$O$ is a point inside $\triangle ABC$, and $\overrightarrow{OA}+2\overrightarrow{OB}+3\overrightarrow{OC}=\overrightarrow{0}$, then the ratio of the area of $\triangle ABC$ to $\triangle AOC$ is $\text{(A)}2\qquad\text{(B)}\frac{3}{2}\qquad\text{(C)}3\qquad\text{(D)}\frac{5}{3}$

Let $ I$ be an ideal of the ring $\mathbb{Z}\left[x\right]$ of all polynomials with integer coefficients such that a) the elements of $ I$ do not have a common divisor of degree greater than $ 0$, and b) $ I$ contains of a polynomial with constant term $ 1$. Prove that $ I$ contains the polynomial $ 1 + x + x^2 + ... + x^{r-1}$ for some natural number $ r$. [i]Gy. Szekeres[/i]

Show that every positive rational number can be written as a quotient of products of factorials of (not necessarily distinct) primes. For example, $ \frac{10}9\equal{}\frac{2!\cdot 5!}{3!\cdot 3!\cdot 3!}.$

Let $n\ge 3$ be an integer. In a game there are $n$ boxes in a circular array. At the beginning, each box contains an object which can be rock, paper or scissors, in such a way that there are no two adjacent boxes with the same object, and each object appears in at least one box. Same as in the game, rock beats scissors, scissors beat paper, and paper beats rock. The game consists on moving objects from one box to another according to the following rule: [i]Two adjacent boxes and one object from each one are chosen in such a way that these are different, and we move the loser object to the box containing the winner object. For example, if we picked rock from box A and scissors from box B, we move scossors to box A.[/i] Prove that, applying the rule enough times, it is possible to move all the objects to the same box. [i]Proposed by Victor de la Fuente[/i]

Consider the set of all polynomials of degree less than or equal to $4$ with rational coefficients. a) Prove that it has a vector space structure over the field of numbers rational. b) Prove that the polynomials $1, x - 2, (x -2)^2, (x - 2)^3$ and $(x -2)^4$ form a base of this space. c) Express the polynomial $7 + 2x - 45x^2 + 3x^4$ in the previous base.

In the adjoining figure of a rectangular solid, $\angle DHG=45^\circ$ and $\angle FHB=60^\circ$. Find the cosine of $\angle BHD$. [asy] size(200); import three;defaultpen(linewidth(0.7)+fontsize(10)); currentprojection=orthographic(1/3+1/10,1-1/10,1/3); real r=sqrt(3); triple A=(0,0,r), B=(0,r,r), C=(1,r,r), D=(1,0,r), E=O, F=(0,r,0), G=(1,0,0), H=(1,r,0); draw(D--G--H--D--A--B--C--D--B--F--H--B^^C--H); draw(A--E^^G--E^^F--E, linetype("4 4")); label("$A$", A, N); label("$B$", B, dir(0)); label("$C$", C, N); label("$D$", D, W); label("$E$", E, NW); label("$F$", F, S); label("$G$", G, W); label("$H$", H, S); triple H45=(1,r-0.15,0.1), H60=(1-0.05, r, 0.07); label("$45^\circ$", H45, dir(125), fontsize(8)); label("$60^\circ$", H60, dir(25), fontsize(8));[/asy] $\textbf {(A) } \frac{\sqrt{3}}{6} \qquad \textbf {(B) } \frac{\sqrt{2}}{6} \qquad \textbf {(C) } \frac{\sqrt{6}}{3} \qquad \textbf {(D) } \frac{\sqrt{6}}{4} \qquad \textbf {(E) } \frac{\sqrt{6}-\sqrt{2}}{4}$

Given is a prime number $p$ and natural $n$ such that $p \geq n \geq 3$. Set $A$ is made of sequences of lenght $n$ with elements from the set $\{0,1,2,...,p-1\}$ and have the following property: For arbitrary two sequence $(x_1,...,x_n)$ and $(y_1,...,y_n)$ from the set $A$ there exist three different numbers $k,l,m$ such that: $x_k \not = y_k$, $x_l \not = y_l$, $x_m \not = y_m$. Find the largest possible cardinality of $A$.

