The linear operator $A$ on a finite-dimensional vector space $V$ is called an involution if $A^{2}=I$, where $I$ is the identity operator. Let $\dim V=n$. i) Prove that for every involution $A$ on $V$, there exists a basis of $V$ consisting of eigenvectors of $A$. ii) Find the maximal number of distinct pairwise commuting involutions on $V$.

Find the maximum value of $f(x)=\int_0^1 t\sin (x+\pi t)\ dt$.

We define $(x_1, x_2, \ldots , x_n) \Delta (y_1, y_2, \ldots , y_n) = \left( \sum_{i=1}^{n}x_iy_{2-i}, \sum_{i=1}^{n}x_iy_{3-i}, \ldots , \sum_{i=1}^{n}x_iy_{n+1-i} \right)$, where the indices are taken modulo $n$. Besides this, if $v$ is a vector, we define $v^k = v$, if $k=1$, or $v^k = v \Delta v^{k-1}$, otherwise. Prove that, if $(x_1, x_2, \ldots , x_n)^k = (0, 0, \ldots , 0)$, for some natural number $k$, then $x_1 = x_2 = \ldots = x_n = 0$.

Let $ R$ be a finite commutative ring. Prove that $ R$ has a multiplicative identity element $ (1)$ if and only if the annihilator of $ R$ is $ 0$ (that is, $ aR\equal{}0, \;a\in R $ imply $ a\equal{}0$).

Prove that the sum of an odd number of vectors of length 1, of common origin $O$ and all situated in the same semi-plane determined by a straight line which goes through $O,$ is at least 1.

A number of robots are placed on the squares of a finite, rectangular grid of squares. A square can hold any number of robots. Every edge of each square of the grid is classified as either passable or impassable. All edges on the boundary of the grid are impassable. You can give any of the commands up, down, left, or right. All of the robots then simultaneously try to move in the specified direction. If the edge adjacent to a robot in that direction is passable, the robot moves across the edge and into the next square. Otherwise, the robot remains on its current square. You can then give another command of up, down, left, or right, then another, for as long as you want. Suppose that for any individual robot, and any square on the grid, there is a finite sequence of commands that will move that robot to that square. Prove that you can also give a finite sequence of commands such that all of the robots end up on the same square at the same time.

From a $n\times (n-1)$ rectangle divided into unit squares, we cut the [i]corner[/i], which consists of the first row and the first column. (that is, the corner has $2n-2$ unit squares). For the following, when we say [i]corner[/i] we reffer to the above definition, along with rotations and symmetry. Consider an infinite lattice of unit squares. We will color the squares with $k$ colors, such that for any corner, the squares in that corner are coloured differently (that means that there are no squares coloured with the same colour). Find out the minimum of $k$. [i]Proposed by S. Berlov[/i]

Given vector $\mathbf{u}=\left(\frac{1}{3}, \frac{1}{3}, \frac{1}{3} \right)\in\mathbb{R}^3$ and recursively defined sequence of vectors $\{\mathbf{v}_n\}_{n\geq 0}$ $$\mathbf{v}_0 = (1,2,3),\quad \mathbf{v}_n = \mathbf{u}\times\mathbf{v}_{n-1}$$ Evaluate the value of infinite series $\sum_{n=1}^\infty (3,2,1)\cdot \mathbf{v}_{2n}$.

Suppose that $V$ is a finite dimensional vector space over the real numbers equipped with an inner product and $S:V\times V \longrightarrow \mathbb R$ is a skew symmetric function that is linear for each variable when others are kept fixed. Prove there exists a linear transformation $T:V \longrightarrow V$ such that $\forall u,v \in V: S(u,v)=<u,T(v)>$. We know that there always exists $v\in V$ such that $W=<v,T(v)>$ is invariant under $T$. (it means $T(W)\subseteq W$). Prove that if $W$ is invariant under $T$ then the following subspace is also invariant under $T$: $W^{\perp}=\{v\in V:\forall u\in W <v,u>=0\}$. Prove that if dimension of $V$ is more than $3$, then there exist a two dimensional subspace $W$ of $V$ such that the volume defined on it by function $S$ is zero!!!! (This is the way that we can define a two dimensional volume for each subspace $V$. This can be done for volumes of higher dimensions.)

We consider graphs with vertices colored black or white. "Switching" a vertex means: coloring it black if it was formerly white, and coloring it white if it was formerly black. Consider a finite graph with all vertices colored white. Now, we can do the following operation: Switch a vertex and simultaneously switch all of its neighbours (i. e. all vertices connected to this vertex by an edge). Can we, just by performing this operation several times, obtain a graph with all vertices colored black? [It is assumed that our graph has no loops (a [i]loop[/i] means an edge connecting one vertex with itself) and no multiple edges (a [i]multiple edge[/i] means a pair of vertices connected by more than one edge).]

On the Cartesian plane consider the set $V$ of all vectors with integer coordinates. Determine all functions $f : V \rightarrow \mathbb{R}$ satisfying the conditions: (i) $f(v) = 1$ for each of the four vectors $v \in V$ of unit length. (ii) $f(v+w) = f(v)+f(w)$ for every two perpendicular vectors $v, w \in V$ (Zero vector is considered to be perpendicular to every vector).

Given $1980$ vectors in the plane, and there are some non-collinear among them. The sum of every $1979$ vectors is collinear to the vector not included in that sum. Prove that the sum of all vectors equals to the zero vector.

If $ a\equal{}\minus{}2$, the largest number in the set $ \left \{ \minus{}3a,4a,\frac{24}{a},a^2,1 \right \}$ is \[ \textbf{(A)}\ \minus{}3a \qquad \textbf{(B)}\ 4a \qquad \textbf{(C)}\ \frac{24}{a} \qquad \textbf{(D)}\ a^2 \qquad \textbf{(E)}\ 1 \]

Consider $ \triangle ABC$ and points $ M \in (AB)$, $ N \in (BC)$, $ P \in (CA)$, $ R \in (MN)$, $ S \in (NP)$, $ T \in (PM)$ such that $ \frac {AM}{MB} \equal{} \frac {BN}{NC} \equal{} \frac {CP}{PA} \equal{} k$ and $ \frac {MR}{RN} \equal{} \frac {NS}{SP} \equal{} \frac {PT}{TN} \equal{} 1 \minus{} k$ for some $ k \in (0, 1)$. Prove that $ \triangle STR \sim \triangle ABC$ and, furthermore, determine $ k$ for which the minimum of $ [STR]$ is attained.