Place 2005 points on the circumference of a circle. Two points $P,Q$ are said to form a pair of neighbours if the chord $PQ$ subtends an angle of at most 10 degrees at the centre. Find the smallest number of pairs of neighbours.

Given an infinite sequence of $0$'s and $1$'s and a fixed integer $k,$ suppose that there are no more than $k$ distinct blocks of $k$ consecutive terms. Show that the sequence is eventually periodic. (For example, the sequence $11011010101$ followed by alternating $0$'s and $1$'s indefinitely, which is periodic beginning with the fifth term.)

Let $ABC$ be an acute triangle with $AT, AS$ respectively are the internal, external angle bisector of $ABC$ and $T, S \in BC$. On the circle with diameter $TS$, take an arbitrary point $P$ that lies inside the triangle ABC. Denote $D, E, F, I$ as the incenter of triangle $PBC, PCA, PAB, ABC$. Prove that four lines $AD, BE, CF$ and $IP$ are concurrent.

An ice ballerina rotates at a constant angular velocity at one particular point. That is, she does not translationally move. Her arms are fully extended as she rotates. Her moment of inertia is $I$. Now, she pulls her arms in and her moment of inertia is now $\frac{7}{10}I$. What is the ratio of the new kinetic energy (arms in) to the initial kinetic energy (arms out)? $ \textbf {(A) } \dfrac {7}{10} \qquad \textbf {(B) } \dfrac {49}{100} \qquad \textbf {(C) } 1 \qquad \textbf {(C) } \dfrac {100}{49} \qquad \textbf {(E) } \dfrac {10}{7} $ [i]Problem proposed by Ahaan Rungta[/i]

A circle centered at $O{}$ passes through the vertices $B{}$ and $C{}$ of the acute-angles triangle $ABC$ and intersects the sides $AC{}$ and $AB{}$ at $D{}$ and $E{}$ respectively. The segments $CE$ and $BD$ intersect at $U{}.$ The ray $OU$ intersects the circumcircle of $ABC$ at $P{}.$ Prove that the incenters of the triangles $PEC$ and $PBD$ coincide.

Morteza and Amir Reza play the following game. First each of them independently roll a dice $100$ times in a row to construct a $100$-digit number with digits $1,2,3,4,5,6$ then they simultaneously shout a number from $1$ to $100$ and write down the corresponding digit to the number other person shouted in their $100$ digit number. If both of the players write down $6$ they both win otherwise they both loose. Do they have a strategy with wining chance more than $\frac{1}{36}$? [i]Proposed by Morteza Saghafian[/i]

Let $S$ be an infinite set of positive integers, such that there exist four pairwise distinct $a,b,c,d \in S$ with $\gcd(a,b) \neq \gcd(c,d)$. Prove that there exist three pairwise distinct $x,y,z \in S$ such that $\gcd(x,y)=\gcd(y,z) \neq \gcd(z,x)$.

How many members are there of the set $\{-79,-76,-73,\dots,98,101\}?$

Let be given $r_1,r_2,\ldots,r_n \in \mathbb R$. Show that there exists a subset $I$ of $\{1,2,\ldots,n \}$ which which has one or two elements in common with the sets $\{i,i + 1,i + 2\} , (1 \leq i \leq n- 2)$ such that \[\left| {\mathop \sum \limits_{i \in I} {r_i}} \right| \geqslant \frac{1}{6}\mathop \sum \limits_{i = 1}^n \left| {{r_i}} \right|.\]

a) On a circumference $8$ points are marked. We say that Juliana does an “T-operation ” if she chooses three of these points and paint the sides of the triangle that they determine, so that each painted triangle has at most one vertex in common with a painted triangle previously. What is the greatest number of “T-operations” that Juliana can do? b) If in part (a), instead of considering $8$ points, $7$ points are considered, what is the greatest number of “T operations” that Juliana can do?

Among the divisors of a natural number $n$, we have numbers such that when they are devided by $2013$, give us remainders $1001, 1002, \ldots, 2012$. Prove that among the divisors of the number $n^2$, there exist numbers such that when they are divided by $2013$, give us reminders $1, 2, 3, \ldots, 2012$.

Let $n$ be a natural number. Prove that \[ \left\lfloor \frac{n+2^0}{2^1} \right\rfloor + \left\lfloor \frac{n+2^1}{2^2} \right\rfloor +\cdots +\left\lfloor \frac{n+2^{n-1}}{2^n}\right\rfloor =n. \] [hide="Remark"]For any real number $x$, the number $\lfloor x \rfloor$ represents the largest integer smaller or equal with $x$.[/hide]

Let $S= \{ a_1, \ldots, a_n\}$ be a set of $n\geq 1$ positive real numbers. For each nonempty subset of $S$ the sum of its elements is written down. Show that all written numbers can be divided into $n$ classes such that in each class the ratio of the greatest number to the smallest number is not greater than $2$.

Let $n$ be a positive integer. A sequence $(a, b, c)$ of $a, b, c \in \{1, 2, . . . , 2n\}$ is called [i]joke [/i] if its shortest term is odd and if only that smallest term, or no term, is repeated. For example, the sequences $(4, 5, 3)$ and $(3, 8, 3)$ are jokes, but $(3, 2, 7)$ and $(3, 8, 8)$ are not. Determine the number of joke sequences in terms of $n$.

