Given a set of $n>2$ planar vectors. A vector from this set is called [i]long[/i], if its length is not less than the length of the sum of other vectors in this set. Prove that if each vector is long, then the sum of all vectors equals to zero. [i]N. Agakhanov[/i]

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

$n$ is a natural number that $\frac{x^{n}+1}{x+1}$ is irreducible over $\mathbb Z_{2}[x]$. Consider a vector in $\mathbb Z_{2}^{n}$ that it has odd number of $1$'s (as entries) and at least one of its entries are $0$. Prove that these vector and its translations are a basis for $\mathbb Z_{2}^{n}$

Let $L$ be a subset in the coordinate plane defined by $L = \{(41x + 2y, 59x + 15y) | x, y \in \mathbb Z \}$, where $\mathbb Z$ is set of integers. Prove that for any parallelogram with center in the origin of coordinate and area $1990$, there exist at least two points of $L$ located in it.

Decompose a $5$-dimensional real linear space into the irreducible invariant subspaces of the group generated by cyclic permutations of the basis vectors.

In a circle of radius 42, two chords of length 78 intersect at a point whose distance from the center is 18. The two chords divide the interior of the circle into four regions. Two of these regions are bordered by segments of unequal lenghts, and the area of either of them can be expressed uniquley in the form $m\pi-n\sqrt{d},$ where $m, n,$ and $d$ are positive integers and $d$ is not divisible by the square of any prime number. Find $m+n+d.$

Three of the edges of a cube are $\overline{AB}, \overline{BC},$ and $\overline{CD},$ and $\overline{AD}$ is an interior diagonal. Points $P, Q,$ and $R$ are on $\overline{AB}, \overline{BC},$ and $\overline{CD},$ respectively, so that $AP=5, PB=15, BQ=15,$ and $CR=10.$ What is the area of the polygon that is the intersection of plane $PQR$ and the cube?

In a regular tetrahedron the centers of the four faces are the vertices of a smaller tetrahedron. The ratio of the volume of the smaller tetrahedron to that of the larger is $m/n$, where $m$ and $n$ are relatively prime positive integers. Find $m+n.$

A lamp is situated at point $A$ and shines inside the cube. A (massive) square is hung on the midpoints of the 4 vertical faces. What's the area of its shadow? [img]http://www.mathlinks.ro/Forum/album_pic.php?pic_id=285[/img]

$f: \mathbb R^{n}\longrightarrow\mathbb R^{n}$ is a bijective map, that Image of every $n-1$-dimensional affine space is a $n-1$-dimensional affine space. 1) Prove that Image of every line is a line. 2) Prove that $f$ is an affine map. (i.e. $f=goh$ that $g$ is a translation and $h$ is a linear map.)

Let $x_1,\cdots, x_n$ be nonzero vectors of a vector space $V$ and $\varphi:V\to V$ be a linear transformation such that $\varphi x_1 = x_1$, $\varphi x_k = x_k - x_{k-1}$ for $k = 2, 3,\ldots,n$. Prove that the vectors $x_1,\ldots,x_n$ are linearly independent.

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$).

Let $P$ be a polynomial with each root is real and each coefficient is either $1$ or $-1$. The degree of $P$ can be at most ? $ \textbf{(A)}\ 5 \qquad\textbf{(B)}\ 4 \qquad\textbf{(C)}\ 3 \qquad\textbf{(D)}\ 2 \qquad\textbf{(E)}\ \text{None} $

Let $m<n$ be two positive integers, let $I$ and $J$ be two index sets such that $|I|=|J|=n$ and $|I\cap J|=m$, and let $u_k$, $k\in I\cup J$ be a collection of vectors in the Euclidean plane such that \[|\sum_{i\in I}u_i|=1=|\sum_{j\in J}u_j|.\] Prove that \[\sum_{k\in I\cup J}|u_k|^2\geq \frac{2}{m+n}\] and find the cases of equality.

Given a prime number $p$ and let $\overline{v_1},\overline{v_2},\dotsc ,\overline{v_n}$ be $n$ distinct vectors of length $p$ with integer coordinates in an $\mathbb{R}^3$ Cartesian coordinate system. Suppose that for any $1\leqslant j<k\leqslant n$, there exists an integer $0<\ell <p$ such that all three coordinates of $\overline{v_j} -\ell \cdot \overline{v_k} $ is divisible by $p$. Prove that $n\leqslant 6$.

