many pairs of integers $(a,b)$ are there such that $0\leq a < b \leq 100$ and such that $\tfrac{2^b-2^a}{2016}$ is an integer?

Given is a prime number $p$. Find the number of positive integer solutions $(a, b, c, d)$ of the system of equations $ac+bd = p(a+c)$ and $bc-ad = p(b-d)$.

[b]p1.[/b] Let $X,Y ,Z$ be nonzero real numbers such that the quadratic function $X t^2 - Y t + Z = 0$ has the unique root $t = Y$ . Find $X$. [b]p2.[/b] Let $ABCD$ be a kite with $AB = BC = 1$ and $CD = AD =\sqrt2$. Given that $BD =\sqrt5$, find $AC$. [b]p3.[/b] Find the number of integers $n$ such that $n -2016$ divides $n^2 -2016$. An integer $a$ divides an integer $b$ if there exists a unique integer $k$ such that $ak = b$. [b]p4.[/b] The points $A(-16, 256)$ and $B(20, 400)$ lie on the parabola $y = x^2$ . There exists a point $C(a,a^2)$ on the parabola $y = x^2$ such that there exists a point $D$ on the parabola $y = -x^2$ so that $ACBD$ is a parallelogram. Find $a$. [b]p5.[/b] Figure $F_0$ is a unit square. To create figure $F_1$, divide each side of the square into equal fifths and add two new squares with sidelength $\frac15$ to each side, with one of their sides on one of the sides of the larger square. To create figure $F_{k+1}$ from $F_k$ , repeat this same process for each open side of the smallest squares created in $F_n$. Let $A_n$ be the area of $F_n$. Find $\lim_{n\to \infty} A_n$. [img]https://cdn.artofproblemsolving.com/attachments/8/9/85b764acba2a548ecc61e9ffc29aacf24b4647.png[/img] [b]p6.[/b] For a prime $p$, let $S_p$ be the set of nonnegative integers $n$ less than $p$ for which there exists a nonnegative integer $k$ such that $2016^k -n$ is divisible by $p$. Find the sum of all $p$ for which $p$ does not divide the sum of the elements of $S_p$ . [b]p7. [/b] Trapezoid $ABCD$ has $AB \parallel CD$ and $AD = AB = BC$. Unit circles $\gamma$ and $\omega$ are inscribed in the trapezoid such that circle $\gamma$ is tangent to $CD$, $AB$, and $AD$, and circle $\omega$ is tangent to $CD$, $AB$, and $BC$. If circles $\gamma$ and $\omega$ are externally tangent to each other, find $AB$. [b]p8.[/b] Let $x, y, z$ be real numbers such that $(x+y)^2+(y+z)^2+(z+x)^2 = 1$. Over all triples $(x, y, z)$, find the maximum possible value of $y -z$. [b]p9.[/b] Triangle $\vartriangle ABC$ has sidelengths $AB = 13$, $BC = 14$, and $CA = 15$. Let $P$ be a point on segment $BC$ such that $\frac{BP}{CP} = 3$, and let $I_1$ and $I_2$ be the incenters of triangles $\vartriangle ABP$ and $\vartriangle ACP$. Suppose that the circumcircle of $\vartriangle I_1PI_2$ intersects segment $AP$ for a second time at a point $X \ne P$. Find the length of segment $AX$. [b]p10.[/b] For $1 \le i \le 9$, let Ai be the answer to problem i from this section. Let $(i_1,i_2,... ,i_9)$ be a permutation of $(1, 2,... , 9)$ such that $A_{i_1} < A_{i_2} < ... < A_{i_9}$. For each $i_j$ , put the number $i_j$ in the box which is in the $j$th row from the top and the $j$th column from the left of the $9\times 9$ grid in the bonus section of the answer sheet. Then, fill in the rest of the squares with digits $1, 2,... , 9$ such that $\bullet$ each bolded $ 3\times 3$ grid contains exactly one of each digit from $ 1$ to $9$, $\bullet$ each row of the $9\times 9$ grid contains exactly one of each digit from $ 1$ to $9$, and $\bullet$ each column of the $9\times 9$ grid contains exactly one of each digit from $ 1$ to $9$. PS. You had better use hide for answers.

How many different integer solutions to the inequality $|x| + |y| < 100$ are there?

Find the largest integer $x<1000$ such that $\left(\begin{array}{c}1515 \\ x\end{array}\right)$ and $\left(\begin{array}{c}1975 \\ x\end{array}\right)$ are both odd.

A triangle $ABC$ is given on the plane, such that all its vertices have integer coordinates. Does there necessarily exist a straight line which intersects the straight lines $AB,$ $BC,$ and $AC$ at three distinct points with integer coordinates?

Positive integers $d_1,d_2,...,d_n$ are divisors of $1995$. Prove that there exist $d_i$ and $d_j$ among them, such the denominator of the reduced fraction $d_i/d_j$ is at least $n$

Prove that all the positive integer numbers , except for the powers of $2$, can be written as the sum of (at least two) consecutive natural numbers .

Show that $n^n + (n + 1)^{n + 1}$ is composite for infinitely many positive integers $n$.

An integer is square-free if it is not divisible by $a^2$ for any integer $a>1$. Let $S$ be the set of positive square-free integers. Determine, with justification, the value of\[\sum_{k\epsilon S}\left[\sqrt{\frac{10^{10}}{k}}\right]\]where $[x]$ denote the greatest integer less than or equal to $x$

Show that for every integer $n \geq 1$, it is possible to express $5^n$ as the sum of two nonzero squares.

Let $n$ be a positive integer. Determine all positive integers $p$ for which there exist positive integers $x_1 < x_2 <...< x_n$ such that $\frac{1}{x_1}+\frac{2}{x_2}+ ... +\frac{n}{x_n}= p$ Irish Mathematical Olympiad