There is a graph with 30 vertices. If any of 26 of its vertices with their outgoiing edges are deleted, then the remained graph is a connected graph with 4 vertices. What is the smallest number of the edges in the initial graph with 30 vertices?

Prove that $a^9 - a$ is divisible by $6$ for all integers $a$.

Determine all functions $f: \mathbb{R} \to \mathbb{R}$, such that $$f(xf(y)+1)=y+f(f(x)f(y))$$ for all $x, y \in \mathbb{R}$. (Theresia Eisenkölbl)

Let $s_1, s_2, s_3, \dots$ be an infinite, nonconstant sequence of rational numbers, meaning it is not the case that $s_1 = s_2 = s_3 = \dots.$ Suppose that $t_1, t_2, t_3, \dots$ is also an infinite, nonconstant sequence of rational numbers with the property that $(s_i - s_j)(t_i - t_j)$ is an integer for all $i$ and $j$. Prove that there exists a rational number $r$ such that $(s_i - s_j)r$ and $(t_i - t_j)/r$ are integers for all $i$ and $j$.

Determine the smallest natural number $n$ having the following property: For every integer $p, p \geq n$, it is possible to subdivide (partition) a given square into $p$ squares (not necessarily equal).

In cube $ABCD-A_1B_1C_1D_1$, draw a plane $\alpha$ perpendicular to line $AC'$, and $\alpha$ has intersections with any surface of the cube. The area of the cross section is $S$, the perimeter of the cross section is $l$, then $\text{(A)}$ The value of $S$ is fixed, but the value of $l$ is not fixed. $\text{(B)}$ The value of $S$ is not fixed, but the value of $l$ is fixed. $\text{(C)}$ The value of $S$ is fixed, the value of $l$ is fixed as well. $\text{(D)}$ The value of $S$ is not fixed, the value of $l$ is not fixed either.

Let $A, B,$ and $C$ denote the angles of a triangle. If $\sin^2 A + \sin^2 B + \sin^2 C = 2$, prove that the triangle is right-angled.

All divisors of a positive integer $n$ are listed in the ascending order: $1=d_1<d_2< \ldots < d_k=n$. It turned out that the amount of pairs $(d_i,d_{i+1})$ of adjacent divisors such that $d_{i+1}$ is a multiple of $d_i$, is odd. Prove that $n=pm^2$, where $p$ is the smallest prime divisor of $n$, and $m$ is a positive integer.

Find all integers $a,b,c,x,y,z$ such that \[a+b+c=xyz, \; x+y+z=abc, \; a \ge b \ge c \ge 1, \; x \ge y \ge z \ge 1.\]

Let $p_1,p_2,...,p_n$ be $n\ge 2$ fixed positive real numbers. Let $x_1,x_2,...,x_n$ be nonnegative real numbers such that $$x_1p_1+x_2p_2+...+x_np_n=1.$$ Determine the [i](a)[/i] maximal; [i](b)[/i] minimal possible value of $x_1^2+x_2^2+...+x_n^2$.

Let $n$ be a positive integer, and let $a>0$ be a real number. Consider the equation: \[ \sum_{i=1}^{n}(x_i^2+(a-x_i)^2)= na^2 \] How many solutions ($x_1, x_2 \cdots , x_n$) does this equation have, such that: \[ 0 \leq x_i \leq a, i \in N^+ \]

If $x$ and $y$ are acute angles such that $x+y=\pi/4$ and $\tan y=1/6$, find the value of $\tan x$. $\textbf{(A) }\dfrac{27\sqrt2-18}{71}\hspace{11.5em}\textbf{(B) }\dfrac{35\sqrt2-6}{71}\hspace{11.2em}\textbf{(C) }\dfrac{35\sqrt3+12}{33}$ $\textbf{(D) }\dfrac{37\sqrt3+24}{33}\hspace{11.5em}\textbf{(E) }1\hspace{15em}\textbf{(F) }\dfrac57$ $\textbf{(G) }\dfrac37\hspace{15.4em}\textbf{(H) }6\hspace{15em}\textbf{(I) }\dfrac16$ $\textbf{(J) }\dfrac12\hspace{15.7em}\textbf{(K) }\dfrac67\hspace{14.8em}\textbf{(L) }\dfrac47$ $\textbf{(M) }\sqrt3\hspace{14.5em}\textbf{(N) }\dfrac{\sqrt3}3\hspace{14em}\textbf{(O) }\dfrac56$ $\textbf{(P) }\dfrac23\hspace{15.4em}\textbf{(Q) }\dfrac1{2007}$

In a certain regular polygon, the measure of each interior angle is twice the measure of each exterior angle. How many sides does this regular polygon have?

(V.Protasov, 9--10) The Euler line of a non-isosceles triangle is parallel to the bisector of one of its angles. Determine this angle (There was an error in published condition of this problem).

