Suppose we have some proteins that each protein is a sequence of 7 "AMINO-ACIDS" $A,\ B,\ C,\ H,\ F,\ N$. For example $AFHNNNHAFFC$ is a protein. There are some steps that in each step an amino-acid will change to another one. For example with the step $NA\rightarrow N$ the protein $BANANA$ will cahnge to $BANNA$("in Persian means workman"). We have a set of allowed steps that each protein can change with these steps. For example with the set of steps: $\\ 1)\ AA\longrightarrow A\\ 2)\ AB\longrightarrow BA\\ 3)\ A\longrightarrow \mbox{null}$ Protein $ABBAABA$ will change like this: $\\ ABB\underline{AA}BA\\ \underline{AB}BABA\\ B\underline{AB}ABA\\ BB\underline{AA}BA\\ BB\underline{AB}A\\ BBB\underline{AA}\\ BBB\underline{A}\\ BBB$ You see after finite steps this protein will finish it steps. Set of allowed steps that for them there exist a protein that may have infinitely many steps is dangerous. Which of the following allowed sets are dangerous? a) $NO\longrightarrow OONN$ b) $\left\{\begin{array}{c}HHCC\longrightarrow HCCH\\ CC\longrightarrow CH\end{array}\right.$ c) Design a set of allowed steps that change $\underbrace{AA\dots A}_{n}\longrightarrow\underbrace{BB\dots B}_{2^{n}}$ d) Design a set of allowed steps that change $\underbrace{A\dots A}_{n}\underbrace{B\dots B}_{m}\longrightarrow\underbrace{CC\dots C}_{mn}$ You see from $c$ and $d$ that we acn calculate the functions $F(n)=2^{n}$ and $G(M,N)=mn$ with these steps. Find some other calculatable functions with these steps. (It has some extra mark.)

For any function $f: \mathbb{R}^2\to \mathbb{R}$ consider the function $\Phi_f:\mathbb{R}^2\to [-\infty,\infty]$ for which $\Phi_f(x,y)=\limsup_{ z \to y} f(x,z)$ for any $(x,y) \in \mathbb{R}^2$. [list=a] [*]Is it true that if $f$ is Lebesgue measurable then $\Phi_f$ is also Lebesgue measurable?[/*] [*]Is it true that if $f$ is Borel measurable then $\Phi_f$ is also Borel measurable?[/*] [/list]

The Fibonacci sequence $f_1,f_2,f_3,\dots$ is defined by $f_1=f_2=1$, $f_{n+2}=f_{n+1}+f_n$. Find all $n$ such that $f_n = n^2$.

Let $a$, $b$ be positive integers such that $b^n+n$ is a multiple of $a^n+n$ for all positive integers $n$. Prove that $a=b$. [i]Proposed by Mohsen Jamali, Iran[/i]

The number $5^{867}$ is between $2^{2013}$ and $2^{2014}$. How many pairs of integers $(m,n)$ are there such that $1\leq m\leq 2012$ and \[5^n<2^m<2^{m+2}<5^{n+1}?\] $\textbf{(A) }278\qquad \textbf{(B) }279\qquad \textbf{(C) }280\qquad \textbf{(D) }281\qquad \textbf{(E) }282\qquad$

prove for all $k> 1$ equation $(x+1)(x+2)...(x+k)=y^{2}$ has finite solutions.

Points $H_1, H_2, ... , H_n$ are arranged in the plane so that each distance $H_iH_j \le 1$. The point $P$ is chosen to minimise $\max (PH_i)$. Find the largest possible value of $\max (PH_i)$ for $n = 3$. Find the best upper bound you can for $n = 4$.

Let $n$ be a positive odd integer , $a_1,a_2,\cdots,a_n$ be any permutation of the positive integers $1,2,\cdots,n$ . Prove that :$(a_1-1)(a^2_2-2)(a^3_3-3)\cdots (a^n_n-n)$ is an even number.

Let $ABCD$ be a quadrilateral with $\angle DAB = \angle ABC = 120^o$. If $AB = 3$, $BC = 2$, and $AD = 4$, what is the length of $CD$?

A point $P$ lies outside a circle. Consider all possible lines drawn through $P$ so that they intersect the circle. Find the locus of the midpoints of the chords — segments the circle intercepts on these lines.

