Find all functions $f: \mathbb{Z}^+\to \mathbb{R}$, which satisfies $f(n+1)\geq f(n)$ for all $n\geq 1$ and $f(mn)=f(m)f(n)$ for all $(m,n)=1$.

The real numbers $a_0,a_1,a_2,\ldots$ satisfy $1=a_0\le a_1\le a_2\le\ldots. b_1,b_2,b_3,\ldots$ are defined by $b_n=\sum_{k=1}^n{1-{a_{k-1}\over a_k}\over\sqrt a_k}$. [b]a.)[/b] Prove that $0\le b_n<2$. [b]b.)[/b] Given $c$ satisfying $0\le c<2$, prove that we can find $a_n$ so that $b_n>c$ for all sufficiently large $n$.

Le $S$ be the set of positive integers greater than or equal to $2$. A function $f: S\rightarrow S$ is italian if $f$ satifies all the following three conditions: 1) $f$ is surjective 2) $f$ is increasing in the prime numbers(that is, if $p_1<p_2$ are prime numbers, then $f(p_1)<f(p_2)$) 3) For every $n\in S$ the number $f(n)$ is the product of $f(p)$, where $p$ varies among all the primes which divide $n$ (For instance, $f(360)=f(2^3\cdot 3^2\cdot 5)=f(2)\cdot f(3)\cdot f(5)$). Determine the maximum and the minimum possible value of $f(2020)$, when $f$ varies among all italian functions.

$ 1 \le n \le 455$ and $ n^3 \equiv 1 \pmod {455}$. The number of solutions is ? $\textbf{(A)}\ 9 \qquad\textbf{(B)}\ 6 \qquad\textbf{(C)}\ 3 \qquad\textbf{(D)}\ 1 \qquad\textbf{(E)}\ \text{None}$

Show that there exists $n>2$ such that $1991 | 1999 \ldots 91$ (with $n$ 9's).

Let $ABC$ be a triangle, and $M$ and $N$ variable points on $AB$ and $AC$ respectively, such that both $M$ and $N$ do not lie on the vertices, and also, $AM \times MB = AN \times NC$. Prove that the perpendicular bisector of $MN$ passes through a fixed point.

On a long metal stick, $1000$ red marbles are embedded in the stick so the stick is equally partitioned into $1001$ parts by them. $1001$ blue marbles are embedded in the stick too, so the stick is equally partitioned into $1002$ parts by them. If you cut the metal stick equally into $2003$ smaller parts, how many of the smaller parts would contain at least one embedded marble?

Determine the minimum value of $$(r-1)^2 + \left(\frac{s}{r}-1 \right)^2 + \left(\frac{t}{s}-1 \right)^{2} + \left( \frac{4}{t} -1 \right)^2$$ for all real numbers $1\leq r \leq s \leq t \leq 4.$

Let $M$ be the midpoint of side $AB$ of the triangle $ABC$. Let$P$ be a point on $AB$ between $A$ and $M$, and let $MD$ be drawn parallel to $PC$ and intersecting $BC$ at $D$. If the ratio of the area of the triangle $BPD$ to that of triangle $ABC$ is denoted by $r$, then $\text{(A)}\ \tfrac{1}{2}<r<1\text{ depending upon the position of }P \qquad\\ \text{(B)}\ r=\tfrac{1}{2}\text{ independent of the position of }P\qquad\\ \text{(C)}\ \tfrac{1}{2}\le r<1\text{ depending upon the position of }P \qquad\\ \text{(D)}\ \tfrac{1}{3}<r<\tfrac{2}{3}\text{ depending upon the position of }P \qquad\\ \text{(E)}\ r=\tfrac{1}{3} \text{ independent of the position of }P$

Hannusya, Petrus and Mykolka drew independently one isosceles triangle $ABC$, all angles of which are measured as a integer number of degrees. It turned out that the bases $AC$ of these triangles are equals and for each of them on the ray $BC$ there is a point $E$ such that $BE=AC$, and the angle $AEC$ is also measured by an integer number of degrees. Is it in necessary that: a) all three drawn triangles are equal to each other? b) among them there are at least two equal triangles?

Find all pairs of integers $(x,y)$ satisfying the following condition: [i]each of the numbers $x^3 + y$ and $x + y^3$ is divisible by $x^2 + y^2$ [/i] Tournament of Towns

For a polynomials $ P\in \mathbb{R}[x]$, denote $f(P)=n$ if $n$ is the smallest positive integer for which is valid $$(\forall x\in \mathbb{R})(\underbrace{P(P(\ldots P}_{n}(x))\ldots )>0),$$ and $f(P)=0$ if such n doeas not exist. Exists polyomial $P\in \mathbb{R}[x]$ of degree $2014^{2015}$ such that $f(P)=2015$? (Serbia)

a) Prove that of $5$ consecutive positive integers one that is relatively prime with the other $4$ can always be selected. b) Prove that of $10$ consecutive positive integers one that is relatively prime with the other $9$ can always be selected.

