If $3x^3-9x^2+kx-12$ is divisible by $x-3$, then it is also divisible by: ${{ \textbf{(A)}\ 3x^2-x+4 \qquad\textbf{(B)}\ 3x^2-4 \qquad\textbf{(C)}\ 3x^2+4 \qquad\textbf{(D)}\ 3x-4 }\qquad\textbf{(E)}\ 3x+4 } $

The diagonals of a convex quadrilateral $ABCD$ intersect in the point $E$. Let $U$ be the circumcenter of the triangle $ABE$ and $H$ be its orthocenter. Similarly, let $V$ be the circumcenter of the triangle $CDE$ and $K$ be its orthocenter. Prove that $E$ lies on the line $UK$ if and only if it lies on the line $VH$.

Find $\alpha$ such that $$\lim_{x\to0^+}x^\alpha I(x)=a\enspace\text{given}\enspace I(x)=\int^\infty_0\sqrt{1+t}\cdot e^{-xt}dt$$ where $a$ is a nonzero real number.

Consider a square-free even integer $n$ and a prime $p$, such that 1) $(n,p)=1$; 2) $p\le 2\sqrt{n}$; 3) There exists an integer $k$ such that $p|n+k^2$. Prove that there exists pairwise distinct positive integers $a,b,c$ such that $n=ab+bc+ca$. [i]Proposed by Hongbing Yu[/i]

$M$ is midpoint of $BC$.$P$ is an arbitary point on $BC$. $C_{1}$ is tangent to big circle.Suppose radius of $C_{1}$ is $r_{1}$ Radius of $C_{4}$ is equal to radius of $C_{1}$ and $C_{4}$ is tangent to $BC$ at P. $C_{2}$ and $C_{3}$ are tangent to big circle and line $BC$ and circle $C_{4}$. [img]http://aycu01.webshots.com/image/4120/2005120338156776027_rs.jpg[/img] Prove : \[r_{1}+r_{2}+r_{3}=R\] ($R$ radius of big circle)

Suppose $f:\mathbb{R} \to \mathbb{R}$ is a function such that \[|f(x+y)-f(x)-f(y)|\le 1\ \ \ \text{for all} \ \ x, y \in\mathbb R.\] Prove that there is a function $g:\mathbb{R}\to\mathbb{R}$ such that $|f(x)-g(x)|\le 1$ and $g(x+y)=g(x)+g(y)$ for all $x,y \in\mathbb R.$

What is the perimeter of the boundary of the region consisting of all points which can be expressed as $(2u-3w,v+4w)$ with $0 \le u \le 1$, $0 \le v \le 1$, and $0 \le w \le 1$? \\ \\ $\textbf{(A) } 10\sqrt{3} \qquad \textbf{(B) } 10 \qquad \textbf{(C) } 12 \qquad \textbf{(D) } 18 \qquad \textbf{(E) } 16$

A square is divided by horizontal and vertical lines that form $n^2$ squares each with side $1$. What is the greatest possible value of $n$ such that it is possible to select $n$ squares such that any rectangle with area $n$ formed by the horizontal and vertical lines would contain at least one of the selected $n$ squares.

$AD$ is the angle bisector of the right triangle $ABC$ with $\angle ABC = 60^o$ and $\angle BCA = 90^o$. $E$ is chosen on $\overline{AB}$ so that the line parallel to $\overline{DE}$ through $C$ bisects $\overline{AE}$. Find $\angle EDB$ in degrees.

Let $SABC$ is pyramid, such that $SA\le 4$, $SB\ge 7$, $SC\ge 9$, $AB=5$, $BC\le 6$ and $AC\le 8$. Find max value capacity(volume) of the pyramid $SABC$.

Let $n$ be a positive integer and $x_1,x_2,\ldots,x_n$ be positive reals. Show that there are numbers $a_1,a_2,\ldots, a_n \in \{-1,1\}$ such that the following holds: \[a_1x_1^2+a_2x_2^2+\cdots+a_nx_n^2 \ge (a_1x_1+a_2x_2 +\cdots+a_nx_n)^2\]

Find the constant $k$ such that the sum of all $x \ge 0$ satisfying $\sqrt{x}(x+12)=17x-k$ is $256.$ [i]Proposed by Michael Tang[/i]

The set of all points $(x,y)$ in the plane satisfying $x<y$ and $x^3-y^3>x^2-y^2$ has area $A$. What is the value of $A$? [i]Proposed by Adam Bertelli[/i]

Let $A$, $B$ and $C$ be $n \times n$ matrices with complex entries satisfying $$A^2=B^2=C^2 \text{ and } B^3 = ABC + 2I.$$ Prove that $A^6=I$.