Let $ x$, $ y$, $ z \in\mathbb{R}^+$ satisfying $ xyz = 1$. Prove that \[ \frac {(x + y - 1)^2}{z} + \frac {(y + z - 1)^2}{x} + \frac {(z + x - 1)^2}{y}\geqslant x + y + z\mbox{.}\]

Find all real $\alpha,\beta$ such that the following limit exists and is finite: \[\lim_{x,y\rightarrow 0^{+}}\frac{x^{2\alpha}y^{2\beta}}{x^{2\alpha}+y^{3\beta}}\]

Let $BCB'C'$ be a rectangle, let $M$ be the midpoint of $B'C'$, and let $A$ be a point on the circumcircle of the rectangle. Let triangle $ABC$ have orthocenter $H$, and let $T$ be the foot of the perpendicular from $H$ to line $AM$. Suppose that $AM=2$, $[ABC]=2020$, and $BC=10$. Then $AT=\frac{m}{n}$, where $m$ and $n$ are positive integers with $\gcd(m,n)=1$. Compute $100m+n$. [i]Proposed by Ankit Bisain[/i]

Let $a$, $b$ and $c$ be given positive integers which are two by two coprime. A positive integer $n$ is called [i]sozopolian[/i], if it [u]can’t[/u] be written as $n=bcx+cay+abz$ where $x$, $y$, $z$ are also positive integers. Find the number of [i]sozopolian[/i] numbers as a function of $a$, $b$ and $c$.

Let $(x, y)$ be an intersection of the equations $y = 4x^2 - 28x + 41$ and $x^2 + 25y^2 - 7x + 100y +\frac{349}{4}= 0$. Find the sum of all possible values of $x$.

Find all integer numbers $a,b,c$ such that $\frac{(a+b)(b+c)(c+a)}{2}+(a+b+c)^{3}=1-abc$.

An [i]annulus[/i] is the region between two concentric circles. The concentric circles in the figure have radii $ b$ and $ c$, with $ b > c$. Let $ \overline{OX}$ be a radius of the larger circle, let $ \overline{XZ}$ be tangent to the smaller circle at $ Z$, and let $ \overline{OY}$ be the radius of the larger circle that contains $ Z$. Let $ a \equal{} XZ$, $ d \equal{} YZ$, and $ e \equal{} XY$. What is the area of the annulus? $ \textbf{(A)}\ \pi a^2 \qquad \textbf{(B)}\ \pi b^2 \qquad \textbf{(C)}\ \pi c^2 \qquad \textbf{(D)}\ \pi d^2 \qquad \textbf{(E)}\ \pi e^2$ [asy]unitsize(1.4cm); defaultpen(linewidth(.8pt)); dotfactor=3; real r1=1.0, r2=1.8; pair O=(0,0), Z=r1*dir(90), Y=r2*dir(90); pair X=intersectionpoints(Z--(Z.x+100,Z.y), Circle(O,r2))[0]; pair[] points={X,O,Y,Z}; filldraw(Circle(O,r2),mediumgray,black); filldraw(Circle(O,r1),white,black); dot(points); draw(X--Y--O--cycle--Z); label("$O$",O,SSW,fontsize(10pt)); label("$Z$",Z,SW,fontsize(10pt)); label("$Y$",Y,N,fontsize(10pt)); label("$X$",X,NE,fontsize(10pt)); defaultpen(fontsize(8pt)); label("$c$",midpoint(O--Z),W); label("$d$",midpoint(Z--Y),W); label("$e$",midpoint(X--Y),NE); label("$a$",midpoint(X--Z),N); label("$b$",midpoint(O--X),SE);[/asy]

A function $f:\mathbb{N} \rightarrow \mathbb{N} $ is called nice if $f^a(b)=f(a+b-1)$, where $f^a(b)$ denotes $a$ times applied function $f$. Let $g$ be a nice function, and an integer $A$ exists such that $g(A+2018)=g(A)+1$. a) Prove that $g(n+2017^{2017})=g(n)$ for all $n \geq A+2$. b) If $g(A+1) \neq g(A+1+2017^{2017})$ find $g(n)$ for $n <A$.

Let $f$ and $g$ be two polynomials with integer coefficients such that the leading coefficients of both the polynomials are positive. Suppose $\deg(f)$ is odd and the sets $\{f(a)\mid a\in \mathbb{Z}\}$ and $\{g(a)\mid a\in \mathbb{Z}\}$ are the same. Prove that there exists an integer $k$ such that $g(x)=f(x+k)$.

Assume that complex numbers $z_1,z_2,...,z_n$ satisfy $|z_i-z_j| \le 1$ for any $1 \le i <j \le n$. Let $$S= \sum_{1 \le i <j \le n} |z_i-z_j|^2.$$ (1) If $n = 6063$, find the maximum value of $S$. (2) If $n= 2021$, find the maximum value of $S$.

The number $2013$ has the property that it includes four consecutive digits ($0$, $1$, $2$, and $3$). How many $4$-digit numbers include $4$ consecutive digits? [i](9 and 0 are not considered consecutive digits.)[/i] $\text{(A) }18\qquad\text{(B) }24\qquad\text{(C) }144\qquad\text{(D) }162\qquad\text{(E) }168$