In a country every two towns are connected by exactly one one-way road. Each road is intended either for cars or for cyclists. The roads cross only in towns, otherwise interchanges are used as road junctions. Show that there is a town from which you can go to any other town without changing the means of transport.

A candle and a man are placed in a dihedral mirror angle. How many reflections can the man see ?

A [i]perfect number[/i] is an integer that equals half the sum of its positive divisors. For example, because $2 \cdot 28 = 1 + 2 + 4 + 7 + 14 + 28$, $28$ is a perfect number. [list] [*] [b](a)[/b] A [i]square-free[/i] integer is an integer not divisible by a square of any prime number. Find all square-free integers that are perfect numbers. [*] [b](b)[/b] Prove that no perfect square is a perfect number.[/list]

Let $ABC$ be a triangle and $D$ is a point inside of the triangle, such that $\angle DBC=60^{\circ}$ and $\angle DCB=\angle DAB=30^{\circ}$. Let $M$ and $N$ be the midpoints of $AC$ and $BC$, respectively. Prove that $\angle DMN=90^{\circ}$.

Let $ABC$ be a scalene acute triangle with incentre $I{}$ and circumcentre $O{}$. Let $AI$ cross $BC$ at $D$. On circle $ABC$, let $X$ and $Y$ be the mid-arc points of $ABC$ and $BCA$, respectively. Let $DX{}$ cross $CI{}$ at $E$ and let $DY{}$ cross $BI{}$ at $F{}$. Prove that the lines $FX, EY$ and $IO$ are concurrent on the external bisector of $\angle BAC$. [i]David-Andrei Anghel[/i]

When the celebrated German mathematician Karl Gauss (1777-1855) was nine years old, he was asked to add all the integers from 1 through 100. He quickly added 1 and 100, 2 and 99, and so on for 50 pairs of numbers each adding in 101. His answer was 50 · 101=5,050. Now find the sum of all the digits in the integers from 1 through 1,000,000 (i.e. all the digits in those numbers, not the numbers themselves).

Determine all pairs $(m,n)$ of positive integers for which $2^{m} + 3^{n}$ is a perfect square.

The three points $A, B, C$ form a triangle. $AB=4, BC=5, AC=6$. Let the angle bisector of $\angle A$ intersect side $BC$ at $D$. Let the foot of the perpendicular from $B$ to the angle bisector of $\angle A$ be $E$. Let the line through $E$ parallel to $AC$ meet $BC$ at $F$. Compute $DF$.

Find all integers $\,a,b,c\,$ with $\,1<a<b<c\,$ such that \[ (a-1)(b-1)(c-1) \] is a divisor of $abc-1.$

Let $a,b,c,d$ are integers such that $(a,b)=(c,d)=1$ and $ad-bc=k>0$. Prove that there are exactly $k$ pairs $(x_{1},x_{2})$ of rational numbers with $0\leq x_{1},x_{2}<1$ for which both $ax_{1}+bx_{2},cx_{1}+dx_{2}$ are integers.

Solve in real numbers the system $$\begin{cases} x^3 + y = 3x + 4 \\ 2y^3 + z = 6y + 6 \\ 3z^3 + x = 9z + 8\end{cases}$$

Find all the positive primes $p$ for which there exist integers $m,n$ satisfying : $p=m^2+n^2$ and $m^3+n^3-4$ is divisible by $p$.

The region consisting of all points in three-dimensional space within $3$ units of line segment $\overline{AB}$ has volume $216\pi$. What is the length $AB$? $\textbf{(A)}\ 6\qquad\textbf{(B)}\ 12\qquad\textbf{(C)}\ 18\qquad\textbf{(D)}\ 20\qquad\textbf{(E)}\ 24$

Find all functions $f : R \to R$ such that $f(x^2)-f(y^2) \le (f(x)+y) (x-f(y))$ for all $x, y \in R$.

Consider the equilateral triangular lattice in the complex plane defined by the Eisenstein integers; let the ordered pair $(x,y)$ denote the complex number $x+y\omega$ for $\omega=e^{2\pi i/3}$. We define an $\omega$-chessboard polygon to be a (non self-intersecting) polygon whose sides are situated along lines of the form $x=a$ or $y=b$, where $a$ and $b$ are integers. These lines divide the interior into unit triangles, which are shaded alternately black and white so that adjacent triangles have different colors. To tile an $\omega$-chessboard polygon by lozenges is to exactly cover the polygon by non-overlapping rhombuses consisting of two bordering triangles. Finally, a [i]tasteful tiling[/i] is one such that for every unit hexagon tiled by three lozenges, each lozenge has a black triangle on its left (defined by clockwise orientation) and a white triangle on its right (so the lozenges are BW, BW, BW in clockwise order). a) Prove that if an $\omega$-chessboard polygon can be tiled by lozenges, then it can be done so tastefully. b) Prove that such a tasteful tiling is unique. [i]Victor Wang.[/i]

For a set $S$, let $|S|$ denote the number of elements in $S$. Let $A$ be a set of positive integers with $|A| = 2001$. Prove that there exists a set $B$ such that (i) $B \subseteq A$; (ii) $|B| \ge 668$; (iii) for any $u, v \in B$ (not necessarily distinct), $u+v \not\in B$.