Let $\mathbb{R}_{>0}$ be the set of all positive real numbers. Find all functions $f:\mathbb{R}_{>0} \to \mathbb{R}_{>0}$ such that for all $x,y\in \mathbb{R}_{>0}$ we have \[f(x) = f(f(f(x)) + y) + f(xf(y)) f(x+y).\]

Let $a > 1$ be a positive integer and $d > 1$ be a positive integer coprime to $a$. Let $x_1=1$, and for $k\geq 1$, define $$x_{k+1} = \begin{cases} x_k + d &\text{if } a \text{ does not divide } x_k \\ x_k/a & \text{if } a \text{ divides } x_k \end{cases}$$ Find, in terms of $a$ and $d$, the greatest positive integer $n$ for which there exists an index $k$ such that $x_k$ is divisible by $a^n$.

Let $ \{X_n\}$ and $ \{Y_n\}$ denote two sequences of integers defined as follows: \begin{align*} X_0 \equal{} 1,\ X_1 \equal{} 1,\ X_{n \plus{} 1} \equal{} X_n \plus{} 2X_{n \minus{} 1} \quad (n \equal{} 1,2,3,\ldots), \\ Y_0 \equal{} 1,\ Y_1 \equal{} 7,\ Y_{n \plus{} 1} \equal{} 2Y_n \plus{} 3Y_{n \minus{} 1} \quad (n \equal{} 1,2,3,\ldots).\end{align*} Prove that, except for the "1", there is no term which occurs in both sequences.

Let O be the center of the circumscribed circle of the triangle ABC. Let AH be the altitude in this triangle, and let P be the base of the perpendicular drawn from point A to the line CO. Prove that the line HP passes through the midpoint of the side AB. (6 points) Egor Bakaev

Let $ABC$ be an equilateral triangle. A regular hexagon $PXQYRZ$ of side length $2$ is placed so that $P, Q,$ and $R$ lie on segments $\overline{BC}, \overline{CA},$ and $\overline{AB}$, respectively. If points $A, X,$ and $Y$ are collinear, compute $BC.$

Consider an inscriptible polygon $ABCDE$. Let $H_1,H_2,H_3,H_4,H_5$ be the orthocenters of the triangles $ABC,BCD,CDE,DEA,EAB$ and let $M_1,M_2,M_3,M_4,M_5$ be the midpoints of $DE,EA,AB,BC$ and $CD$, respectively. Prove that the lines $H_1M_1,H_2M_2,H_3M_3,H_4M_4,H_5M_5$ have a common point. [i]Dinu Serbanescu[/i]

Let $\Omega$ and $O$ be the circumcircle and the circumcenter of an acute triangle $ABC$ $(\overline{AB} < \overline{AC})$. Define $D,E(\neq A)$ be the points such that ray $AO$ intersects $BC$ and $\Omega$. Let the line passing through $D$ and perpendicular to $AB$ intersects $AC$ at $P$ and define $Q$ similarly. Tangents to $\Omega$ on $A,E$ intersects $BC$ at $X,Y$. Prove that $X,Y,P,Q$ lie on a circle.

A circle $ \Gamma$ and a line $ \ell$ is given in a plane such that $ \ell$ doesn't cut $ \Gamma$.Determine the intersection set of the circles has $ [AB]$ as diameter for all pairs of $ \left\{A,B\right\}$ (lie on $ \ell$) and satisfy $ P,Q,R,S \in \Gamma$ such that $ PQ \cap RS\equal{}\left\{A\right\}$ and $ PS \cap QR\equal{}\left\{B\right\}$

Consider the statement, "If $n$ is not prime, then $n-2$ is prime." Which of the following values of $n$ is a counterexample to this statement? $\textbf{(A) } 11 \qquad \textbf{(B) } 15 \qquad \textbf{(C) } 19 \qquad \textbf{(D) } 21 \qquad \textbf{(E) } 27$

Let $ ABC$ be a triangle with $ BC > AC > AB$. Let $ A',B',C'$ be feet of perpendiculars from $ A,B,C$ to $ BC,AC,AB$, such that $ AA' \equal{} BB' \equal{} CC' \equal{} x$. Prove that: a) If $ ABC\sim A'B'C'$ then $ x \equal{} 2r$ b) Prove that if $ A',B'$ and $ C'$ are collinear, then $ x \equal{} R \plus{} d$ or $ x \equal{} R \minus{} d$. (In this problem $ R$ is the radius of circumcircle, $ r$ is radius of incircle and $ d \equal{} OI$)

For a given positive integer $n$, we wish to construct a circle of six numbers as shown below so that the circle has the following properties: (a) The six numbers are different three-digit numbers, none of whose digits is a 0. (b) Going around the circle clockwise, the first two digits of each number are the last two digits, in the same order, of the previous number. (c) All six numbers are divisible by $n$. The example above shows a successful circle for $n = 2$. For each of $n = 3, 4, 5, 6, 7, 8, 9$, either construct a circle that satisfies these properties, or prove that it is impossible to do so. [asy] pair a = (1,0); defaultpen(linewidth(0.7)); draw(a..-a..a); int[] num = {264,626,662,866,486,648}; for (int i=0;i<6;++i) { dot(a); label(format("$%d$",num[i]),a,a); a=dir(60*i+60); }[/asy]

Find all prime numbers $p,q,r,k$ such that $pq+qr+rp = 12k+1$