For each positive integer $n$, a new number $t_n$ is formed from the numbers $2^n$ and $5^n$ which consists of the digits from $2^n$ followed by the digits from $5^n$. For example, $t_4$ is $16625$. How many digits does the number $t_{2008}$ have?

The plan of a picture gallery is a chequered figure where each square is a room, and every room can be reached from each other by moving to adjacent rooms. A custodian in a room can watch all the rooms that can be reached from this room by one move of a chess queen (without leaving the gallery). What minimum number of custodians is sufficient to watch all the rooms in every gallery of $n$ rooms ($n > 2$)?

In the diagram below $ \angle CAB, \angle CBD$, and $\angle CDE$ are all right angles with side lengths $AC = 3$, $BC = 5$, $BD = 12$, and $DE = 84$. The distance from point $E$ to the line $AB$ can be expressed as the ratio of two relatively prime positive integers, $m$ and $n$. Find $m + n$. [asy] size(300); defaultpen(linewidth(0.8)); draw(origin--(3,0)--(0,4)--cycle^^(0,4)--(6,8)--(3,0)--(30,-4)--(6,8)); label("$A$",origin,SW); label("$B$",(0,4),dir(160)); label("$C$",(3,0),S); label("$D$",(6,8),dir(80)); label("$E$",(30,-4),E);[/asy]

Prove that there exists $m \in \mathbb{N}$ such that there exists an integral sequence $\lbrace a_n \rbrace$ which satisfies: [b]I.[/b] $a_0 = 1, a_1 = 337$; [b]II.[/b] $(a_{n + 1} a_{n - 1} - a_n^2) + \frac{3}{4}(a_{n + 1} + a_{n - 1} - 2a_n) = m, \forall$ $n \geq 1$; [b]III. [/b]$\frac{1}{6}(a_n + 1)(2a_n + 1)$ is a perfect square $\forall$ $n \geq 1$.

Find the largest positive integer $n$ ($n \ge 3$), so that there is a convex $n$-gon, the tangent of each interior angle is an integer.

Find all triples of positive integers $(a,b,c)$ such that the following equations are both true: I- $a^2+b^2=c^2$ II- $a^3+b^3+1=(c-1)^3$

Prove that the infinite series $ 1\minus{}\frac{1}{x(x\plus{}1)}\minus{}\frac{x\minus{}1}{2!x^2(2x\plus{}1)}\minus{}\frac{(x\minus{}1)(2x\minus{}1)}{3!(x^3(3x\plus{}1))}\minus{}\frac{(x\minus{}1)(2x\minus{}1)(3x\minus{}1)}{4!x^4(4x\plus{}1)}\minus{}\cdots$ is convergent for every positive $ x$. Denoting its sum by $ F(x)$, find $ \lim_{x\to \plus{}0}F(x)$ and $ \lim_{x\to \infty}F(x)$.

The circle $S_1$ intersects the hyperbola $y=\frac1x$ at four points $A$, $B$, $C$, and $D$, and the other circle $S_2$ intersects the same hyperbola at four points $A$, $B$, $F$, and $G$. It's known that the radii of circles $S_1$ and $S_2$ are equal. Prove that the points $C$, $D$, $F$, and $G$ are the vertices of the parallelogram.

Consider a $100 \times 100$ table, and identify the cell in row $a$ and column $b$, $1 \leq a, b \leq 100$, with the ordered pair $(a, b)$. Let $k$ be an integer such that $51 \leq k \leq 99$. A $k$-knight is a piece that moves one cell vertically or horizontally and $k$ cells to the other direction; that is, it moves from $(a, b)$ to $(c, d)$ such that $(|a-c|, |b - d|)$ is either $(1, k)$ or $(k, 1)$. The $k$-knight starts at cell $(1, 1)$, and performs several moves. A sequence of moves is a sequence of cells $(x_0, y_0)= (1, 1)$, $(x_1, y_1), (x_2, y_2)$, $\ldots, (x_n, y_n)$ such that, for all $i = 1, 2, \ldots, n$, $1 \leq x_i , y_i \leq 100$ and the $k$-knight can move from $(x_{i-1}, y_{i-1})$ to $(x_i, y_i)$. In this case, each cell $(x_i, y_i)$ is said to be reachable. For each $k$, find $L(k)$, the number of reachable cells.

The sum of the squares of the first $ n$ positive integers is given by the expression $ \frac{n(n \plus{} c)(2n \plus{} k)}{6}$, if $ c$ and $ k$ are, respectively: $ \textbf{(A)}\ {1}\text{ and }{2} \qquad \textbf{(B)}\ {3}\text{ and }{5}\qquad \textbf{(C)}\ {2}\text{ and }{2}\qquad \textbf{(D)}\ {1}\text{ and }{1}\qquad \textbf{(E)}\ {2}\text{ and }{1}$

Let $S = \{10^n + 1000: n = 0, 1, \dots\}$. Compute the largest positive integer not expressible as the sum of (not necessarily distinct) elements of $S$. [i]Proposed by Ankan Bhattacharya[/i]