Consider all pairs of positive integers $(a,b)$, with $a<b$, such that $\sqrt{a} +\sqrt{b} = \sqrt{2,160}$ Determine all possible values of $a$.

Prove for $ 0 < k \leq 1$ and $ a_i \in \mathbb{R}^\plus{},$ $ i \equal{} 1,2 \ldots, n$ the following inequality holds: \[ \left( \frac{a_1}{a_2 \plus{} \ldots \plus{} a_n} \right)^k \plus{} \ldots \plus{} \left( \frac{a_n}{a_1 \plus{} \ldots \plus{} a_{n\minus{}1}} \right)^k \geq \frac{n}{(n\minus{}1)^k}.\]

Let $n,k$ be positive integers such that $n$ is not divisible by $3$ and $k\ge n$. Prove that there is an integer $m$ divisible by $n$ whose sum of digits in base $10$ equals $k$.

The following is known about the quadrilateral $ABCD$: triangles $ABC$ and $CDA$ are equal in area, the area of triangle $BCD$ is $k$ times greater than the area of triangle $DAB$, the bisectors of angles $ABC$ and $CDA$ intersect on the diagonal $AC$, straight lines $AC$ and $BD$ are not perpendicular. Find the ratio $AC/BD$.

Find all complex roots (with multiplicities) of the polynomial $$p(x)=\sum_{n=1}^{2008}(1004-|1004-n|)x^n.$$

Prove that if $p$ and $q$ are two prime numbers, such that $$p+p^2+p^3+...+p^q=q+q^2+q^3+...+q^p,$$ then $p=q$.

Let $\triangle A_1A_2A_3$ be an equilateral triangle with unit side length. For $k = 1$, $2$, and $3$, let $B_k$ be the point on the boundary of $\triangle A_1A_2A_3$ located $1/3$ unit away from $A_k$ in the clockwise direction and let $C_k$ be the point on the boundary of $\triangle A_1A_2A_3$ located $1/3$ unit away from $A_k$ in the counterclockwise direction. What fraction of the area of $\triangle A_1A_2A_3$ is the area of the intersection of $\triangle B_1B_2B_3$ and $\triangle C_1C_2C_3$?

Consider $\triangle \natural\flat\sharp$. Let $\flat\sharp$, $\sharp\natural$ and $\natural\flat$ be the answers to problems $4$, $5$, and $6$, respectively. If the incircle of $\triangle \natural\flat\sharp$ touches $\natural\flat$ at $\odot$, find $\flat\odot$. [i]Proposed by Evan Chen[/i]

Given triangle $ABC$. Points $A_1,B_1$ and $C_1$ are symmetric to its vertices with respect to opposite sides. $C_2$ is the intersection point of lines $AB_1$ and $BA_1$. Points$ A_2$ and $B_2$ are defined similarly. Prove that the lines $A_1 A_2, B_1 B_2$ and $C_1 C_2$ are parallel. (A. Zaslavsky)

Segment $AB$ of length $13$ is the diameter of a semicircle. Points $C$ and $D$ are located on the semicircle but not on segment $AB$. Segments $AC$ and $BD$ both have length $5$. Given that the length of $CD$ can be expressed as $\frac{a}{b}$ where $a$ and $b$ are relatively prime positive integers, find $a +b$.

A segment of length $1$ is drawn such that its endpoints lie on a unit circle, dividing the circle into two parts. Compute the area of the larger region.

If $a\pm bi~(b\neq 0)$ are imaginary roots of the equation $x^3+qx+r=0$ where $a,~b,~q,$ and $r$ are real numbers, then $q$ in terms of $a$ and $b$ is $\textbf{(A) }a^2+b^2\qquad\textbf{(B) }2a^2-b^2\qquad\textbf{(C) }b^2-a^2\qquad\textbf{(D) }b^2-2a^2\qquad \textbf{(E) }b^2-3a^2$

If $a,b,c,d$ be any four positive real numbers, then prove that \[ \frac{a}{b} + \frac{b}{c} + \frac{c}{d} + \frac{d}{a} \geq 4. \]

Four semicircles of radius $1$ are placed in a square, as shown below. The diameters of these semicircles lie on the sides of the square and each semicircle touches a vertex of the square. Find the absolute difference between the shaded area and the "hatched" area. [asy] import patterns; add("hatch",hatch(1.2mm)); add("checker",checker(2mm)); real r = 1 + sqrt(3); filldraw((0,0)--(r,0)--(r,r)--(0,r)--cycle,gray(0.4),linewidth(1.5)); fill((1,0)--(r,1)--(r-1,r)--(0,r-1)--cycle,white); fill((1,0)--(r,1)--(r-1,r)--(0,r-1)--cycle,pattern("hatch")); filldraw(arc((1,0),1,0,180)--(0,0)--cycle,white,linewidth(1.5)); filldraw(arc((r,1),1,90,270)--(r,0)--cycle,white,linewidth(1.5)); filldraw(arc((r-1,r),1,180,360)--(r,r)--cycle,white,linewidth(1.5)); filldraw(arc((0,r-1),1,270,450)--(0,r)--cycle,white,linewidth(1.5)); [/asy] [i]Proposed by Connor Gordon[/i]

Point $D$ is chosen on the side $AC$ of an acute-angled triangle $ABC$. The median $AM$ intersects the altitude $CH$ and the segment $BD$ at points $N$ and $K$ respectively. Prove that if $AK = BK$, then $AN = 2KM$.

For how many ordered triplets of three positive integers is it true that their product is four more than twice their sum?

Laura and Daniel play with quadratic polynomials. First Laura says a nonzero real number $r$. Then Daniel says a nonzero real number $s$, and then again Laura says another nonzero real number $t$. Finally. Daniel writes the polynomial $P(x) = ax^2 + bx + c$ where $a,b$, and $c$ are $r,s$, and $t$ in some order Daniel chooses. Laura wins if the equation $P(x) = 0$ has two different real solutions, and Daniel wins otherwise. Determine who has a winning strategy and describe that strategy.