[b](a)[/b] Let $n$ be a positive integer, prove that \[ \sqrt{n+1} - \sqrt{n} < \frac{1}{2 \sqrt n}\] [b](b)[/b] Find a positive integer $n$ for which \[ \bigg\lfloor 1 +\frac{1}{\sqrt 2} +\frac{1}{\sqrt 3} +\frac{1}{\sqrt 4} + \cdots +\frac{1}{\sqrt n} \bigg\rfloor =12\]

Find all functions $f : \mathbb{Z}\rightarrow \mathbb{Z}$ such that for all prime $p$ the following condition holds: $$p \mid ab + bc + ca \iff p \mid f(a)f(b) + f(b)f(c) + f(c)f(a)$$ [i]Proposed by Anzo Teh Zhao Yang[/i]

In a country there are $100$ airports. Super-Air operates direct flights between some pairs of airports (in both directions). The [i]traffic[/i] of an airport is the number of airports it has a direct Super-Air connection with. A new company, Concur-Air, establishes a direct flight between two airports if and only if the sum of their traffics is at least $100.$ It turns out that there exists a round-trip of Concur-Air flights that lands in every airport exactly once. Show that then there also exists a round-trip of Super-Air flights that lands in every airport exactly once.

The points $A, B, C, D$ lie in this order on a circle with center $O$. Furthermore, the straight lines $AC$ and $BD$ should be perpendicular to each other. The base of the perpendicular from $O$ on $AB$ is $F$. Prove $CD = 2 OF$. [i](Karl Czakler)[/i]

Let $\mathcal{K}$ be a circle centered in $A$. Let $p$ be a line tangent to $\mathcal{K}$ in $B$ and let a line parallel to $p$ intersect $\mathcal{K}$ in $C$ and $D$. Let the line $AD$ intersect $p$ in $E$ and let $F$ be the intersection of the lines $CE$ and $AB$. Prove that the line through $D$, parallel to the tangent through $A$ to the circumcircle of $AFD$ intersects the line $CF$ on $\mathcal{K}$.

Magnesium chloride dissolves in water to form: $ \textbf{(A)}\hspace{.05in}\text{hydrated MgCl}_2 \text{molecules}\qquad$ $\textbf{(B)}\hspace{.05in}\text{hydrated Mg}^{2+} \text{ions and hydrated Cl}^- \text{ions} \qquad$ $\textbf{(C)}\hspace{.05in}\text{hydrated Mg}^{2+} \text{ions and hydrated Cl}_2 ^{2-} \text{ions}\qquad$ $\textbf{(D)}\hspace{.05in}\text{hydrated Mg atoms and hydrated Cl}_2 \text{molecules}\qquad$

Find all triples $(x, y, z) \in R^3$ satisfying the equations $\begin{cases} x^2 + 4y^2 = 4zx \\ y^2 + 4z^2 = 4xy \\ z^2 + 4x^2 = 4yz \end{cases}$

Determine the greatest common divisor of the elements of the set \[\{n^{13}-n \; \vert \; n \in \mathbb{Z}\}.\]

Find all functions $f:\mathbb{Z}^+ \rightarrow \mathbb{Z}^+$ (where $\mathbb{Z}^+$ is the set of positive integers) such that $f(n!) = f(n)!$ for all positive integers $n$ and such that $m-n$ divides $f(m) - f(n)$ for all distinct positive integers $m, n$.

We are given a regular decagon with all diagonals drawn. The number "$+ 1$ " is attached to each vertex and to each point where diagonals intersect (we consider only internal points of intersection). We can decide at any time to simultaneously change the sign of all such numbers along a given side or a given diagonal . Is it possible after a certain number of such operations to have changed all the signs to negative?

In a given triangle $ABC$, $O$ is its circumcenter, $D$ is the midpoint of $AB$ and $E$ is the centroid of the triangle $ACD$. Show that the lines $CD$ and $OE$ are perpendicular if and only if $AB=AC$.

Do there exist $ 14$ positive integers, upon increasing each of them by $ 1$,their product increases exactly $ 2008$ times?

Let $G$ be the centroid of triangle $ABC$, and $A'$ the symmetric of $A$ wrt $C$. Show that $G, B, C, A'$ are concyclic if and only if $GA \perp GC$.

