A triangle is bounded by the lines $a_1 x+ b_1 y +c_1=0$, $a_2 x+ b_2 y +c_2=0$ and $a_2 x+ b_2 y +c_2=0$. Show that its area, disregarding sign, is $$\frac{\Delta^{2}}{2(a_2 b_3- a_3 b_2)(a_3 b_1- a_1 b_3)(a_1 b_2- a_2 b_1)},$$ where $\Delta$ is the discriminant of the matrix $$M=\begin{pmatrix} a_1 & b_1 &c_1\\ a_2 & b_2 &c_2\\ a_3 & b_3 &c_3 \end{pmatrix}.$$

In the figure, the outer equilateral triangle has area $16$, the inner equilateral triangle has area $1$, and the three trapezoids are congruent. What is the area of one of the trapezoids? [asy] size((70)); draw((0,0)--(7.5,13)--(15,0)--(0,0)); draw((1.88,3.25)--(9.45,3.25)); draw((11.2,0)--(7.5,6.5)); draw((9.4,9.7)--(5.6,3.25)); [/asy] $\textbf{(A)}\ 3 \qquad \textbf{(B)}\ 4 \qquad \textbf{(C)}\ 5 \qquad \textbf{(D)}\ 6 \qquad \textbf{(E)}\ 7$

A positive integer $m$ is perfect if the sum of all its positive divisors, $1$ and $m$ inclusive, is equal to $2m$. Determine the positive integers $n$ such that $n^n + 1$ is a perfect number.

Given $1000$ squares on the plane with their sides parallel to the coordinate axes. Let $M$ be the set of those squares centres. Prove that you can mark some squares in such a way, that every point of $M$ will be contained not less than in one and not more than in four marked squares

Let $ABC$ be a triangle with symmedian point $K$, and let $\theta = \angle AKB-90^{\circ}$. Suppose that $\theta$ is both positive and less than $\angle C$. Consider a point $K'$ inside $\triangle ABC$ such that $A,K',K,$ and $B$ are concyclic and $\angle K'CB=\theta$. Consider another point $P$ inside $\triangle ABC$ such that $K'P\perp BC$ and $\angle PCA=\theta$. If $\sin \angle APB = \sin^2 (C-\theta)$ and the product of the lengths of the $A$- and $B$-medians of $\triangle ABC$ is $\sqrt{\sqrt{5}+1}$, then the maximum possible value of $5AB^2-CA^2-CB^2$ can be expressed in the form $m\sqrt{n}$ for positive integers $m,n$ with $n$ squarefree. Compute $100m+n$. [i]Proposed by Vincent Huang[/i]

Let $m \geq 2$ be a fixed positive integer, and $(a_n)_{n\geq 1}$ be a sequence of nonnegative real numbers such that, for all $n\geq 1$, we have that $a_{n+1} \leq a_n - a_{mn}$. a) Prove that the sequence $b_n = \sum_{k=1}^{n} a_k$ is bounded above. b) Prove that the sequence $c_n = \sum_{k=1}^{n} k^2 a_k$ is bounded above.

Prove that there exists an infinite sequence of positive integers $ a_1, a_2, ... $ such that for all positive integers $ i $, \\ i) $ a_{i + 1} $ is divisible by $ a_{i} $.\\ ii) $ a_i $ is not divisible by $ 3 $.\\ iii) $ a_i $ is divisible by $ 2^{i + 2} $ but not $ 2^{i + 3} $.\\ iv) $ 6a_i + 1 $ is a prime power.\\ v) $ a_i $ can be written as the sum of the two perfect squares.

For every n = 2; 3; : : : , we put $$A_n = \left(1 - \frac{1}{1+2}\right) X \left(1 - \frac{1}{1+2+3}\right)X \left(1 - \frac{1}{1+2+3+...+n}\right) $$ Determine all positive integer $ n (n \geq 2)$ such that $\frac{1}{A_n}$ is an integer.

The continuous function and twice differentiable function $f: \mathbb{R}\rightarrow\mathbb{R}$ satisfies $2007^{2}\cdot f(x)+f''(x)=0$. Prove that there exist two such real numbers $k$ and $l$ such that $f(x)=l\cdot\sin(2007x)+k\cdot\cos(2007x)$.