Define a positive integer $ n$ to be a factorial tail if there is some positive integer $ m$ such that the decimal representation of $ m!$ ends with exactly $ n$ zeroes. How many positive integers less than $ 1992$ are not factorial tails?

Anna and Orjan play the following game: they start with a positive integer $n>1$, Anna writes it as the sum of two other positive integers, $n = n_1+n_2$. Orjan deletes one of them, $n_1$ or $n_2$. If the remaining number is larger than $1$, the process is repeated, i.e. Anna writes it as the sum of two positive integers, $ n_3+n_4$, Orjan deletes one of them etc. The game ends when the last number is $1$. Orjan is the winner if there are two equal numbers among the numbers he has deleted, otherwise Anna wins. Who is winning the game if n = 2008 and they both play optimally?

The two pairs of consecutive natural numbers $(8, 9)$ and $(288, 289)$ have the following property: in each pair, each number contains each of its prime factors to a power not less than $2$. Prove that there are infinitely many such pairs. (A Andjans, Riga)

Rectangle $ABCD$ is folded in half so that the vertices $D$ and $B$ coincide, creating the crease $\overline{EF}$, with $E$ on $\overline{AD}$ and $F$ on $\overline{BC}$. Let $O$ be the midpoint of $\overline{EF}$. If triangles $DOC$ and $DCF$ are congruent, what is the ratio $BC : CD$?

Let $ f:[0,1]\longrightarrow [0,1] $ a function with the property that, for all $ y\in [0,1] $ and $ \varepsilon >0, $ there exists a $ x\in [0,1] $ such that $ |f(x)-y|<\varepsilon . $ [b]a)[/b] Prove that if $ \left. f\right|_{[0,1]} $ is continuos, then $ f $ is surjective. [b]b)[/b] Give an example of a function with the given property, but which isn´t surjective.

Let $f(x) =\frac{x+a}{x+b}$ satisfy $f(f(f(x))) = x$ for real numbers $a, b$. If the maximum value of a is $p/q$, where $p, q$ are relatively prime integers, what is $|p| + |q|$?

For real numer $ c$ for which $ cx^2\geq \ln (1\plus{}x^2)$ for all real numbers $ x$, find the value of $ c$ such that the area of the figure bounded by two curves $ y\equal{}cx^2$ and $ y\equal{}\ln (1\plus{}x^2)$ and two lines $ x\equal{}1,\ x\equal{}\minus{}1$ is 4.

The sum of $n$ consecutive integers is divisible by $n$ for some $n > 1$. For which $n$ is this always true? $\textbf{(A) }\text{even }n\qquad\textbf{(B) }\text{odd }n\text{ divisible by }3\qquad\textbf{(C) }\text{odd }n\qquad\textbf{(D) }\text{prime }n\qquad\textbf{(E) }\text{no such }n\text{ exists}$

Let $P$ a convex regular polygon with $n$ sides, having the center $O$ and $\angle xOy$ an angle of measure $a$, $a \in (0,k)$. Let $S$ be the area of the common part of the interiors of the polygon and the angle. Find, as a function of $n$, the values of $a$ such that $S$ remains constant when $\angle xOy$ is rotating around $O$.

Real numbers $a, b, c$ satisfy $(2b - a)^2 + (2b - c)^2 = 2(2b^2 - ac)$. Prove that $a + c = 2b$.

There are $n$ line segments on the plane, no three intersecting at a point, and each pair intersecting once in their respective interiors. Tony and his $2n - 1$ friends each stand at a distinct endpoint of a line segment. Tony wishes to send Christmas presents to each of his friends as follows: First, he chooses an endpoint of each segment as a “sink”. Then he places the present at the endpoint of the segment he is at. The present moves as follows : $\bullet$ If it is on a line segment, it moves towards the sink. $\bullet$ When it reaches an intersection of two segments, it changes the line segment it travels on and starts moving towards the new sink. If the present reaches an endpoint, the friend on that endpoint can receive their present. Prove that Tony can send presents to exactly $n$ of his $2n - 1$ friends.