Find the number of triples of nonnegative integers $(a,b,c)$ satisfying $a + b + c = 300$ and \[ a^2b + a^2c + b^2a + b^2c + c^2a + c^2b = 6{,}000{,}000.\]

Show that all complex roots of the polynomial $P(z)=a_0z^n+a_1z^{n-1}+\ldots+a_{n-1}z+a_n$, where $0<a_0<\ldots<a_n$, satisfy $|z|>1$.

From the vertices of a regular 27-gon, seven are chosen arbitrarily. Prove that among these seven points there are three points that form an isosceles triangle or four points that form an isosceles trapezoid.

[b](1)[/b] Find the number of positive integer solutions $(m,n,r)$ of the indeterminate equation $mn+nr+mr=2(m+n+r)$. [b](2)[/b] Given an integer $k (k>1)$, prove that indeterminate equation $mn+nr+mr=k(m+n+r)$ has at least $3k+1$ positive integer solutions $(m,n,r)$.

A square is dissected into $n$ congruent non-convex polygons whose sides are parallel to the sides of the square, and no two of these polygons are parallel translates of each other. What is the maximum value of $n$? (4)

What are the possible areas of a hexagon with all angles equal and sides $1, 2, 3, 4, 5$, and $6$, in some order?

Alan is standing in the middle of a very long straight road. In addition, there is typically British fog. And he lost his bomb somewhere on the street. He doesn't know how far they are from him away or in which direction it is, and could not see it until it would be no more than $10$ meters away from him. Since he wants to be efficient, he only wants to search at most ten times the distance that the bomb was initially away from him. How he was able to to accomplish this? [hide=original wording]Alan steht mitten auf einer sehr langen geraden Straße. Zudem herrscht typisch britischer Nebel und er hat seine Bombe irgendwo auf der Straße verloren. Er weiß nicht, wie weit sie von ihm entfernt ist oder in welcher Richtung sie liegt, und k¨onnte sie auch erst sehen, wenn sie h¨ochstens 10 Meter von ihm entfernt w¨are. Da er effizient sein will, m¨ochte er maximal eine zehnmal so hohe Distanz auf der Suche zuru¨cklegen, wie die Bombe anfangs von ihm entfernt war. Wie k¨onnte er dies bewerkstelligen¿[/hide]

$n$ is a natural number. for every positive real numbers $x_{1},x_{2},...,x_{n+1}$ such that $x_{1}x_{2}...x_{n+1}=1$ prove that: $\sqrt[x_{1}]{n}+...+\sqrt[x_{n+1}]{n} \geq n^{\sqrt[n]{x_{1}}}+...+n^{\sqrt[n]{x_{n+1}}}$

Let $n$ be an integer greater than $1$.Find all pairs of integers $(s,t)$ such that equations: $x^n+sx=2007$ and $x^n+tx=2008$ have at least one common real root.

For what positive integers $n, k$ (with $k < n$) are the binomial coefficients $${n \choose k- 1} \,\,\, , \,\,\, {n \choose k} \,\,\, , \,\,\, {n \choose k + 1}$$ three successive terms of an arithmetic progression?

The number $10!$ $(10$ is written in base $10)$, when written in the base $12$ system, ends in exactly $k$ zeroes. The value of $k$ is $\textbf{(A) }1\qquad\textbf{(B) }2\qquad\textbf{(C) }3\qquad\textbf{(D) }4\qquad \textbf{(E) } 5$

Let $\triangle ABC$ have $AB=6$, $BC=7$, and $CA=8$, and denote by $\omega$ its circumcircle. Let $N$ be a point on $\omega$ such that $AN$ is a diameter of $\omega$. Furthermore, let the tangent to $\omega$ at $A$ intersect $BC$ at $T$, and let the second intersection point of $NT$ with $\omega$ be $X$. The length of $\overline{AX}$ can be written in the form $\tfrac m{\sqrt n}$ for positive integers $m$ and $n$, where $n$ is not divisible by the square of any prime. Find $100m+n$. [i]Proposed by David Altizio[/i]

In right-angle triangle $ABC$, $\angle C=90$°, Point $D$ is the midpoint of side $AB$. Points $M$ and $C$ lie on the same side of $AB$ such that $MB\bot AB$, line $MD$ intersects side $AC$ at $N$, line $MC$ intersects side $AB$ at $E$. Show that $\angle DBN=\angle BCE$.

