Given strictly increasing sequence $ a_1<a_2<\dots$ of positive integers such that each its term $ a_k$ is divisible either by 1005 or 1006, but neither term is divisible by $ 97$. Find the least possible value of maximal difference of consecutive terms $ a_{i\plus{}1}\minus{}a_i$.

[color=darkred]Let $f\colon [0,\infty)\to\mathbb{R}$ be a continuous function such that $\int_0^nf(x)f(n-x)\ \text{d}x=\int_0^nf^2(x)\ \text{d}x$ , for any natural number $n\ge 1$ . Prove that $f$ is a periodic function.[/color]

Two players play the following game starting with one pile of at least two stones. A player in turn chooses one of the piles and divides it into two or three nonempty piles. The player who cannot make a legal move loses the game. Which player has a winning strategy?

An integer is between $0$ and $999999$ (inclusive) is chosen, and the digits of its decimal representation are summed. What is the probability that the sum will be $19$?

Find all positive integers $k \geq 2$ for which there exists some positive integer $n$ such that the last $k$ digits of the decimal representation of $10^{10^n} - 9^{9^n}$ are the same. [i]Proposed by Andrew Wen[/i]

Let $a_1<a_2 \dots <a_n$ be positive integers, with $n\geq 2$. An invisible frog lies on the real line, at a positive integer point. Initially, the hunter chooses a number $k$, and then, once every minute, he can check if the frog currently lies in one of $k$ points of his choosing, after which the frog goes from its point $x$ to one of the points $x+a_1, x+a_2 \dots x+a_n$. Based on the values of $a_1, a_2 \dots a_n$, what is the smallest value of $k$ such that the hunter can guarantee to find the frog within a finite number of minutes, no matter where it initially started? [i]Proposed by David Anghel[/i]

In quadrilateral $ ABCD$, $ AB \equal{} 5$, $ BC \equal{} 17$, $ CD \equal{} 5$, $ DA \equal{} 9$, and $ BD$ is an integer. What is $ BD$? [asy]unitsize(4mm); defaultpen(linewidth(.8pt)+fontsize(8pt)); dotfactor=4; pair C=(0,0), B=(17,0); pair D=intersectionpoints(Circle(C,5),Circle(B,13))[0]; pair A=intersectionpoints(Circle(D,9),Circle(B,5))[0]; pair[] dotted={A,B,C,D}; draw(D--A--B--C--D--B); dot(dotted); label("$D$",D,NW); label("$C$",C,W); label("$B$",B,E); label("$A$",A,NE);[/asy]$ \textbf{(A)}\ 11 \qquad \textbf{(B)}\ 12 \qquad \textbf{(C)}\ 13 \qquad \textbf{(D)}\ 14 \qquad \textbf{(E)}\ 15$

Let \( n \) be a natural number. The numbers \( 1, 2, \ldots, n \) are written in a row in some order. For each pair of adjacent numbers, their greatest common divisor (GCD) is calculated and written on a sheet. What is the maximum possible number of distinct values among the \( n - 1 \) GCDs obtained? \\

Let $M$ be the centroid of $\Delta ABC$ Prove the inequality $\sin \angle CAM + \sin\angle CBM \le \frac{2}{\sqrt 3}$  (a) if the circumscribed circle of $\Delta AMC$ is tangent to the line $AB$ (b) for any $\Delta ABC$

A plastic ball with a radius of 45 mm has a circular hole made in it. The hole is made to fit a ball with a radius of 35 mm, in such a way that the distance between their centers is 60 mm. Calculate the radius of the hole.

How many positive integers $n$ less than $1000$ have the property that the number of positive integers less than $n$ which are coprime to $n$ is exactly $\tfrac n3$?

Consider the triangle $ABC{}$ and let $I_A{}$ be its $A{}$-excenter. Let $M,N$ and $P{}$ be the projections of $I_A{}$ onto the lines $AC,BC{}$ and $AB{}$ respectively. Prove that if $\overrightarrow{I_AM}+\overrightarrow{I_AP}=\overrightarrow{I_AN}$ then $ABC{}$ is an equilateral triangle.

Find all integers $n\geq 2$ for which there exist the real numbers $a_k, 1\leq k \leq n$, which are satisfying the following conditions: \[\sum_{k=1}^n a_k=0, \sum_{k=1}^n a_k^2=1 \text{ and } \sqrt{n}\cdot \Bigr(\sum_{k=1}^n a_k^3\Bigr)=2(b\sqrt{n}-1), \text{ where } b=\max_{1\leq k\leq n} \{a_k\}.\]

Put $ M\equal{}\begin{pmatrix}p&q&r\\ r&p&q\\q&r&p\end{pmatrix}$ where $ p,q,r>0$ and $ p\plus{}q\plus{}r\equal{}1$. Prove that $ \lim_{n\rightarrow \infty}M^n\equal{}\begin{bmatrix}\frac13&\frac13&\frac13\\ \frac13&\frac13&\frac13\\\frac13&\frac13&\frac13\end{bmatrix}$

Let $a$ and $b$ be integer solutions to $17a+6b=13$. What is the smallest possible positive value for $a-b$?

