Let $P$ and $P^{\prime }$ be two parallelograms with equal area, and let their sidelengths be $a,$ $b$ and $a^{\prime },$ $b^{\prime }.$ Assume that $a^{\prime }\leq a\leq b\leq b^{\prime },$ and moreover, it is possible to place the segment $b^{\prime }$ such that it completely lies in the interior of the parallelogram $P.$ Show that the parallelogram $P$ can be partitioned into four polygons such that these four polygons can be composed again to form the parallelogram $% P^{\prime }.$

We define an operation $\oplus$ on the set $\{0, 1\}$ by \[ 0 \oplus 0 = 0 \,, 0 \oplus 1 = 1 \,, 1 \oplus 0 = 1 \,, 1 \oplus 1 = 0 \,.\] For two natural numbers $a$ and $b$, which are written in base $2$ as $a = (a_1a_2 \ldots a_k)_2$ and $b = (b_1b_2 \ldots b_k)_2$ (possibly with leading 0's), we define $a \oplus b = c$ where $c$ written in base $2$ is $(c_1c_2 \ldots c_k)_2$ with $c_i = a_i \oplus b_i$, for $1 \le i \le k$. For example, we have $7 \oplus 3 = 4$ since $ 7 = (111)_2$ and $3 = (011)_2$. For a natural number $n$, let $f(n) = n \oplus \left[ n/2 \right]$, where $\left[ x \right]$ denotes the largest integer less than or equal to $x$. Prove that $f$ is a bijection on the set of natural numbers.

Let $P$ be a fixed point and $T$ a given triangle that contains the point $P$. Translate the triangle $T$ by a given vector $\bold{v}$ and denote by $T'$ this new triangle. Let $r, R$, respectively, be the radii of the smallest disks centered at $P$ that contain the triangles $T , T'$, respectively. Prove that $r + |\bold{v}| \leq 3R$ and find an example to show that equality can occur.

For each pair of integers $(i,k)$ such that $1\le i\le k$, the linear transformation $P_{i,k}:\mathbb{R}^k\to\mathbb{R}^k$ is defined as: $P_{i,k}(a_1,\cdots,a_{i-1},a_i,a_{i+1},\cdots,a_k)=(a_1,\cdots,a_{i-1},0,a_{i+1},\cdots,a_k)$ Prove that for all $n\ge2$ and for every set of $n-1$ linearly independent vectors $v_1,\cdots,v_{n-1}$ in $\mathbb{R}^n$, there is an integer $k$ such that $1\le k\le n$ and such that the vectors $P_{k,n}(v_1),\cdots,P_{k,n}(v_{n-1})$ are linearly independent.

Let $ABC$ be a triangle with orthocenter $H$, incenter $I$ and centroid $S$, and let $d$ be the diameter of the circumcircle of triangle $ABC$. Prove the inequality \[9\cdot HS^2+4\left(AH\cdot AI+BH\cdot BI+CH\cdot CI\right)\geq 3d^2,\] and determine when equality holds.

Let $F:\mathbb{R}^2\to\mathbb{R}$ and $g:\mathbb{R}\to\mathbb{R}$ be twice continuously differentiable functions with the following properties: • $F(u,u)=0$ for every $u\in\mathbb{R};$ • for every $x\in\mathbb{R},g(x)>0$ and $x^2g(x)\le 1;$ • for every $(u,v)\in\mathbb{R}^2,$ the vector $\nabla F(u,v)$ is either $\mathbf{0}$ or parallel to the vector $\langle g(u),-g(v)\rangle.$ Prove that there exists a constant $C$ such that for every $n\ge 2$ and any $x_1,\dots,x_{n+1}\in\mathbb{R},$ we have \[\min_{i\ne j}|F(x_i,x_j)|\le\frac{C}{n}.\]

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 $BCD$.

Prove that on the coordinate plane it is impossible to draw a closed broken line such that [i](i)[/i] the coordinates of each vertex are rational; [i](ii)[/i] the length each of its edges is 1; [i](iii)[/i] the line has an odd number of vertices.

Let’s consider three pairwise non-parallel straight constant lines in the plane. Three points are moving along these lines with different nonzero velocities, one on each line (we consider the movement to have taken place for infinite time and continue infinitely in the future). Is it possible to determine these straight lines, the velocities of each moving point and their positions at some “zero” moment in such a way that the points never were, are or will be collinear?

Triangle $ABC$ lies in the Cartesian Plane and has an area of 70. The coordinates of $B$ and $C$ are $(12,19)$ and $(23,20)$, respectively, and the coordinates of $A$ are $(p,q)$. The line containing the median to side $BC$ has slope $-5$. Find the largest possible value of $p+q$.

On $xy$plane the parabola $K: \ y=\frac{1}{d}x^{2}\ (d: \ positive\ constant\ number)$ intersects with the line $y=x$ at the point $P$ that is different from the origin. Assumed that the circle $C$ is touched to $K$ at $P$ and $y$ axis at the point $Q.$ Let $S_{1}$ be the area of the region surrounded by the line passing through two points $P,\ Q$ and $K,$ or $S_{2}$ be the area of the region surrounded by the line which is passing through $P$ and parallel to $x$ axis and $K.$ Find the value of $\frac{S_{1}}{S_{2}}.$

Given regular $1987$ -gon on plane with vertices $A_1, A_2,..., A_{1987}$. Find locus of points M of the plane sych that $$\left|\overrightarrow{MA_1}+\overrightarrow{MA_2}+...+\overrightarrow{MA_{1987}}\right| \le 1987$$.

