A sequence infinity $a_{1}, a_{2},...,$ is $generadora$ if: $a_{1}=1,2$ and $a_{n+1}$ is obtained by placing a digit 1 on the left or a digit 2 on the right for all natural n. Prove that there is an infinite $generadora$ sequence such that it does not contain any multiples of 7.

Let $n$ be a positive integer. Suppose for any $\mathcal{S} \subseteq \{1, 2, \cdots, n\}$, $f(\mathcal{S})$ is the set containing all positive integers at most $n$ that have an odd number of factors in $\mathcal{S}$. How many subsets of $\{1, 2, \cdots, n\}$ can be turned into $\{1\}$ after finitely many (possibly zero) applications of $f$?

Determine all positive integers that are expressible in the form \[a^{2}+b^{2}+c^{2}+c,\] where $a$, $b$, $c$ are integers.

Find an integer $n$, where $100 \leq n \leq 1997$, such that \[\frac{2^{n}+2}{n}\] is also an integer.

Let $n$ be a positive integer. Navinim writes down all positive square numbers that divide $n$ on a blackboard. For each number $k$ on the blackboard, Navagem replaces it with $d(k)$. Show that the sum of all numbers on the blackboard now is a perfect square. (Note: $d(k)$ denotes the number of divisors of $k$.) [i](Proposed by Ivan Chan Guan Yu)[/i]

Let $a,b$ be two positive integers and $a>b$.We know that $\gcd(a-b,ab+1)=1$ and $\gcd(a+b,ab-1)=1$. Prove that $(a-b)^2+(ab+1)^2$ is not a perfect square.

A triple of numbers $(a_1,a_2,a_3)=(3,4,12)$ is given. The following operation is performed a finite number of times: choose two numbers $a,b$ from the triple and replace them by $0.6x-0.8y$ and $0.8x+0.6y$. Is it possible to obtain the (unordered) triple $(2,8,10)$?

Consider the fourteen numbers, $1^4,2^4,...,14^4$. The smallest natural numebr $n$ such that they leave distinct remainders when divided by $n$ is:

We consider $n$ real variables $x_i(1 \le i \le n)$, where $n$ is an integer and $n \ge 2$. The product of these variables will be denoted by $p$, their sum by $s$, and the sum of their squares by $S$. Furthermore, let $\alpha$ be a positive constant. We now study the inequality $ps \le S\alpha$. Prove that it holds for every $n$-tuple $(x_i)$ if and only if $\alpha=\frac{n+1}{2}$

No matter how $n$ real numbers on the interval $[1,2013]$ are selected, if it is possible to find a scalene polygon such that its sides are equal to some of the numbers selected, what is the least possible value of $n$? $ \textbf{(A)}\ 14 \qquad\textbf{(B)}\ 13 \qquad\textbf{(C)}\ 12 \qquad\textbf{(D)}\ 11 \qquad\textbf{(E)}\ 10 $

A function $f$ is defined recursively by $f(1)=f(2)=1$ and $$f(n)=f(n-1)-f(n-2)+n$$ for all integers $n \geq 3$. What is $f(2018)$? $\textbf{(A)} \text{ 2016} \qquad \textbf{(B)} \text{ 2017} \qquad \textbf{(C)} \text{ 2018} \qquad \textbf{(D)} \text{ 2019} \qquad \textbf{(E)} \text{ 2020}$

A point object of mass $M$ hangs from the ceiling of a car from a massless string of length $L$. It is observed to make an angle $\theta$ from the vertical as the car accelerates uniformly from rest. Find the acceleration of the car in terms of $\theta$, $M$, $L$, and $g$. [asy] size(250); import graph; // Left draw((-3,0)--(-23,0),linewidth(1.5)); draw((-13,0)--(-13,-14)); filldraw(circle((-13,-15),2),gray); draw((-13,-15)--(-21,-15),dashed); draw((-21,-14)--(-21,-1),EndArrow(size=5)); draw((-21,-1)--(-21,-14),EndArrow(size=5)); label(scale(1.5)*"$L$",(-21,-7.5),2*E); // Right draw((3,0)--(23,0),linewidth(1.5)); draw((13,0)--(13,-19),dashed); draw((13,0)--(5,-12)); filldraw(circle((3.89,-13.66),2),gray); label(scale(1.5)*"$\theta$",(12,-9),1.5*W); real f(real x){ return 5x^2/12-95x/12+25; } draw(graph(f,12,7),Arrows); [/asy] (A) $Mg \sin \theta$ (B) $MgL \tan \theta$ (C) $g \tan \theta$ (D) $g \cot \theta$ (E) $Mg \tan \theta$

Let $ a_1$, $ a_2$, $ \ldots$, $ a_n$ be positive real numbers ($ n \ge 2$) whose sum is $ S$. Show that \[ \sum_{i\equal{}1}^n\frac{a_i^{2^{k}}}{\left(S\minus{}a_i\right)^{2^t\minus{}1}}\ge\frac{S^{1\plus{}2^k\minus{}2^t}}{\left(n\minus{}1\right)^{2^t\minus{}1}n^{2^k\minus{}2^t}}\] for any nonnegative integers $ k$, $ t$ with $ k \ge t$. When does equality occur?

Show that among any nine complex numbers whose affixes in the complex plane lie on the unit circle, there are at least two of them such that the modulus of their sum is greater than $ \sqrt 2. $ [i]Ion Tecu[/i]

Find the smallest possible area of a convex pentagon whose vertexes are lattice points in a plane.