Denote by $ \mathcal{H}$ a set of sequences $ S\equal{}\{s_n\}_{n\equal{}1}^{\infty}$ of real numbers having the following properties: (i) If $ S\equal{}\{s_n\}_{n\equal{}1}^{\infty}\in \mathcal{H}$, then $ S'\equal{}\{s_n\}_{n\equal{}2}^{\infty}\in \mathcal{H}$; (ii) If $ S\equal{}\{s_n\}_{n\equal{}1}^{\infty}\in \mathcal{H}$ and $ T\equal{}\{t_n\}_{n\equal{}1}^{\infty}$, then $ S\plus{}T\equal{}\{s_n\plus{}t_n\}_{n\equal{}1}^{\infty}\in \mathcal{H}$ and $ ST\equal{}\{s_nt_n\}_{n\equal{}1}^{\infty}\in \mathcal{H}$; (iii) $ \{\minus{}1,\minus{}1,\dots,\minus{}1,\dots\}\in \mathcal{H}$. A real valued function $ f(S)$ defined on $ \mathcal{H}$ is called a quasi-limit of $ S$ if it has the following properties: If $ S\equal{}{c,c,\dots,c,\dots}$, then $ f(S)\equal{}c$; If $ s_i\geq 0$, then $ f(S)\geq 0$; $ f(S\plus{}T)\equal{}f(S)\plus{}f(T)$; $ f(ST)\equal{}f(S)f(T)$, $ f(S')\equal{}f(S)$ Prove that for every $ S$, the quasi-limit $ f(S)$ is an accumulation point of $ S$.

Let $n$ be a positive integer and let $a_1, a_2, \cdots a_k$ be all numbers less than $n$ and coprime to $n$ in increasing order. Find the set of values the function $f(n)=gcd(a_1^3-1, a_2^3-1, \cdots, a_k^3-1)$.

We have a regular polygon $P$ with 2019 vertices, and in each vertex there is a coin. Two players [i]Azul[/i] and [i]Rojo[/i] take turns alternately, beginning with Azul, in the following way: first, Azul chooses a triangle with vertices in $P$ and colors its interior with blue, then Rojo selects a triangle with vertices in $P$ and colors its interior with red, so that the triangles formed in each move don't intersect internally the previous colored triangles. They continue playing until it's not possible to choose another triangle to be colored. Then, a player wins the coin of a vertex if he colored the greater quantity of triangles incident to that vertex (if the quantities of triangles colored with blue or red incident to the vertex are the same, then no one wins that coin and the coin is deleted). The player with the greater quantity of coins wins the game. Find a winning strategy for one of the players. [i]Note.[/i] Two triangles can share vertices or sides.

Let $f:\mathbb{R}\rightarrow \mathbb{R}$ a continuous function such that for any $x\in \mathbb{R}$, the limit $\lim_{h\to 0} \left|\frac{f(x+h)-f(x)}{h}\right|$ exists and it is finite. Prove that in any real point, $f$ is differentiable or it has finite one-side derivates, of the same modul, but different signs.

Let $f(x) = \frac{1}{1-\frac{3x}{16}}$. Consider the sequence $\{ 0, f(0), f(f(0)), f^3(0), \dots \}$ Find the smallest $L$ such that $f^n(0) \leq L$ for all $n$. If the sequence is unbounded, write none as your answer.

A configuration of $4027$ points in the plane is called Colombian if it consists of $2013$ red points and $2014$ blue points, and no three of the points of the configuration are collinear. By drawing some lines, the plane is divided into several regions. An arrangement of lines is good for a Colombian configuration if the following two conditions are satisfied: i) No line passes through any point of the configuration. ii) No region contains points of both colors. Find the least value of $k$ such that for any Colombian configuration of $4027$ points, there is a good arrangement of $k$ lines. Proposed by [i]Ivan Guo[/i] from [i]Australia.[/i]

A positive number $a$ is given, such that $a$ could be expressed as difference of two inverses of perfect squares ($a=\frac1{n^2}-\frac1{m^2}$). Is it possible for $2a$ to be expressed as difference of two perfect squares?

In an acute-angled and not isosceles triangle $ABC,$ we draw the median $AM$ and the height $AH.$ Points $Q$ and $P$ are marked on the lines $AB$ and $AC$, respectively, so that the $QM \perp AC$ and $PM \perp AB$. The circumcircle of $PMQ$ intersects the line $BC$ for second time at point $X.$ Prove that $BH = CX.$ M. Didin

Let $ABCD$ be an isosceles trapezoid with $AB\parallel CD$. Let $E$ be the midpoint of $AC$. Denote by $\omega$ and $\Omega$ the circumcircles of the triangles $ABE$ and $CDE$, respectively. Let $P$ be the crossing point of the tangent to $\omega$ at $A$ with the tangent to $\Omega$ at $D$. Prove that $PE$ is tangent to $\Omega$. [i]Jakob Jurij Snoj, Slovenia[/i]

In a tournament, every team plays exactly once against every other team. One won match earns $3$ points for the winner and $0$ for the loser. With a draw both teams receive $1$ point each. At the end of the tournament it appears that all teams together have achieved $15$ points. The last team on the final list scored exactly $1$ point. The second to last team has not lost a match. a) How many teams participated in the tournament? b) How many points did the team score in second place in the final ranking?