How many integers less than $2502$ are equal to the square of a prime number?

a) Does there exist a triangle in which the shortest median is longer that the longest bisectrix? b) Does there exist a triangle in which the shortest bisectrix is longer that the longest altitude?

Let $x$ and $y$ be real numbers such that $\frac{\sin{x}}{\sin{y}} = 3$ and $\frac{\cos{x}}{\cos{y}} = \frac{1}{2}$. The value of $\frac{\sin{2x}}{\sin{2y}} + \frac{\cos{2x}}{\cos{2y}}$ can be expressed in the form $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find $p + q$.

The rays $l_1,l_2,\ldots,l_{n-1}$ divide a given angle $ABC$ into $n$ equal parts. A line $l$ intersects $AB$ at $A_1$, $BC$ at $A_{n+1}$, and $l_i$ at $A_{i+1}$ for $i=1,\ldots,n-1$. Show that the quantity $$\left(\frac1{BA_1}+\frac1{BA_{n+1}}\right)\left(\frac1{BA_1}+\frac1{BA_2}+\ldots+\frac1{BA_{n+1}}\right)^{-1}$$is independent of the line $l$, and compute its value if $\angle ABC=\phi$.

Fix a sequence $a_1,a_2,a_3\ldots$ of integers satisfying the following condition:for all prime numbers $p$ and all positive integers $k$,we have $a_{pk+1}=pa_k-3a_p+13$.Determine all possible values of $a_{2013}$.

Let $c$ be a positive integer. The sequence $\{f_n\}$ is defined as follows: \[f_1 = 1, f_2 = c, f_{n+1} = 2f_n - f_{n-1} + 2 \quad (n \geq 2).\] Show that for each $k \in \mathbb N$ there exists $r \in \mathbb N$ such that $f_kf_{k+1}= f_r.$

Given an integer $n\ge3$ consider positive integers $x_1,\ldots,x_n$ such that $x_1<x_2<\cdots<x_n<2x_1$. If $p$ is a prime and $r$ is a positive integer such that $p^r$ divides the product $x_1\cdots x_n$, prove that $$\frac{x_1\cdots x_n}{p^r}>n!.$$

a) Prove that one cannot assign to each vertex of a cube $ 8$ distinct numbers from the set $\{0, 1, 2, 3, . . . , 11, 12\}$ such that, for every edge, the sum of the two numbers assigned to its vertices is even. b) Prove that one can assign to each vertex of a cube $8$ distinct numbers from the set $\{0, 1, 2, 3, . . . , 11, 12\}$ such that, for every edge, the sum of the two numbers assigned to its vertices is divisible by $3$.

Let $n$ ($n \ge 1$) be an integer. Consider the equation $2\cdot \lfloor{\frac{1}{2x}}\rfloor - n + 1 = (n + 1)(1 - nx)$, where $x$ is the unknown real variable. (a) Solve the equation for $n = 8$. (b) Prove that there exists an integer $n$ for which the equation has at least $2021$ solutions. (For any real number $y$ by $\lfloor{y} \rfloor$ we denote the largest integer $m$ such that $m \le y$.)

The integers greater than one are arranged in five columns as follows: \[ \begin{tabular}{c c c c c} \ & 2 & 3 & 4 & 5 \\ 9 & 8 & 7 & 6 & \ \\ \ & 10 & 11 & 12 & 13 \\ 17 & 16 & 15 & 14 & \ \\ \ & . & . & . & . \\ \end{tabular} \] (Four consecutive integers appear in each row; in the first, third and other odd numbered rows, the integers appear in the last four columns and increase from left to right; in the second, fourth and other even numbered rows, the integers appear in the first four columns and increase from right to left.) In which column will the number $1,000$ fall? $ \textbf{(A)}\ \text{first} \qquad\textbf{(B)}\ \text{second} \qquad\textbf{(C)}\ \text{third} \qquad\textbf{(D)}\ \text{fourth} \qquad\textbf{(E)}\ \text{fifth} $

Let $a$ and $b$ be positive integers with the property that $\frac{a}{b} > \sqrt2$. Prove that $$\frac{a}{b} - \frac{1}{2ab} > \sqrt2$$

Let $F$ be an intersection point of altitude $CD$ and internal angle bisector $AE$ of right angled triangle $ABC$, $\angle ACB = 90^{\circ}$. Let $G$ be an intersection point of lines $ED$ and $BF$. Prove that area of quadrilateral $CEFG$ is equal to area of triangle $BDG$