Let $Z,W,\lambda$ be complex numbers, $|\lambda|\neq1$. Which statements are correct about the equation $\overline{Z}-\lambda Z=W$? I. $Z=\frac{\overline{\lambda}W+\overline{W}}{1-|\lambda|^2}$ is a solution to the equation. II. The equation has only one solution. III. The equation has two solutions. IV. The equation has infinitely many solutions. $\text{(A)}$ Only I and II. $\text{(B)}$ Only I and III. $\text{(C)}$ Only I and IV. $\text{(D)}$ None of $\text{(A)(B)(C)}$.

Determine whether the polynomial $P(x)=(x^2-2x+5)(x^2-4x+20)+1$ is irreducible over $\mathbb{Z}[X]$.

If $4^{4^4} =\sqrt[128]{2^{2^{2^n}}}$ , find $n$.

For real numbers $a_0,a_1,\cdots,a_n(a_0\neq a_1)$, we have$a_{i-1}+a_{i+1}=2a_i$ for $i=1,2,\cdots,n-1$. Prove that $P(x)=a_0\text{C}_n^0(1-x)^n+a_1\text{C}_n^1x(1-x)^{n-1}+\cdots+a_n\text{C}_n^nx^n$ is a linear polynomial.

Let $a$, $b$, $c$ be positive real numbers such that $a + b + c = 1$. Prove that $$\sqrt[3]{\left (\frac{1+a}{b+c}\right )^{\frac{1-a}{bc}}\left (\frac{1+b}{c+a}\right )^{\frac{1-b}{ca}}\left (\frac{1+c}{a+b}\right )^{\frac{1-c}{ab}}} \geq 64 $$

Find the union of all projections of a given line segment $AB$ to all lines passing through a given point $O$.

Let a, b, c be natural numbers , so that c> b> a> 0. Show that, among any 2c consecutive natural numbers, there are three distinct numbers x, y, z so abc divides xyz.

Let $D$ be an arbitrary point on the side $BC$ of triangle $ABC$. Points $E$ and $F$ are on $CA$ and $BA$ are such that $CD=CE$ and $BD=BF$. Lines $BE$ and $CF$ intersect at point $P$. Prove that when point $D$ varies along the line $BC$, $PD$ passes through a fixed point.

Let $n$ be a positive integer. For a finite set $M$ of positive integers and each $i \in \{0,1,..., n-1\}$, we denote $s_i$ the number of non-empty subsets of $M$ whose sum of elements gives remainder $i$ after division by $n$. We say that $M$ is "$n$-balanced" if $s_0 = s_1 =....= s_{n-1}$. Prove that for every odd number $n$ there exists a non-empty $n$-balanced subset of $\{0,1,..., n\}$. For example if $n = 5$ and $M = \{1,3,4\}$, we have $s_0 = s_1 = s_2 = 1, s_3 = s_4 = 2$ so $M$ is not $5$-balanced.(Czech Republic)

Let $ABC$ be an acute, non isosceles triangle with $O,H$ are circumcenter and orthocenter, respectively. Prove that the nine-point circles of $AHO,BHO,CHO$ has two common points.

A circle of length $10$ is inscribed in a convex polygon with perimeter $15$. What part of the area of this polygon is occupied by the resulting circle?

Find the sum of all real solutions for $x$ to the equation $(x^2 + 2x + 3)^{(x^2+2x+3)^{(x^2+2x+3)}} = 2012$.