Let $S=2+4+6+ \cdots +2N$, where $N$ is the smallest positive integer such that $S>1,000,000$. Then the sum of the digits of $N$ is: $\textbf{(A)}\ 27 \qquad\textbf{(B)}\ 12 \qquad\textbf{(C)}\ 6 \qquad\textbf{(D)}\ 2 \qquad\textbf{(E)}\ 1$

Suppose $ \,G\,$ is a connected graph with $ \,k\,$ edges. Prove that it is possible to label the edges $ 1,2,\ldots ,k\,$ in such a way that at each vertex which belongs to two or more edges, the greatest common divisor of the integers labeling those edges is equal to 1. [b]Note: Graph-Definition[/b]. A [b]graph[/b] consists of a set of points, called vertices, together with a set of edges joining certain pairs of distinct vertices. Each pair of vertices $ \,u,v\,$ belongs to at most one edge. The graph $ G$ is connected if for each pair of distinct vertices $ \,x,y\,$ there is some sequence of vertices $ \,x \equal{} v_{0},v_{1},v_{2},\cdots ,v_{m} \equal{} y\,$ such that each pair $ \,v_{i},v_{i \plus{} 1}\;(0\leq i < m)\,$ is joined by an edge of $ \,G$.

In the plane we have $16$ lines(not parallel and not concurrents), we have $120$ point(s) of intersections of this lines. Sebastian has to paint this $120$ points such that in each line all the painted points are with colour differents, find the minimum(quantity) of colour(s) that Sebastian needs to paint this points. If we have have $15$ lines(in this situation we have $105$ points), what's the minimum(quantity) of colour(s)?

An angle bisector $AD$ was drawn in triangle $ABC$. It turned out that the center of the inscribed circle of triangle $ABC$ coincides with the center of the inscribed circle of triangle $ABD$. Find the angles of the original triangle.

Let $ABC$ be a triangle such that $AB=9$, $BC=6$, and $AC=10$. $2$ points $D_1,D_2$ are labeled on $BC$ such that $BC$ is subdivided into $3$ equal segments; $4$ points $E_1,E_2,\dots,E_4$ are labeled on $AC$ such that $AC$ is subdivided into $5$ equal segments; and $8$ points $F_1,F_2,\dots,F_8$ are labeled on $AB$ such that $AB$ is subdivided into $9$ equal segments. All possible cevians are drawn from $A$ to each $D_i$; from $B$ to each $E_j$; and from $C$ to each $F_k$. At how many points in the interior of $\triangle ABC$ do at least $2$ cevians intersect?

Find all polynomials $P(x)$ with real coefficients that satisfy the relation $$1+P(x)=\frac{P(x-1)+P(x+1)}2.$$

On an exam with $80$ problems, Roger solved $68$ of them. What percentage of the problems did he solve?

Let $n\ge 2$ be an integer. Elwyn is given an $n\times n$ table filled with real numbers (each cell of the table contains exactly one number). We define a [i]rook set[/i] as a set of $n$ cells of the table situated in $n$ distinct rows as well as in n distinct columns. Assume that, for every rook set, the sum of $n$ numbers in the cells forming the set is nonnegative.\\ \\ By a move, Elwyn chooses a row, a column, and a real number $a,$ and then he adds $a$ to each number in the chosen row, and subtracts $a$ from each number in the chosen column (thus, the number at the intersection of the chosen row and column does not change). Prove that Elwyn can perform a sequence of moves so that all numbers in the table become nonnegative.

Let $a$ and $b$ be real numbers such that $ \frac {ab}{a^2 + b^2} = \frac {1}{4} $. Find all possible values of $ \frac {|a^2-b^2|}{a^2+b^2} $.

Let $ABCDE$ be a convex pentagon with $AB\parallel CE$, $BC\parallel AD$, $AC\parallel DE$, $\angle ABC=120^\circ$, $AB=3$, $BC=5$, and $DE=15$. Given that the ratio between the area of triangle $ABC$ and the area of triangle $EBD$ is $m/n$, where $m$ and $n$ are relatively prime positive integers, find $m+n$.

Let $A$ be a non empty subset of positive reals such that for every $a,b,c \in A$, the number $ab+bc+ca$ is rational. Prove that $\frac{a}{b}$ is a rational for every $a,b \in A$.

Tarik wants to choose some distinct numbers from the set $S = \{2,...,111\}$ in such a way that each of the chosen numbers cannot be written as the product of two other distinct chosen numbers. What is the maximum number of numbers Tarik can choose ?