$1000\times 1993 \times 0.1993 \times 10 = $ $\text{(A)}\ 1.993\times 10^3 \qquad \text{(B)}\ 1993.1993 \qquad \text{(C)}\ (199.3)^2 \qquad \text{(D)}\ 1,993,001.993 \qquad \text{(E)}\ (1993)^2$

Find $$\sum^{100}_{i=1}i \gcd(i ,100).$$

Let $n$ be an odd positive integer. Prove that if the equation $\frac1x+\frac1y=\frac4n$ has a solution in positive integers $x,y$, then $n$ has at least one divisor of the form $4k-1$, $k\in\mathbb N$.

Evaluate: $\lim_{n\rightarrow\infty}\sum_{k=n^2}^{(n+1)^2} \dfrac{1}{\sqrt{k}}$

For each positive integer $n$ we denote by $p(n)$ the greatest square less than or equal to $n$. [list=a] [*]Find all pairs of positive integers $( m,n)$, with $m\leq n$, for which \[ p( 2m+1) \cdot p( 2n+1) =400 \] [*]Determine the set $\mathcal{P}=\{ n\in\mathbb{N}^{\ast}\vert n\leq100\text{ and }\dfrac{p(n+1)}{p(n)}\notin\mathbb{N}^{\ast}\}$[/list]

Let $\mathbb N$ denote the set of positive integers, and for a function $f$, let $f^k(n)$ denote the function $f$ applied $k$ times. Call a function $f : \mathbb N \to \mathbb N$ [i]saturated[/i] if \[ f^{f^{f(n)}(n)}(n) = n \] for every positive integer $n$. Find all positive integers $m$ for which the following holds: every saturated function $f$ satisfies $f^{2014}(m) = m$. [i]Proposed by Evan Chen[/i]

The sides $ AB$,$ BC$ and $ CA$ of the triangle $ ABC$ are tangent to the incircle of the triangle $ ABC$ with center $ I$ at the points $ C_1$,$ A_1$ and $ B_1$, respectively.Let $ B_2$ be the midpoint of the side $ AC$.Prove that the lines $ B_1I$, $ A_1C_1$ and $ BB_2$ are concurrent.

Yet another trapezoid $ABCD$ has $AD$ parallel to $BC$. $AC$ and $BD$ intersect at $P$. If $[ADP]=[BCP] = 1/2$, find $[ADP]/[ABCD]$. (Here the notation $[P_1 ...P_n]$ denotes the area of the polygon $P_1 ...P_n$.)

[b]a)[/b] Prove that for any natural number $ n\ge 1, $ there is a set $ \mathcal{M} $ of $ n $ points from the Cartesian plane such that the barycenter of every subset of $ \mathcal{M} $ has integral coordinates (both coordinates are integer numbers). [b]b)[/b] Show that if a set $ \mathcal{N} $ formed by an infinite number of points from the Cartesian plane is given such that no three of them are collinear, then there exists a finite subset of $ \mathcal{N} , $ the barycenter of which has non-integral coordinates.

Suppose $x>1$ is a real number such that $x+\tfrac 1x = \sqrt{22}$. What is $x^2-\tfrac1{x^2}$?

Let $ 0 \leq c \leq 1$, and let $ \eta$ denote the order type of the set of rational numbers. Assume that with every rational number $ r$ we associate a Lebesgue-measurable subset $ H_r$ of measure $ c$ of the interval $ [0,1]$. Prove the existence of a Lebesgue-measurable set $ H \subset [0,1]$ of measure $ c$ such that for every $ x \in H$ the set \[ \{r : \;x \in H_r\ \}\] contains a subset of type $ \eta$. [i]M. Laczkovich[/i]

Assign to each positive real number a color, either red or blue. Let $D$ be the set of all distances $d>0$ such that there are two points of the same color at distance $d$ apart. Recolor the positive reals so that the numbers in $D$ are red and the numbers not in $D$ are blue. If we iterate the recoloring process, will we always end up with all the numbers red after a finite number of steps?

$HOW,BOW,$ and $DAH$ are equilateral triangles in a plane such that $WO=7$ and $AH=2$. Given that $D,A,B$ are collinear in that order, find the length of $BA$.

Let $ABC$ be an acute-angled triangle, and $M$ be the midpoint of the minor arc $BC$ of its circumcircle. A circle $\omega$ touches the side $AB, AC$ at points $P, Q$ respectively and passes through $M$. Prove that $BP + CQ = PQ$.

Show that the number \[\sqrt[n]{\sqrt{2019} + \sqrt{2018}} + \sqrt[n]{\sqrt{2019} - \sqrt{2018}}\] is irrational for any $n\ge 2$.