Let \(L_3 = \{3\}\), \(L_n = \{3, 4, \ldots, h\}\) (where \(h > 3\)). For any given integer \(n \geq 3\), consider a graph \(G\) with \(n\) vertices that contains a Hamiltonian cycle \(C\) and has more than \(\frac{n^2}{4}\) edges. For which lengths \(l \in L_n\) must the graph \(G\) necessarily contain a cycle of length \(l\)?

Define a rod to be a 1 by $n$ rectangle for any integer $n$. An $8 \times 8$ board is tiled with 13 rods so that all of it is covered without overlap. Find the maximum possible value of the product of the lengths of the 13 rods. [i]Lightning 4.3[/i]

Sara makes a staircase out of toothpicks as shown:[asy] size(150); defaultpen(linewidth(0.8)); path h = ellipse((0.5,0),0.45,0.015), v = ellipse((0,0.5),0.015,0.45); for(int i=0;i<=2;i=i+1) { for(int j=0;j<=3-i;j=j+1) { filldraw(shift((i,j))*h,black); filldraw(shift((j,i))*v,black); } }[/asy] This is a 3-step staircase and uses 18 toothpicks. How many steps would be in a staircase that used 180 toothpicks? $\textbf{(A)}\ 10\qquad\textbf{(B)}\ 11\qquad\textbf{(C)}\ 12\qquad\textbf{(D)}\ 24\qquad\textbf{(E)}\ 30$

Let $n$ be a positive integer. Let $B_n$ be the set of all binary strings of length $n$. For a binary string $s_1\hdots s_n$, we define it's twist in the following way. First, we count how many blocks of consecutive digits it has. Denote this number by $b$. Then, we replace $s_b$ with $1-s_b$. A string $a$ is said to be a [i]descendant[/i] of $b$ if $a$ can be obtained from $b$ through a finite number of twists. A subset of $B_n$ is called [i]divided[/i] if no two of its members have a common descendant. Find the largest possible cardinality of a divided subset of $B_n$. [i]Remark.[/i] Here is an example of a twist: $101100 \rightarrow 101000$ because $1\mid 0\mid 11\mid 00$ has $4$ blocks of consecutive digits. [i]Viktor Simjanoski[/i]

Pasha and Vova play the following game, making moves in turn; Pasha moves first. Initially, they have a large piece of plasticine. By a move, Pasha cuts one of the existing pieces into three(of arbitrary sizes), and Vova merges two existing pieces into one. Pasha wins if at some point there appear to be $100$ pieces of equal weights. Can Vova prevent Pasha's win?

Let $n$ be a positive integer and let $(1+iT)^n=f(T)+ig(T)$ where $i$ is the square root of $-1$, and $f$ and $g$ are polynomials with real coefficients. Show that for any real number $k$ the equation $f(T)+kg(T)=0$ has only real roots.

Prove that in any convex polygon with $4n+2$ sides ($n\geq 1$) there exist two consecutive sides which form a triangle of area at most $\frac 1{6n}$ of the area of the polygon.

There are $10{}$ apples, each with a with a weight which is no more than $100{}$ g. There is a weighing scale with two plates which shows the difference between the weights on the plates. Prove that 1) It is possible to put some (more than one) apples on the plates of the scale such that the difference between the weights on the plates will be less than $1$ g. 2) It is possible to put an equal amount (more than one) of apples on each plate of the scale such that the difference between the weights on the plates will be less than $2$ g.

$\displaystyle 2n$ students $\displaystyle (n \geq 5)$ participated at table tennis contest, which took $\displaystyle 4$ days. In every day, every student played a match. (It is possible that the same pair meets twice or more times, in different days) Prove that it is possible that the contest ends like this: - there is only one winner; - there are $\displaystyle 3$ students on the second place; - no student lost all $\displaystyle 4$ matches. How many students won only a single match and how many won exactly $\displaystyle 2$ matches? (In the above conditions)