Let $f(n)$ be the number of ordered pairs $(k, \ell)$ of positive integers such that $n = (2\ell-1)\cdot 2^k - k$, and let $g(n)$ be the number of ordered pairs $(k, \ell)$ of positive integers such that $n = \ell \cdot 2^{k+1}-k$. Compute $\sum_{i=1}^{\infty}\frac{f(i) - g(i)}{2^i}$. .

Pete has some trouble slicing a 20-inch (diameter) pizza. His first two cuts (from center to circumference of the pizza) make a 30º slice. He continues making cuts until he has gone around the whole pizza, each time trying to copy the angle of the previous slice but in fact adding 2º each time. That is, he makes adjacent slices of 30º, 32º, 34º, and so on. What is the area of the smallest slice?

Consider a binary matrix $M$(all entries are $0$ or $1$) on $r$ rows and $c$ columns, where every row and every column contain at least one entry equal to $1$. Prove that there exists an entry $M(i,j) = 1$, such that the corresponding row-sum $R(i)$ and column-sum $C(j)$ satisfy $r R(i)\ge c C(j)$. (Proposed by Gerhard Woeginger, Austria)

Determine the smallest positive real number $ k$ with the following property. Let $ ABCD$ be a convex quadrilateral, and let points $ A_1$, $ B_1$, $ C_1$, and $ D_1$ lie on sides $ AB$, $ BC$, $ CD$, and $ DA$, respectively. Consider the areas of triangles $ AA_1D_1$, $ BB_1A_1$, $ CC_1B_1$ and $ DD_1C_1$; let $ S$ be the sum of the two smallest ones, and let $ S_1$ be the area of quadrilateral $ A_1B_1C_1D_1$. Then we always have $ kS_1\ge S$. [i]Author: Zuming Feng and Oleg Golberg, USA[/i]

A function $f : \Bbb{N}_0 \to \Bbb{R}$ satisfies $f(1) = 3$ and \[f(m + n) + f(m - n) - m + n - 1 =\frac{f(2m) + f(2n)}{2},\] for any non-negative integers $m$ and $n$ with $m \geq n.$ Find all such functions $f$.

In a triangle $ABC, O$ is the center of the circumscribed circle. Line $a$ passes through the midpoint of the altitude of the triangle from the vertex $A$ and is parallel to $OA$. Similarly, the straight lines $b$ and $c$ are defined. Prove that these three lines intersect at one point.

Let $p$ be an odd prime number. Let $S$ be the sum of all primitive roots modulo $p$. Show that if $p-1$ isn't squarefree (i. e., if there exist integers $k$ and $m$ with $k>1$ and $p-1=k^2m$), then $S \equiv 0 \mod p$. If not, then what is $S$ congruent to $\mod p$ ?

We are given a family of discs in the plane, with pairwise disjoint interiors. Each disc is tangent to at least six other discs of the family. Show that the family is infinite.

Let $c$ be a positive real and $a_1, a_2, \dots$ be a sequence of nonnegative integers satisfying the following conditions for every positive integer $n$: [b](i)[/b]$\frac{2^{a_1}+2^{a_2}+\cdots+2^{a_n}}{n}$ is an integer; [b](ii)[/b]$\textbullet 2^{a_n}\leq cn$. Prove that the sequence $a_1, a_2, \dots$ is eventually constant.

What is the value of the gravitational potential energy of the two star system? (A) $-\frac{GM^2}{d}$ (B) $\frac{3GM^2}{d}$ (C) $-\frac{GM^2}{d^2}$ (D) $-\frac{3GM^2}{d}$ (E) $-\frac{3GM^2}{d^2}$

Tsrutsuna starts in the bottom left cell of a 7 × 7 square table, while Tsuna is in the upper right cell. The center cell of the table contains cheese. Tsrutsuna wants to reach Tsuna and bring a piece of cheese with him. From a cell Tsrutsuna can only move to the right or the top neighboring cell. Determine the number of different paths Tsrutsuna can take from the lower left cell to the upper right cell, such that he passes through the center cell. [i]Proposed by Giorgi Arabidze, Georgia[/i]

The sum of three numbers is $ 98$. The ratio of the first to the second is $ \frac {2}{3}$, and the ratio of the second to the third is $ \frac {5}{8}$. The second number is: $ \textbf{(A)}\ 15 \qquad\textbf{(B)}\ 20 \qquad\textbf{(C)}\ 30 \qquad\textbf{(D)}\ 32 \qquad\textbf{(E)}\ 33$

How many ways are there (in terms of power) to represent the number 1 as a finite number or an infinite sum of some subset of the set: {$\phi^{-n} | n \in Z^+$} $\phi=\frac{1+\sqrt5}{2}$