Color the positive integers by four colors $c_1,c_2,c_3,c_4$. (1)Prove that there exists a positive integer $n$ and $i,j\in\{1,2,3,4\}$,such that among all the positive divisors of $n$, the number of divisors with color $c_i$ is at least greater than the number of divisors with color $c_j$ by $3$. (2)Prove that for any positive integer $A$,there exists a positive integer $n$ and $i,j\in\{1,2,3,4\}$,such that among all the positive divisors of $n$, the number of divisors with color $c_i$ is at least greater than the number of divisors with color $c_j$ by $A$.

In a basketball tournament any two of the $n$ teams $S_1,S_2,...,S_n$ play one match (no draws). Denote by $v_i$ and $p_i$ the number of victories and defeats of team $S_i$ ($i = 1,2,...,n$), respectively. Prove that $v^2_1 +v^2_2 +...+v^2_n = p^2_1 +p^2_2 +...+p^2_n$

Square $ BDEC$ with center $ F$ is constructed to the out of triangle $ ABC$ such that $ \angle A \equal{} 90{}^\circ$, $ \left|AB\right| \equal{} \sqrt {12}$, $ \left|AC\right| \equal{} 2$. If $ \left[AF\right]\bigcap \left[BC\right] \equal{} \left\{G\right\}$ , then $ \left|BG\right|$ will be $\textbf{(A)}\ 6 \minus{} 2\sqrt {3} \qquad\textbf{(B)}\ 2\sqrt {3} \minus{} 1 \qquad\textbf{(C)}\ 2 \plus{} \sqrt {3} \\ \qquad\textbf{(D)}\ 4 \minus{} \sqrt {3} \qquad\textbf{(E)}\ 5 \minus{} 2\sqrt {2}$

To get to the Stromboli Volcano from the observatory, one has to take a road and a passway, each taking $4$ hours. There are two craters on the top. The first crater erupts for $1$ hour, stays silent for $17$ hours, then repeats the cycle. The second crater erupts for $1$ hour, stays silent for $9$ hours, erupts for $1$ hour, stays silent for $17$ hours, and then repeats the cycle. During the eruption of the first crater, it is dangerous to take both the passway and the road, but the second crater is smaller, so it is still safe to take the road. At noon, scout Vanya saw both craters erupting simultaneously. Will it ever be possible for him to mount the top of the volcano without risking his life?

Let $ABC$ be a triangle with incenter $I$, and let $D$ be a point on line $BC$ satisfying $\angle AID=90^{\circ}$. Let the excircle of triangle $ABC$ opposite the vertex $A$ be tangent to $\overline{BC}$ at $A_1$. Define points $B_1$ on $\overline{CA}$ and $C_1$ on $\overline{AB}$ analogously, using the excircles opposite $B$ and $C$, respectively. Prove that if quadrilateral $AB_1A_1C_1$ is cyclic, then $\overline{AD}$ is tangent to the circumcircle of $\triangle DB_1C_1$. [i]Ankan Bhattacharya[/i]

Let $ABC$ be an acute-angle triangle. Suppose that $M$ be the midpoint of $BC$ and $H$ be the orthocenter of $ABC$. Let $F\equiv BH\cap AC$ and $E\equiv CH\cap AB$. Suppose that $X$ be a point on $EF$ such that $\angle XMH=\angle HAM$ and $A,X$ are in the distinct side of $MH$. Prove that $AH$ bisects $MX$.

Evaluate $2010^2-2009\cdot2011.$

Circles $ C_1$ and $ C_2$ with centers at $ O_1$ and $ O_2$ respectively meet at points $ A$ and $ B$. The radii $ O_1B$ and $ O_2B$ meet $ C_1$ and $ C_2$ at $ F$ and$ E$. The line through $ B$ parallel to $ EF$ intersects $ C_1$ again at $ M$ and $ C_2$ again at $ N$. Prove that $ MN \equal{} AE \plus{} AF$.

For each positive integer $n$, put $$p_n =\left(1+\frac{1}{n}\right)^{n},\; P_n =\left(1+\frac{1}{n}\right)^{n+1}, \; h_n = \frac{2 p_n P_{n}}{ p_n + P_n }.$$ Prove that $h_1 < h_2 < h_3 <\ldots$

Let $ABCD$ be a rectangle with area $1$, and let $E$ lie on side $CD$. What is the area of the triangle formed by the centroids of triangles $ABE$, $BCE$, and $ADE$?

We say that a natural number $n$ is interesting if it can be written in the form \[ n = \left\lfloor \frac{1}{a} \right\rfloor + \left\lfloor \frac{1}{b} \right\rfloor + \left\lfloor \frac{1}{c} \right\rfloor, \] where $a,b,c$ are positive real numbers such that $a + b + c = 1.$ Determine all interesting numbers. ( $\lfloor x \rfloor$ denotes the greatest integer not greater than $x$.)

