For any natural number $m$, we denote by $\phi (m)$ the number of integers $k$ relatively prime to $m$ and satisfying $1 \le k \le m$. Determine all positive integers $n$ such that for every integer $k > n^2$, we have $n | \phi (nk + 1)$. (Daniel Harrer)

Consider all parabolas of the form $ y\equal{}x^2\plus{}2px\plus{}q$ for $ p,q \in \mathbb{R}$ which intersect the coordinate axes in three distinct points. For such $ p,q$, denote by $ C_{p,q}$ the circle through these three intersection points. Prove that all circles $ C_{p,q}$ have a point in common.

On the plane are given $ k\plus{}n$ distinct lines , where $ k>1$ is integer and $ n$ is integer as well.Any three of these lines do not pass through the same point . Among these lines exactly $ k$ are parallel and all the other $ n$ lines intersect each other.All $ k\plus{}n$ lines define on the plane a partition of triangular , polygonic or not bounded regions. Two regions are colled different, if the have not common points or if they have common points only on their boundary.A regions is called ''good'' if it contained in a zone between two parallel lines . If in a such given configuration the minimum number of ''good'' regionrs is $ 176$ and the maximum number of these regions is $ 221$, find $ k$ and $ n$. Babis

Let $A_1,A_2,\cdots , A_n$ be arithmetic progressions of integers, each of $k$ terms, such that any two of these arithmetic progressions have at least two common elements. Suppose $b$ of these arithmetic progressions have common difference $d_1$ and the remaining arithmetic progressions have common difference $d_2$ where $0<b<n$. Prove that \[b \le 2\left(k-\frac{d_2}{gcd(d_1,d_2)}\right)-1.\]

Find the sum of the numbers written with two digits $\overline{ab}$ for which the equation $3^{x + y} =3^x + 3^y + \overline{ab}$ has at least one solution $(x, y)$ in natural numbers.

Every phone number in an area consists of eight digits and starts with digit $ 8$. Mr Edy, who has just moved to the area, apply for a new phone number. What is the chance that Mr Edy gets a phone number which consists of at most five different digits?

$1390$ ants are placed near a line, such that the distance between their heads and the line is less than $1\text{cm}$ and the distance between the heads of two ants is always larger than $2\text{cm}$. Show that there is at least one pair of ants such that the distance between their heads is at least $10$ meters (consider the head of an ant as point).

Let $p$ be some prime number. a) Prove that there exist positive integers $a$ and $b$ such that $a^2 + b^2 + 2018$ is multiple of $p$. b) Find all $p$ for which the $a$ and $b$ from a) can be chosen in such way that both these numbers aren’t multiples of $p$.

Find the smallest integer $n$ such that each subset of $\{1,2,\ldots, 2004\}$ with $n$ elements has two distinct elements $a$ and $b$ for which $a^2-b^2$ is a multiple of $2004$.

The positive integers from 1 to 100 are painted into three colors: 50 integers are red, 25 integers are yellow and 25 integers are green. The red and yellow integers can be divided into 25 triples such that each triple includes two red integers and one yellow integer which is greater than one of the red integers and smaller than another one. The same assertion is valid for the red and green integers. Is it necessarily possible to divide all the 100 integers into 25 quadruples so that each quadruple includes two red integers, one yellow integer and one green integer such that the yellow and the green integer lie between the red ones? [i]Alexandr Gribalko[/i]

Write a list of the first $10$ positive integers in increasing order. Erase any number adjacent to a prime; if two primes are adjacent, do not erase either prime. Apply this process twice. How many positive integers remain in the list? $\textbf{(A) }2\qquad\textbf{(B) }3\qquad\textbf{(C) }4\qquad\textbf{(D) }5\qquad\textbf{(E) }6$

In boxes labeled $0$, $1$, $2$, $\dots$, we place integers according to the following rules: $\bullet$ If $p$ is a prime number, we place it in box $1$. $\bullet$ If $a$ is placed in box $m_a$ and $b$ is placed in box $m_b$, then $ab$ is placed in the box labeled $am_b+bm_a$. Find all positive integers $n$ that are placed in the box labeled $n$.

Consider the polynomial \[P(x)=x^3+3x^2+6x+10.\] Let its three roots be $a$, $b$, $c$. Define $Q(x)$ to be the monic cubic polynomial with roots $ab$, $bc$, $ca$. Compute $|Q(1)|$. [i]Proposed by Nathan Xiong[/i]

Let $ f_1,f_2,\dots,f_n$ be regular functions on a domain of the complex plane, linearly independent over the complex field. Prove that the functions $ f_i\overline{f}_k, \;1 \leq i,k \leq n$, are also linearly independent. [i]L. Lempert[/i]

Let $ABCD$ be a rectangle with $AB<BC$ and circumcircle $\Gamma$. Let $P$ be a point on the arc $BC$ (not containing $A$) and let $Q$ be a point on the arc $CD$ (not containing $A$) such that $BP=CQ$. The circle with diameter $AQ$ intersects $AP$ again in $S$. The perpendicular to $AQ$ through $B$ intersects $AP$ in $X$. (a) Show that $XS=PS$. (b) Show that $AX=DQ$.

Let $\phi(n,m), m \neq 1$, be the number of positive integers less than or equal to $n$ that are coprime with $m.$ Clearly, $\phi(m,m) = \phi(m)$, where $\phi(m)$ is Euler’s phi function. Find all integers $m$ that satisfy the following inequality: \[\frac{\phi(n,m)}{n} \geq \frac{\phi(m)}{m}\] for every positive integer $n.$