Define a sequence $a_n$ such that $a_n = a_{n-1} - a_{n-2}$. Let $a_1 = 6$ and $a_2 = 5$. Find $\Sigma_{n=1}^{1000}a_n$.

Prove that if $ n $ is an integer greater than $ 4 $, then $ 2^n $ is greater than $ n^2 $.

For real numbers $x,$ let $T_x$ be the triangle with vertices $(5, 5^3),$ $(8, 8^3),$ and $(x, x^3)$ in $\mathbb{R}^2.$ Over all $x$ in the interval $[5, 8],$ the area of the triangle $T_x$ is maximized at $x = \sqrt{n},$ for some positive integer $n.$ Compute $n.$

A $(3n + 1) \times (3n + 1)$ table $(n \in \mathbb{N})$ is given. Prove that deleting any one of its squares yields a shape cuttable into pieces of the following form and its rotations: ''L" shape formed by cutting one square from a $2 \times 2$ squares.

Ana chose some unit squares of a $50 \times 50$ board and placed a chip on each of them. Prove that Beto can always choose at most $99$ empty unit squares and place a chip on each so that each row and each column of the board contains an even number of chips.

Prove that the numbers $1,2,...,1998$ cannot be separated into three classes whose sums of elements are divisible by $2000,3999$, and $5998$, respectively.

Prove that for every acute angle $\alpha$, $\sin (\cos \alpha) < \cos(\sin \alpha)$.

Let $n$ be a positive integer relatively prime to $6$. We paint the vertices of a regular $n$-gon with three colours so that there is an odd number of vertices of each colour. Show that there exists an isosceles triangle whose three vertices are of different colours.

Let $f(x)=x^3+ax^2+bx$ for some $a,b$. For some $c$, $f(c)$ achieves a local maximum of $539$ (in other words, $f(c)$ is the maximum value of $f$ for some open interval around $c$). In addition, at some $d$, $f(d)$ achieves a local minimum of $-325$. Given that $c$ and $d$ are integers, compute $a+b$.

A football is made by sewing together some black and white leather patches. The black patches are regular pentagons of the same size. The white patches are regular hexagons of the same size. Each pentagon is bordered by 5 hexagons. Each hexagons is bordered by $3$ pentagons and $3$ hexagons. We need $12$ pentagons to make one football. How many hexagons are needed to make one football?

Blondes, brunettes, redheads and brown-haired women participate in the Olympiad. There are twice as many redheads as brown-haired. Blondes and redheads make up a quarter of the total number of participants, and Brown-haired and Blondes one fifth part. Prove that the number of Brunettes is divisible by $7$.

In the cells of a $5 \times 5$ grid there are some lamps. If a lamp is touched, it is turned on and it lights up all of its neighbouring cells, including its own cell. If a cell is lit up and there is a lamp in it, the lamp is also turned on and lights up its neighbouring cells, including its own. What is the smallest number of lamps needed to light up all of the cells with just one touch? [i](Note: Two cells are neighbours if they have a common side or vertex.)[/i]

Let $(G,\cdot)$ be a group and $H\neq G$ be a subgroup so that $x^2=y^2$ for all $x,y\in G\setminus H.$ Show that $(H,\cdot)$ is an Abelian group.

[url=http://www.mathlinks.ro/Forum/viewtopic.php?t=107262]IMO 2007 HKTST 1[/url] Problem 4 (a) Given five points on a plane such that no three of the points are collinear. Show that among the triangles which are drwan using any three of these five points as vertices, at least three of the triangles formed are not acute-angled triangles. (An acute-angled triangle is one in which all the three interior angles are acute angles.) (b) Given any 100 points on a plane such that no three of the points are collinear. SHow that among the triangles which are drawn using any three of these 100 points as vertices, at least 30% of the trinagles are not acute-angled triangles.

Find all polynomials $P(x, y)$ with real coefficients which for all real numbers $x$ and $y$ satisfy $P(x + y, x - y) = 2P(x, y)$.

Prove that $1280000401$ is composite.

Let $a$, $b$ and $c$ be pairwise distinct nonnegative real numbers. Prove that \[ (a + b + c) \left( \frac{a}{(b - c)^2} + \frac{b}{(c - a)^2} + \frac{c}{(a - b)^2} \right) > 4. \] [i](Karl Czakler)[/i]

The regular octagon $ABCDEFGH$ has its center at $J$. Each of the vertices and the center are to be associated with one of the digits $1$ through $9$, with each digit used once, in such a way that the sums of the numbers on the lines $AJE$, $BJF$, $CJG$, and $DJH$ are equal. In how many ways can this be done? $\textbf{(A) }384\qquad \textbf{(B) }576\qquad \textbf{(C) }1152\qquad \textbf{(D) }1680\qquad \textbf{(E) }3546\qquad$ [asy] size(175); defaultpen(linewidth(0.8)); path octagon; string labels[]={"A","B","C","D","E","F","G","H","I"}; for(int i=0;i<=7;i=i+1) { pair vertex=dir(135-45/2-45*i); octagon=octagon--vertex; label("$"+labels[i]+"$",vertex,dir(origin--vertex)); } draw(octagon--cycle); dot(origin); label("$J$",origin,dir(0)); [/asy]

Calculate the value of $\sqrt{\dfrac{11^4+100^4+111^4}{2}}$ and answer in the form of an integer.

For each real parameter $a$, find the number of real solutions to the system $$\begin{cases} |x|+|y| = 1 , \\ x^2 +y^2 = a \end{cases}$$

Suppose that for some positive integer $n$, the first two digits of $5^n$ and $2^n$ are identical. Suppose the first two digits are $a$ and $b$ in this order. Find the two-digit number $\overline{ab}$.

Calculate the following indefinite integrals. [1] $\int (x^2+4x-3)^2(x+2)dx$ [2] $\int \frac{\ln x}{x(\ln x+1)}dx$ [3] $\int \frac{\sin \ (\pi \log _2 x)}{x}dx$ [4] $\int \frac{dx}{\sin x\cos ^ 2 x}$ [5] $\int \sqrt{1-3x}\ dx$

Given $k > 0$, the sequence $a_n$ is defined by its first two members and \[ a_{n+2} = a_{n+1} + \frac{k}{n}a_n \] a)For which $k$ can we write $a_n$ as a polynomial in $n$? b) For which $k$ can we write $\frac{a_{n+1}}{a_n} = \frac{p(n)}{q(n)}$? ($p,q$ are polynomials in $\mathbb R[X]$).

Consider $f(x)=x^4+ax^3+bx^2+cx+d\; (a,b,c,d\in\mathbb{R})$. It is known that $f$ intersects X-axis in at least $3$ (distinct) points. Show either $f$ has $4$ $\mathbf{distinct}$ real roots or it has $3$ $\mathbf{distinct}$ real roots and one of them is a point of local maxima or minima.

Compute the smallest positive integer $n$ such that $2016^n$ does not divide $2016!$.