\(ABCD\) is a convex quadrilateral, \(AC\) bisects the angle \(\angle BAD\). Points \(E\) and \(F\) are on the sides \(BC\) and \(CD\) respectively such that \(EF \parallel BD\). Extend \(FA\) and \(EA\) to points \(P\) and \(Q\) respectively, such that the circle \(\omega_1\) passing through points \(A\), \(B\), \(P\) and the circle \(\omega_2\) passing through points \(A\), \(D\), \(Q\) are both tangent to line \(AC\). Prove that the points \(B\), \(P\), \(Q\), \(D\) are concyclic.

In a triangle $ ABC$, $ M$ is the midpoint of $ AB$ and $ P$ an arbitrary point on side $ AC$. Using only a straight edge, construct point $ Q$ on $ BC$ such that $ P$ and $ Q$ are at equal distance from $ CM$.

Prove that of any six four-digit numbers, mutual prime in total, you can always choose five numbers that are also relatively prime in total. [hide=original wording]Докажите, что из любых шести четырехзначных чисел, взаимно простых в совокупности, всегда можно выбратьпя ть чисел, также взаимно простых в совокупности.[/hide]

Determine a polynomial of non-negative real coefficients that satisfies the following two conditions: $$p(0) = 0, p(|z|) \le x^4 + y^4,$$ being $|z|$ the module of the complex number $z = x + iy$ .

If $x$ men working $x$ hours a day for $x$ days produce $x$ articles, then the number of articles (not necessarily an integer) produced by $y$ men working $y$ hours a day for $y$ days is: ${{ \textbf{(A)}\ \frac{x^3}{y^2} \qquad\textbf{(B)}\ \frac{y^3}{x^2} \qquad\textbf{(C)}\ \frac{x^2}{y^3} \qquad\textbf{(D)}\ \frac{y^2}{x^3} }\qquad\textbf{(E)}\ y} $

1. Let $ABC$ be an acute-angled triangle with $AB\neq AC$. The circle with diameter $BC$ intersects the sides $AB$ and $AC$ at $M$ and $N$ respectively. Denote by $O$ the midpoint of the side $BC$. The bisectors of the angles $\angle BAC$ and $\angle MON$ intersect at $R$. Prove that the circumcircles of the triangles $BMR$ and $CNR$ have a common point lying on the side $BC$.

If the radius of a circle is a rational number, its area is given by a number which is: $ \textbf{(A)}\ \text{rational} \qquad\textbf{(B)}\ \text{irrational} \qquad\textbf{(C)}\ \text{integral} \qquad\textbf{(D)}\ \text{a perfect square}$ $ \textbf{(E)}\ \text{none of these}$

In triangle $ABC$, $AB = 52$, $BC = 34$ and $CA = 50$. We split $BC$ into $n$ equal segments by placing $n-1$ new points. Among these points are the feet of the altitude, median and angle bisector from $A$. What is the smallest possible value of $n$?

Find all ordered pairs of integers $(x, y)$ with $y \ge 0$ such that $x^2 + 2xy + y! = 131$.

Solve the following equation ${{2}^{{{\sin }^{4}}x-{{\cos }^{2}}x}}-{{2}^{{{\cos }^{4}}x-{{\sin }^{2}}x}}=\cos 2x$

Consider a cyclic quadrilateral $ABCD$ in which $BC = CD$ and $AB < AD$. Let $E$ be a point on the side $AD$ and $F$ a point on the line $BC$ such that $AE = AB = AF$. Prove that $EF \parallel BD$.

Define $f : \mathbb{Z}_+ \to \mathbb{Z}_+$ such that $f(1) = 1$ and $f(n) $ is the greatest prime divisor of $n$ for $n > 1$. Aino and Väinö play a game, where each player has a pile of stones. On each turn the player to turn with $m$ stones in his pile may remove at most $f(m)$ stones from the opponent's pile, but must remove at least one stone. (The own pile stays unchanged.) The first player to clear the opponent's pile wins the game. Prove that there exists a positive integer $n$ such that Aino loses, when both players play optimally, Aino starts, and initially both players have $n$ stones.

Let $ t(n)$ for $ n \equal{} 3, 4, 5, \ldots,$ represent the number of distinct, incongruent, integer-sided triangles whose perimeter is $ n;$ e.g., $ t(3) \equal{} 1.$ Prove that \[ t(2n\minus{}1) \minus{} t(2n) \equal{} \left[ \frac{6}{n} \right] \text{ or } \left[ \frac{6}{n} \plus{} 1 \right].\]

Given any set $S$ of positive integers, show that at least one of the following two assertions holds: (1) There exist distinct finite subsets $F$ and $G$ of $S$ such that $\sum_{x\in F}1/x=\sum_{x\in G}1/x$; (2) There exists a positive rational number $r<1$ such that $\sum_{x\in F}1/x\neq r$ for all finite subsets $F$ of $S$.