Suppose that the function $h:\mathbb{R}^2\to\mathbb{R}$ has continuous partial derivatives and satisfies the equation \[h(x,y)=a\frac{\partial h}{\partial x}(x,y)+b\frac{\partial h}{\partial y}(x,y)\] for some constants $a,b.$ Prove that if there is a constant $M$ such that $|h(x,y)|\le M$ for all $(x,y)$ in $\mathbb{R}^2,$ then $h$ is identically zero.

[i]Introductory part [/i] We call an $n$-tuple $x = (x_1, x_2, ... , x_n)$, with $x_k \in R$ (or respectively with all $x_k \in Z$) a real vector (or respectively an integer vector). The set of all real vectors (respectively all integer vectors) is usually denoted by $R^n$ (respectively $Z^n$). A vector $x$ is null if and only if $x_k = 0$, for all $k \in \{1, 2,... , n\}$. Also let $U_n$ be the set of all real vectors $x = (x_1, x_2, ... , x_n)$, such that $x^2_1 + x^2_2 + ...+ x^2_n = 1$. For two vectors $x = (x_1, ... , x_n), y = (y_1, ..., y_n)$ we define the scalar product as the real number $x\cdot y = x_1y_1 + x_2y_2 +...+ x_ny_n$. We define the norm of the vector $x$ as $||x|| =\sqrt{x^2_1 + x^2_2 + ...+ x^2_n}$ [i]The problem[/i] Let $A(k, r) = \{x \in U_n |$ for all $z \in Z^n$ we have either $|x \cdot z| \ge \frac{k}{||z||^r}$ or $z$ is null $\}$. Prove that if $r > n - 1$ the we can find a positive number $k$ such that $A(k, r)$ is not empty, and if $r < n - 1$ we cannot find such a positive number $k$.

For a prime $ p$ an a positive integer $ n$, denote by $ \nu_p(n)$ the exponent of $ p$ in the prime factorization of $ n!$. Given a positive integer $ d$ and a finite set $ \{p_1,p_2,\ldots, p_k\}$ of primes, show that there are infinitely many positive integers $ n$ such that $ \nu_{p_i}(n) \equiv 0 \pmod d$, for all $ 1\leq i \leq k$.

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 $ p(x)\equal{}a_0 \plus{}a_1 x\plus{}...\plus{}a_n x^n$ be a polynomial with nonnegative real coefficients. Suppose that $ p(4)\equal{}2$ and $ p(16)\equal{}8$. Prove that $ p(8) \le 4$ and find all such $ p$ with $ p(8)\equal{}4$.

The points $S(i, j)$ with integer Cartesian coordinates $0 < i \leq n, 0 < j \leq m, m \leq n$, form a lattice. Find the number of: [b](a)[/b] rectangles with vertices on the lattice and sides parallel to the coordinate axes; [b](b)[/b] squares with vertices on the lattice and sides parallel to the coordinate axes; [b](c)[/b] squares in total, with vertices on the lattice.

Let $ABCD$ be a tetrahedron such that edges $AB$, $AC$, and $AD$ are mutually perpendicular. Let the areas of triangles $ABC$, $ACD$, and $ADB$ be denoted by $x$, $y$, and $z$, respectively. In terms of $x$, $y$, and $z$, find the area of triangle $BCD$.

Let $\mathbb{F}_p$ denote the field of integers modulo a prime $p,$ and let $n$ be a positive integer. Let $v$ be a fixed vector in $\mathbb{F}_p^n,$ let $M$ be an $n\times n$ matrix with entries in $\mathbb{F}_p,$ and define $G:\mathbb{F}_p^n\to \mathbb{F}_p^n$ by $G(x)=v+Mx.$ Let $G^{(k)}$ denote the $k$-fold composition of $G$ with itself, that is, $G^{(1)}(x)=G(x)$ and $G^{(k+1)}(x)=G(G^{(k)}(x)).$ Determine all pairs $p,n$ for which there exist $v$ and $M$ such that the $p^n$ vectors $G^{(k)}(0),$ $k=1,2,\dots,p^n$ are distinct.

$ABCDEF$ is a convex hexagon, such that in it $AC \parallel DF$, $BD \parallel AE$ and $CE \parallel BF$. Prove that \[AB^2+CD^2+EF^2=BC^2+DE^2+AF^2.\] [i]N. Sedrakyan[/i]

Let $ABC$ be a triangle. Triangles $PAB$ and $QAC$ are constructed outside of triangle $ABC$ such that $AP = AB$ and $AQ = AC$ and $\angle{BAP}= \angle{CAQ}$. Segments $BQ$ and $CP$ meet at $R$. Let $O$ be the circumcenter of triangle $BCR$. Prove that $AO \perp PQ.$