We consider an acute angled triangle $ABC$ (with $AB<AC$) and its circumcircle $c(O,R) $(with center $O$ and semidiametre $R$).The altitude $AD$ cuts the circumcircle at the point $E$ ,while the perpedicular bisector $(m)$ of the segment $AB$,cuts $AD$ at the point $L$.$BL$ cuts $AC$ at the point $M$ and the circumcircle $c(O,R)$ at the point $N$.Finally $EN$ cuts the perpedicular bisector $(m)$ at the point $Z$.Prove that: \[ MZ \perp BC \iff \left(CA=CB \;\; \text{or} \;\; Z\equiv O \right) \]

Find all pairs $(a,b)$ of integers $a$ and $b$ satisfying \[(b^2+11(a-b))^2=a^3 b\]

Let three circles $ \Gamma_1, \Gamma_2, \Gamma_3$, which are non-overlapping and mutually external, be given in the plane. For each point $ P$ in the plane, outside the three circles, construct six points $ A_1, B_1, A_2, B_2, A_3, B_3$ as follows: For each $ i \equal{} 1, 2, 3$, $ A_i, B_i$ are distinct points on the circle $ \Gamma_i$ such that the lines $ PA_i$ and $ PB_i$ are both tangents to $ \Gamma_i$. Call the point $ P$ exceptional if, from the construction, three lines $ A_1B_1, A_2 B_2, A_3 B_3$ are concurrent. Show that every exceptional point of the plane, if exists, lies on the same circle.

Isosceles $\triangle$ $ABC$ has equal side lengths $AB$ and $BC$. In the figure below, segments are drawn parallel to $\overline{AC}$ so that the shaded portions of $\triangle$ $ABC$ have the same area. The heights of the two unshaded portions are 11 and 5 units, respectively. What is the height of $h$ of $\triangle$ $ABC$? [asy] size(12cm); draw((5,10)--(5,6.7),dashed+gray+linewidth(.5)); draw((5,3)--(5,5.3),dashed+gray+linewidth(.5)); filldraw((1.5,3)--(8.5,3)--(10,0)--(0,0)--cycle,lightgray); draw((0,0)--(10,0)--(5,10)--cycle,linewidth(1.3)); dot((0,0)); dot((5,10)); dot((10,0)); label(scale(.8)*"$11$", (5,6.5),S); dot((17.5,0)); dot((27.5,0)); dot((22.5,10)); draw((22.5,1.3)--(22.5,0),dashed+gray+linewidth(.5)); draw((22.5,2.5)--(22.5,3.6),dashed+gray+linewidth(.5)); draw((17.5,0)--(27.5,0)--(22.5,10)--cycle,linewidth(1.3)); filldraw((19.3,3.6)--(25.7,3.6)--(22.5,10)--cycle,lightgray); label(scale(.8)*"$5$", (22.5,1.9)); draw((5,10)--(22.5,10),dashed+gray+linewidth(.5)); draw((10,0)--(17.5,0),dashed+gray+linewidth(.5)); draw((13.75,4.3)--(13.75,0),dashed+gray+linewidth(.5)); draw((13.75,5.7)--(13.75,10),dashed+gray+linewidth(.5)); label(scale(.8)*"$h$", (13.75,5)); label(scale(.7)*"$A$", (0,0), S); label(scale(.7)*"$C$", (10,0), S); label(scale(.7)*"$B$", (5,10), N); label(scale(.7)*"$A$", (17.5,0), S); label(scale(.7)*"$C$", (27.5,0), S); label(scale(.7)*"$B$", (22.5,10), N); [/asy] $\textbf{(A) } 14.6 \qquad \textbf{(B) } 14.8 \qquad \textbf{(C) } 15 \qquad \textbf{(D) } 15.2 \qquad \textbf{(E) } 15.4$

The volume of a pyramid whose base is an equilateral triangle of side length 6 and whose other edges are each of length $ \sqrt{15}$ is $ \textbf{(A)}\ 9 \qquad \textbf{(B)}\ 9/2 \qquad \textbf{(C)}\ 27/2 \qquad \textbf{(D)}\ \frac{9\sqrt3}{2} \qquad \textbf{(E)}\ \text{none of these}$

An ant in the $xy$-plane is at the origin facing in the positive $x$-direction. The ant then begins a progression of moves, on the $n^{th}$ of which it first walks $\frac{1}{5^n}$ units in the direction it is facing and then turns $60^o$ degrees to the left. After a very large number of moves, the ant’s movements begins to converge to a certain point; what is the $y$-value of this point?