We attach to the vertices of a regular hexagon the numbers $1$, $0$, $0$, $0$, $0$, $0$. Now, we are allowed to transform the numbers by the following rules: (a) We can add an arbitrary integer to the numbers at two opposite vertices. (b) We can add an arbitrary integer to the numbers at three vertices forming an equilateral triangle. (c) We can subtract an integer $t$ from one of the six numbers and simultaneously add $t$ to the two neighbouring numbers. Can we, just by acting several times according to these rules, get a cyclic permutation of the initial numbers? (I. e., we started with $1$, $0$, $0$, $0$, $0$, $0$; can we now get $0$, $1$, $0$, $0$, $0$, $0$, or $0$, $0$, $1$, $0$, $0$, $0$, or $0$, $0$, $0$, $1$, $0$, $0$, or $0$, $0$, $0$, $0$, $1$, $0$, or $0$, $0$, $0$, $0$, $0$, $1$ ?)

Suppose that $T,U:V\longrightarrow V$ are two linear transformations on the vector space $V$ such that $T+U$ is an invertible transformation. Prove that $TU=UT=0 \Leftrightarrow \operatorname{rank} T+\operatorname{rank} U=n$.

Given two integers $m,n$ which are greater than $1$. $r,s$ are two given positive real numbers such that $r<s$. For all $a_{ij}\ge 0$ which are not all zeroes,find the maximal value of the expression \[f=\frac{(\sum_{j=1}^{n}(\sum_{i=1}^{m}a_{ij}^s)^{\frac{r}{s}})^{\frac{1}{r}}}{(\sum_{i=1}^{m})\sum_{j=1}^{n}a_{ij}^r)^{\frac{s}{r}})^{\frac{1}{s}}}.\]

Let $n \geq 2$ be a positive integer and, for all vectors with integer entries $$X=\begin{pmatrix} x_1 \\ x_2 \\ \vdots \\ x_n \end{pmatrix}$$ let $\delta(X) \geq 0$ be the greatest common divisor of $x_1,x_2, \dots, x_n.$ Also, consider $A \in \mathcal{M}_n(\mathbb{Z}).$ Prove that the following statements are equivalent: $\textbf{i) }$ $|\det A | = 1$ $\textbf{ii) }$ $\delta(AX)=\delta(X),$ for all vectors $X \in \mathcal{M}_{n,1}(\mathbb{Z}).$ [i]Romeo Raicu[/i]

A polygon $\prod$ is given in the $OXY$ plane and its area exceeds $n.$ Prove that there exist $n+1$ points $P_{1}(x_1, y_1), P_{2}(x_2, y_2), \ldots, P_{n+1}(x_{n+1}, y_{n+1})$ in $\prod$ such that $\forall i,j \in \{1, 2, \ldots, n+1\}$, $x_j - x_i$ and $y_j - y_i$ are all integers.

Let $ a_1$, $ a_2$, ..., $ a_n$ and $ b_1$, $ b_2$, ..., $ b_n$ be $ 2 \cdot n$ real numbers. Prove that the following two statements are equivalent: [b]i)[/b] For any $ n$ real numbers $ x_1$, $ x_2$, ..., $ x_n$ satisfying $ x_1 \leq x_2 \leq \ldots \leq x_ n$, we have $ \sum^{n}_{k \equal{} 1} a_k \cdot x_k \leq \sum^{n}_{k \equal{} 1} b_k \cdot x_k,$ [b]ii)[/b] We have $ \sum^{s}_{k \equal{} 1} a_k \leq \sum^{s}_{k \equal{} 1} b_k$ for every $ s\in\left\{1,2,...,n\minus{}1\right\}$ and $ \sum^{n}_{k \equal{} 1} a_k \equal{} \sum^{n}_{k \equal{} 1} b_k$.

At a certain mathematical conference, every pair of mathematicians are either friends or strangers. At mealtime, every participant eats in one of two large dining rooms. Each mathematician insists upon eating in a room which contains an even number of his or her friends. Prove that the number of ways that the mathematicians may be split between the two rooms is a power of two (i.e., is of the form $ 2^k$ for some positive integer $ k$).