Let $a, b$ and $c$ be natural numbers. Determine the smallest value that the following expression can take: $$\frac{a}{gcd\,\,(a + b, a - c)} + \frac{b}{gcd\,\,(b + c, b - a)} + \frac{c}{gcd\,\,(c + a, c - b)}.$$ . Remark: $gcd \,\, (6, 0) = 6$ and $gcd\,\,(3, -6) = 3$.

Let $ r$ and $ s$ two orthogonal lines that does not lay on the same plane. Let $ AB$ be their common perpendicular, where $ A\in{}r$ and $ B\in{}s$(*).Consider the sphere of diameter $ AB$. The points $ M\in{r}$ and $ N\in{s}$ varies with the condition that $ MN$ is tangent to the sphere on the point $ T$. Find the locus of $ T$. Note: The plane that contains $ B$ and $ r$ is perpendicular to $ s$.

Let $m,n,k$ be positive integers and $1+m+n \sqrt 3=(2+ \sqrt 3)^{2k+1}$. Prove that $m$ is a perfect square.

A list of numbers $a_1,a_2,\ldots,a_m$ contains an arithmetic trio $a_i, a_j, a_k$ if $i < j < k$ and $2a_j = a_i + a_k$. Let $n$ be a positive integer. Show that the numbers $1, 2, 3, \ldots, n$ can be reordered in a list that does not contain arithmetic trios.

a)Prove that $\frac{1}{2}+\frac{1}{3}+...+\frac{1}{{{2}^{m}}}<m$, for any $m\in {{\mathbb{N}}^{*}}$. b)Let ${{p}_{1}},{{p}_{2}},...,{{p}_{n}}$ be the prime numbers less than ${{2}^{100}}$. Prove that $\frac{1}{{{p}_{1}}}+\frac{1}{{{p}_{2}}}+...+\frac{1}{{{p}_{n}}}<10$

Let $ A$ represent the set of all functions $ f : \mathbb{N} \rightarrow \mathbb{N}$ such that for all $ k \in \overline{1, 2007}$, $ f^{[k]} \neq \mathrm{Id}_{\mathbb{N}}$ and $ f^{[2008]} \equiv \mathrm{Id}_{\mathbb{N}}$. a) Prove that $ A$ is non-empty. b) Find, with proof, whether $ A$ is infinite. c) Prove that all the elements of $ A$ are bijective functions. (Denote by $ \mathbb{N}$ the set of the nonnegative integers, and by $ f^{[k]}$, the composition of $ f$ with itself $ k$ times.)

A positive integer $a$ is called a [i]double number[/i] if it has an even number of digits (in base 10) and its base 10 representation has the form $a = a_1a_2 \cdots a_k a_1 a_2 \cdots a_k$ with $0 \le a_i \le 9$ for $1 \le i \le k$, and $a_1 \ne 0$. For example, $283283$ is a double number. Determine whether or not there are infinitely many double numbers $a$ such that $a + 1$ is a square and $a + 1$ is not a power of $10$.

On the same side of a straight line three circles are drawn as follows: a circle with a radius of $4$ inches is tangent to the line, the other two circles are equal, and each is tangent to the line and to the other two circles. The radius of the equal circles is: $ \textbf{(A)}\ 24 \qquad\textbf{(B)}\ 20\qquad\textbf{(C)}\ 18\qquad\textbf{(D)}\ 16\qquad\textbf{(E)}\ 12 $

Points $P$ and $Q$ on sides $AB$ and $AC$ of triangle $ABC$ are such that $PB = QC$. Prove that $PQ < BC$.

Let $ABCDE$ be a convex pentagon with $AB = BC =CA$ and $CD = DE = EC$. Let $T$ be the centroid of $\vartriangle ABC$, and $N$ be the midpoint of $AE$. Compute $\angle NT D$

Let $K$, $L$, $M$, $N$ be the midpoints of sides $BC$, $CD$, $DA$, $AB$ respectively of a convex quadrilateral $ABCD$. The common points of segments $AK$, $BL$, $CM$, $DN$ divide each of them into three parts. It is known that the ratio of the length of the medial part to the length of the whole segment is the same for all segments. Does this yield that $ABCD$ is a parallelogram?

Let $d(n)$ be the number of positive divisors of a positive integer $n$ (including $1$ and $n$). Find all values of $n$ such that $n + d(n) = d(n)^2$.