Consider two points $A(x_1, y_1)$ and $B(x_2, y_2)$ on the graph of the function $y = \frac{1}{x}$ such that $0 < x_1 < x_2$ and $AB = 2 \cdot OA$, where $O = (0, 0)$. Let $C$ be the midpoint of the segment $AB$. Prove that the angle between the $x$-axis and the ray $OA$ is equal to three times the angle between the $x$-axis and the ray $OC$.

Let $\Theta_1$ and $\Theta_2$ be circumferences with centers $O_1$ and $O_2$, exteriorly tangents. Let $A$ and $B$ be points in $\Theta_1$ and $\Theta_2$, respectively, such that $AB$ is common external tangent to $\Theta_1$ and $\Theta_2$. Let $C$ and $D$ be points on the semiplane determined by $AB$ that does not contain $O_1$ and $O_2$ such that $ABCD$ is a square. If $O$ is the center of this square, compute the possible values for the angle $\angle O_1OO_2$.

$N$ horsemen are riding in the same direction along a circular road. Their speeds are constant and pairwise distinct. There is a single point on the road where the horsemen can surpass one another. Can they ride in this fashion for arbitrarily long time? Consider the cases: $(a) N = 3;$ $(b) N = 10.$

Find all polynomials $P(x)$ such that \[ P(x^3+1)=P(x^3)+P(x^2). \] [i]Proposed by A. Golovanov[/i]

Find all functions $f : \mathbb{Q}^{+} \rightarrow \mathbb{Q}^{+}$, where $\mathbb{Q}^{+}$ is the set of positive rationals, such that $f(x+1) = f(x) + 1$ and $f(x^3) = f(x)^3$ for all $x$.

Let $a$ be a positive integer. Determine all $a$ such that the equation $$ \biggl( 1+\frac{1}{x} \biggr) \cdot \biggl( 1+\frac{1}{x+1} \biggr) \cdots \biggl( 1+\frac{1}{x+a} \biggr)=a-x$$ has at least one integer solution for $x$. For every such $a$ state the respective solutions. (Richard Henner)

Let $ABC$ be an equilateral triangle. $M$ is the midpoint of segment $AB$ and $N$ is the midpoint of segment $BC$. Let $P$ be the point outside $ABC$ such that the triangle $ACP$ is isosceles and right in $P$. $PM$ and $AN$ are cut in $I$. Prove that $CI$ is the bisector of the angle $MCA$ .

Let $n=>2$ be a natural number. A set $S$ of natural numbers is called $complete$ if, for any integer $0<=x<n$, there is a subset of $S$ with the property that the remainder of the division by $n$ of the sum of the elements in the subset is $x$. The sum of the elements of the empty set is considered to be $0$. Show that if a set $S$ is $complete$, then there is a subset of $S$ which has at most $n-1$ elements and which is still $complete$.

In a $2^n \times 2^n$ square with $n$ positive integer is covered with at least two non-overlapping rectangle pieces with integer dimensions and a power of two as surface. Prove that two rectangles of the covering have the same dimensions (Two rectangles have the same dimensions as they have the same width and the same height, wherein they, not allowed to be rotated.)

19) Each cell of a $2$ × $5$ grid of unit squares is to be colored white or black. Compute the number of such colorings for which no $2$ × $2$ square is a single color. 20) Let $n$ be a three-digit integer with nonzero digits, not all of which are the same. Define $f(n)$ to be the greatest common divisor of the six integers formed by any permutation of $n$s digits. For example, $f(123) = 3$, because $gcd(123, 132, 213, 231, 312, 321) = 3$. Let the maximum possible value of $f(n)$ be $k$. Find the sum of all $n$ for which $f(n) = k$. 21) Consider a $2$ × $2$ grid of squares. Each of the squares will be colored with one of $10$ colors, and two colorings are considered equivalent if one can be rotated to form the other. How many distinct colorings are there? 22) Find all the roots of the polynomial $x^5 - 5x^4 + 11x^3 -13x^2+9x-3$ 23) Compute the smallest positive integer $n$ for which $0 < \sqrt[4]{n} - \left \lfloor{\sqrt[4]{n}}\right \rfloor < \dfrac{1}{2015}$. 24) Three ants begin on three different vertices of a tetrahedron. Every second, they choose one of the three edges connecting to the vertex they are on with equal probability and travel to the other vertex on that edge. They all stop when any two ants reach the same vertex at the same time. What is the probability that all three ants are at the same vertex when they stop? 25) Let $ABC$ be a triangle that satisfies $AB = 13$, $BC = 14$, $AC = 15$. Given a point $P$ in the plane, let $PA$, $PB$, $PC$ be the reflections of $A$, $B$, $C$ across $P$. Call $P$ [i]good[/i] if the circumcircle of $P_A P_B P_C$ intersects the circumcircle of $ABC$ at exactly 1 point. The locus of good points $P$ encloses a region $S$. Find the area of $S$. 26. Let $f : \mathbb{R}^+ \rightarrow \mathbb{R}$ be a continuous function satisfying $f(xy) = f(x) + f(y) + 1$ for all positive reals ${x,y}$. If $f(2) = 0$, compute $f(2015)$. 27) Let $ABCD$ be a quadrilateral with $A = (3,4)$, $B=(9,-40)$, $C = (-5,-12)$, $D = (-7,24)$. Let $P$ be a point in the plane (not necessarily inside the quadrilateral). Find the minimum possible value of $\overline{AP} + \overline{BP} + \overline{CP} + \overline{DP}$.