Let $f(x)=x^3+ax^2+bx+c$ and $g(x)=x^3+bx^2+cx+a$, where $a,b,c$ are integers with $c\not=0$. Suppose that the following conditions hold: [list=a][*]$f(1)=0$, [*]the roots of $g(x)=0$ are the squares of the roots of $f(x)=0$.[/list] Find the value of $a^{2013}+b^{2013}+c^{2013}$.

A Sudoku matrix is defined as a $ 9\times9$ array with entries from $ \{1, 2, \ldots , 9\}$ and with the constraint that each row, each column, and each of the nine $ 3 \times 3$ boxes that tile the array contains each digit from $ 1$ to $ 9$ exactly once. A Sudoku matrix is chosen at random (so that every Sudoku matrix has equal probability of being chosen). We know two of the squares in this matrix, as shown. What is the probability that the square marked by ? contains the digit $ 3$? $ \setlength{\unitlength}{6mm} \begin{picture}(9,9)(0,0) \multiput(0,0)(1,0){10}{\line(0,1){9}} \multiput(0,0)(0,1){10}{\line(1,0){9}} \linethickness{1.2pt} \multiput(0,0)(3,0){4}{\line(0,1){9}} \multiput(0,0)(0,3){4}{\line(1,0){9}} \put(0,8){\makebox(1,1){1}} \put(1,7){\makebox(1,1){2}} \put(3,6){\makebox(1,1){?}} \end{picture}$

Let $ n \geq 2$ be a positive integer and $ \lambda$ a positive real number. Initially there are $ n$ fleas on a horizontal line, not all at the same point. We define a move as choosing two fleas at some points $ A$ and $ B$, with $ A$ to the left of $ B$, and letting the flea from $ A$ jump over the flea from $ B$ to the point $ C$ so that $ \frac {BC}{AB} \equal{} \lambda$. Determine all values of $ \lambda$ such that, for any point $ M$ on the line and for any initial position of the $ n$ fleas, there exists a sequence of moves that will take them all to the position right of $ M$.

For every positive integer $ n$ let \[ a_n\equal{}\sum_{k\equal{}n}^{2n}\frac{(2k\plus{}1)^n}{k}\] Show that there exists no $ n$, for which $ a_n$ is a non-negative integer.

Show that there is a subset of $A$ from $\{1,2, 3,... , 2014\}$ such that : (i) $|A| = 12$ (ii) for each coloring number in $A$ with red or white , we can always find some numbers colored in $A$ whose sum is $2015$.

A train travels a certain distance at a constant speed. Whose speed is increased by $10$ kilometers per hour, the trip can be made $40$ minutes faster. If, on the other hand, the speed is reduced by $10$ kilometers per hour, the trip takes $1$ hour further. How long is the distance traveled?

In one of the adjoining figures a square of side $2$ is dissected into four pieces so that $E$ and $F$ are the midpoints of opposite sides and $AG$ is perpendicular to $BF$. These four pieces can then be reassembled into a rectangle as shown in the second figure. The ratio of height to base, $XY$ / $YZ$, in this rectangle is [asy] size(180); defaultpen(linewidth(0.7)+fontsize(10)); pair A=(0,1), B=(0,-1), C=(2,-1), D=(2,1), E=(1,-1), F=(1,1), G=(.8,.6); pair X=(4,sqrt(5)), Y=(4,-sqrt(5)), Z=(4+2/sqrt(5),-sqrt(5)), W=(4+2/sqrt(5),sqrt(5)), T=(4,0), U=(4+2/sqrt(5),-4/sqrt(5)), V=(4+2/sqrt(5),1/sqrt(5)); draw(A--B--C--D--A^^B--F^^E--D^^A--G^^rightanglemark(A,G,F)); draw(X--Y--Z--W--X^^T--V--X^^Y--U); label("A", A, NW); label("B", B, SW); label("C", C, SE); label("D", D, NE); label("E", E, S); label("F", F, N); label("G", G, E); label("X", X, NW); label("Y", Y, SW); label("Z", Z, SE); label("W", W, NE); [/asy] $ \textbf{(A)}\ 4\qquad\textbf{(B)}\ 1+2\sqrt{3}\qquad\textbf{(C)}\ 2\sqrt{5}\qquad\textbf{(D)}\ \frac{8+4\sqrt{3}}{3}\qquad\textbf{(E)}\ 5 $

Let $n \geq 2$ be a positive integer. There are $2n$ cities $M_1, M_2, \ldots, M_{2n}$ in the country of Mathlandia. Currently there roads only between $M_1$ and $M_2, M_3, \ldots, M_n$ and the king wants to build more roads so that it is possible to reach any city from every other city. The cost to build a road between $M_i$ and $M_j$ is $k_{i, j}>0$. Let $$K=\sum_{j=n+1}^{2n} k_{1,j}+\sum_{2 \leq i<j \leq 2n} k_{i, j}.$$Prove that the king can fulfill his plan at cost no more than $\frac{2K}{3n-1}$.

One of the sides of the triangle is divided by the ratio $p: q$, and the other by $m: n: k$. The obtained division points of the sides are connected to the opposite vertices of the triangle by straight lines. Find the ratio of the area of ​​this triangle to the area of ​​the quadrilateral formed by three such lines and one of the sides of the triangle.

A “number triangle” $(t_{n, k}) (0 \le k \le n)$ is defined by $t_{n,0} = t_{n,n} = 1 (n \ge 0),$ \[t_{n+1,m} =(2 -\sqrt{3})^mt_{n,m} +(2 +\sqrt{3})^{n-m+1}t_{n,m-1} \quad (1 \le m \le n)\] Prove that all $t_{n,m}$ are integers.