Let $n$ be a given positive integer. Sisyphus performs a sequence of turns on a board consisting of $n + 1$ squares in a row, numbered $0$ to $n$ from left to right. Initially, $n$ stones are put into square $0$, and the other squares are empty. At every turn, Sisyphus chooses any nonempty square, say with $k$ stones, takes one of these stones and moves it to the right by at most $k$ squares (the stone should say within the board). Sisyphus' aim is to move all $n$ stones to square $n$. Prove that Sisyphus cannot reach the aim in less than \[ \left \lceil \frac{n}{1} \right \rceil + \left \lceil \frac{n}{2} \right \rceil + \left \lceil \frac{n}{3} \right \rceil + \dots + \left \lceil \frac{n}{n} \right \rceil \] turns. (As usual, $\lceil x \rceil$ stands for the least integer not smaller than $x$. )

In triangle $ABC$, we have $\angle ABC=\angle ACB=44^o$. Point $M$ is in its interior such that $\angle MBC=16^o$ and $\angle MCB=30^o$. Prove that $\angle MAC=\angle MBC$. [i] (Andra Elena Mircea)[/i]

Suppose $z_1, z_2 , \cdots z_n$ are $n$ complex numbers such that $min_{j \not= k} | z_{j} - z_{k} | \geq max_{1 \leq j \leq n} |z_j|$. Find the maximum possible value of $n$. Further characterise all such maximal configurations.

Points $B = B_1 , B_2, \cdots , B_n , B_{n+1} = C$ are chosen on side $BC$ of a triangle $ABC$ in that order. Let $r_j$ be the inradius of triangle $AB_jB_{j+1}$ for $j = 1, \cdots, n$ , and $r$ be the inradius of $\triangle ABC$. Show that there is a constant $\lambda$ independent of $n$ such that : \[(\lambda -r_1)(\lambda -r_2)\cdots (\lambda -r_n) =\lambda^{n-1}(\lambda -r)\]

$2017$ chess players participated in the chess tournament, each of them played exactly one chess game with each other. Let's call a trio of chess players $A, B, C$ a [i]principled [/i]one, if $A$ defeated $B$, $B$ defeated $C$, and $C$ defeated $A$. What is the largest possible number of threes of principled chess players?

Mr. Jones has eight children of different ages. On a family trip his oldest child, who is 9, spots a license plate with a 4-digit number in which each of two digits appears two times. "Look, daddy!" she exclaims. "That number is evenly divisible by the age of each of us kids!" "That's right," replies Mr. Jones, "and the last two digits just happen to be my age." Which of the following is not the age of one of Mr. Jones's children? $ \textbf{(A) } 4 \qquad \textbf{(B) } 5 \qquad \textbf{(C) } 6 \qquad \textbf{(D) } 7 \qquad \textbf{(E) } 8$

There are $2019$ coins on a table. Some are placed with head up and others tail up. A group of $2019$ persons perform the following operations: the first person chooses any one coin and then turns it over, the second person choses any two coins and turns them over and so on and the $2019$-th person turns over all the coins. Prove that no matter which sides the coins are up initially, the $2019$ persons can come up with a procedure for turning the coins such that all the coins have smae side up at the end of the operations.

The yearly changes in the population census of a town for four consecutive years are, respectively, $25\%$ increase, $25\%$ increase, $25\%$ decrease, $25\%$ decrease. The net change over the four years, to the nearest percent, is: ${{ \textbf{(A)}\ -12 \qquad\textbf{(B)}\ -1 \qquad\textbf{(C)}\ 0 \qquad\textbf{(D)}\ 1}\qquad\textbf{(E)}\ 12} $

If $xyz=1$ find the value of $\left(\frac{1}{1+x+\frac{1}{y}}+\frac{1}{1+y+\frac{1}{z}}+\frac{1}{1+z+\frac{1}{x}}\right)^2$.

Consider an equilateral triangle with every side divided by $n$ points into $n+1$ equal parts. We put a marker on every of the $3n$ division points. We draw lines parallel to the sides of the triangle through the division points, and this way divide the triangle into $(n+1)^2$ smaller ones. Consider the following game: if there is a small triangle with exactly one vertex unoccupied, we put a marker on it and simultaneously take markers from the two its occupied vertices. We repeat this operation as long as it is possible. (a) If $n\equiv1\pmod3$, show that we cannot manage that only one marker remains. (b) If $n\equiv0$ or $n\equiv2\pmod3$, prove that we can finish the game leaving exactly one marker on the triangle.

Po picks $100$ points $P_1,P_2,\cdots, P_{100}$ on a circle independently and uniformly at random. He then draws the line segments connecting $P_1P_2,P_2P_3,\ldots,P_{100}P_1.$ Find the expected number of regions that have all sides bounded by straight lines.

In a $5 \times 9$ checkered rectangle, the middle row and middle column are colored gray. You leave the corner cell and move to the cell adjacent to the side with each move. For each transition from a gray cell to a gray one you need to pay a ruble. What is the smallest number of rubles you need to pay to go around all the squares of the board exactly once (it is not necessary to return to the starting square)?

Prove that for all polynomials $P \in \mathbb{R}[x]$ and positive integers $n$, $P(x)-x$ divides $P^n(x)-x$ as polynomials.