For each natural number $n$ we denote $a_n$ as the greatest perfect square less than or equal to $n$ and $b_n$ as the least perfect square greater than $n$. For example $a_9=3^2$, $b_9=4^2$ and $a_{20}=4^2$, $b_{20}=5^2$. Calculate: $$\frac{1}{a_1b_1}+\frac{1}{a_2b_2}+\frac{1}{a_3b_3}+\ldots +\frac{1}{a_{600}b_{600}}$$

We’re playing a game with a sequence of $2008$ non-negative integers. A move consists of picking a integer $b$ from that sequence, of which the neighbours $a$ and $c$ are positive. We then replace $a, b$ and $c$ by $a - 1, b + 7$ and $c - 1$ respectively. It is not allowed to pick the first or the last integer in the sequence, since they only have one neighbour. If there is no integer left such that both of its neighbours are positive, then there is no move left, and the game ends. Prove that the game always ends, regardless of the sequence of integers we begin with, and regardless of the moves we make.

Prove that the equation $m!n! = k!$ has infinitely many solutions in which $m, n$ and $k$ are natural numbers greater than unity .

Let $x_0, x_1, \ldots$ be a sequence of real numbers such that $x_0=1$ and $x_{n+1}=\sin(x_n)+\frac{\pi} {2}-1$ for all $n \geq 0$. Show that the sequence converges and find its limit.

Determine twice differentiable functions $f: \mathbb{R} \rightarrow \mathbb{R}$ which verify relation \[ \left( f'(x) \right)^2 + f''(x) \leq 0, \forall x \in \mathbb{R}. \]

An ant is walking on a sidewalk and discovers $12$ sidewalk panels with leaves inscribed in them, as shown below. Find the number of ways in which the ant can traverse from point $A$ to point $B$ if it can only move [list] [*] up, down, or right (along the border of a sidewalk panel), or [*] up-right (along one of two margin halves of a leaf) [/list] and cannot visit any border or margin half more than once (an example path is highlighted in red). [asy] unitsize(1cm); int r = 4; int c = 5; for (int i = 0; i < r; ++i) { for (int j = 0; j < c; ++j) { pair A = (j,i); } } for (int i = 0; i < r; ++i) { for (int j = 0; j < c; ++j) { if (j != c-1) { draw((j,i)--(j+1,i)); } if (i != r-1) { draw((j,i)--(j,i+1)); } } } for (int i = 1; i < r+1; ++i) { for (int j = 0; j < c-2; ++j) { fill(arc((i,j),1,90,180)--cycle,deepgreen); fill(arc((i-1,j+1),1,270,360)--cycle,deepgreen); draw((i-1,j)--(i,j+1), heavygreen+linewidth(0.5)); draw((i-2/3,j+1/3)--(i-2/3,j+1/3+0.1), heavygreen); draw((i-1/3,j+2/3)--(i-1/3,j+2/3+0.1), heavygreen); draw((i-2/3,j+1/3)--(i-2/3+0.1,j+1/3), heavygreen); draw((i-1/3,j+2/3)--(i-1/3+0.1,j+2/3), heavygreen); draw(arc((i,j),1,90,180)); draw(arc((i-1,j+1),1,270,360)); } } draw((0,3)--(0,1), red+linewidth(1.5)); draw((0,3)--(0,1), red+linewidth(1.5)); draw(arc((1,1),1,90,180), red+linewidth(1.5)); draw((1,2)--(1,1)--(2,1), red+linewidth(1.5)); draw(arc((2,2),1,270,360), red+linewidth(1.5)); draw(arc((4,2),1,90,180), red+linewidth(1.5)); draw((4,3)--(4,0), red+linewidth(1.5)); dot((0,3)); dot((4,0)); label("$A$", (0,3), NW); label("$B$", (4,0), SE); [/asy]

Can we find a convex quadrilateral $ABCD$ with an interior point $P$ satisfying \[AB=AP, \quad BC=BP, \quad CD=CP, \quad \text{and} \quad DA=DP \quad ?\]

If $x > 0$ and $x^2 +\frac{1}{x^2}= 14$, find $x^5 +\frac{1}{x^5}$.

If we divide number $19250$ with one number, we get remainder $11$. If we divide number $20302$ with the same number, we get the reamainder $3$. Which number is that?

For a sale, a store owner reduces the price of a $10$ dollar scarf by $20\%$. Later the price is lowered again, this time by one-half the reduced price. The price is now $ \text{(A)}\ 2.00\text{ dollars}\qquad\text{(B)}\ 3.75\text{ dollars}\qquad\text{(C)}\ 4.00\text{ dollars}\qquad\text{(D)}\ 4.90\text{ dollars}\qquad\text{(E)}\ 6.40\text{ dollars} $