Suppose $V= \mathbb{Z}_2^n$ and for a vector $x=(x_1,..x_n)$ in $V$ and permutation $\sigma$.We have $x_{\sigma}=(x_{\sigma(1)},...,x_{\sigma(n)})$ Suppose $ n=4k+2,4k+3$ and $f:V \to V$ is injective and if $x$ and $y$ differ in more than $n/2$ places then $f(x)$ and $f(y)$ differ in more than $n/2$ places. Prove there exist permutaion $\sigma$ and vector $v$ that $f(x)=x_{\sigma}+v$

Zeroes and ones are arranged in all the squares of $n\times n$ table. All the squares of the left column are filled by ones, and the sum of numbers in every figure of the form [asy]size(50); draw((2,1)--(0,1)--(0,2)--(2,2)--(2,0)--(1,0)--(1,2));[/asy] (consisting of a square and its neighbours from left and from below) is even. Prove that no two rows of the table are identical. [i]Proposed by O. Vanyushina[/i]

Find the index of the singular point $0$ of the vector field with components \[(x^4+y^4+z^4,x^3y-xy^3,xyz^2)\]

A calculating ruler is a ruler for doing algebric calculations. This ruler has three arms, two of them are sationary and one can move freely right and left. Each of arms is gradient. Gradation of each arm depends on the algebric operation ruler does. For eaxample the ruler below is designed for multiplying two numbers. Gradations are logarithmic. [img]http://aycu05.webshots.com/image/5604/2000468517162383885_rs.jpg[/img] For working with ruler, (e.g for calculating $x.y$) we must move the middle arm that the arrow at the beginning of its gradation locate above the $x$ in the lower arm. We find $y$ in the middle arm, and we will read the number on the upper arm. The number written on the ruler is the answer. 1) Design a ruler for calculating $x^{y}$. Grade first arm ($x$) and ($y$) from 1 to 10. 2) Find all rulers that do the multiplication in the interval $[1,10]$. 3) Prove that there is not a ruler for calculating $x^{2}+xy+y^{2}$, that its first and second arm are grade from 0 to 10.

Let $ABCD$ be a convex quadrilateral and points $E$ and $F$ on sides $AB,CD$ such that \[\tfrac{AB}{AE}=\tfrac{CD}{DF}=n\] If $S$ is the area of $AEFD$ show that ${S\leq\frac{AB\cdot CD+n(n-1)AD^2+n^2DA\cdot BC}{2n^2}}$

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.

Suppose that $a, b, c$ are real numbers such that for all positive numbers $x_1,x_2,\dots,x_n$ we have $(\frac{1}{n}\sum_{i=1}^nx_i)^a(\frac{1}{n}\sum_{i=1}^nx_i^2)^b(\frac{1}{n}\sum_{i=1}^nx_i^3)^c\ge 1$ Prove that vector $(a, b, c)$ is a nonnegative linear combination of vectors $(-2,1,0)$ and $(-1,2,-1)$.

There are $ n \plus{} 1$ cells in a row labeled from $ 0$ to $ n$ and $ n \plus{} 1$ cards labeled from $ 0$ to $ n$. The cards are arbitrarily placed in the cells, one per cell. The objective is to get card $ i$ into cell $ i$ for each $ i$. The allowed move is to find the smallest $ h$ such that cell $ h$ has a card with a label $ k > h$, pick up that card, slide the cards in cells $ h \plus{} 1$, $ h \plus{} 2$, ... , $ k$ one cell to the left and to place card $ k$ in cell $ k$. Show that at most $ 2^n \minus{} 1$ moves are required to get every card into the correct cell and that there is a unique starting position which requires $ 2^n \minus{} 1$ moves. [For example, if $ n \equal{} 2$ and the initial position is 210, then we get 102, then 012, a total of 2 moves.]

Find the parallel displacement of a vector pointing north at Leningrad (latitude $60^{\circ}$) from west to east along a closed parallel.

Let $ ABC$ be a right triangle with $ \angle A \equal{} 90^\circ$. Let $ D$ be the midpoint of $ AB$ and let $ E$ be a point on segment $ AC$ such that $ AD \equal{} AE$. Let $ BE$ meet $ CD$ at $ F$. If $ \angle BFC \equal{} 135^\circ$, determine $ BC / AB$.

Given two triangles and such that the lengths of the sides of the first triangle are the lengths of the medians of the second triangle. Determine the ratio of the areas of these triangles.

$n$ 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 $\mathbb Z_{2}^{n}$

$S$ is a sphere center $O. G$ and $G'$ are two perpendicular great circles on $S$. Take $A, B, C$ on $G$ and $D$ on $G'$ such that the altitudes of the tetrahedron $ABCD$ intersect at a point. Find the locus of the intersection.