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.

Square $ABCD$ has side length $68$. Let $E$ be the midpoint of segment $\overline{CD}$, and let $F$ be the point on segment $\overline{AB}$ a distance $17$ from point $A$. Point $G$ is on segment $\overline{EF}$ so that $\overline{EF}$ is perpendicular to segment $\overline{GD}$. The length of segment $\overline{BG}$ can be written as $m\sqrt{n}$ where $m$ and $n$ are positive integers, and $n$ is not divisible by the square of any prime. Find $m+n$.

Do there exist polynomials $a(x)$, $b(x)$, $c(y)$, $d(y)$ such that \[1 + xy + x^2y^2= a(x)c(y) + b(x)d(y)\] holds identically?

Let $G$ be a group, with operation $*$. Suppose that (i) $G$ is a subset of $\mathbb{R}^3$ (but $*$ need not be related to addition of vectors); (ii) For each $\mathbf{a},\mathbf{b}\in G,$ either $\mathbf{a}\times\mathbf{b}=\mathbf{a}*\mathbf{b}$ or $\mathbf{a}\times\mathbf{b}=\mathbf{0}$ (or both), where $\times$ is the usual cross product in $\mathbb{R}^3.$ Prove that $\mathbf{a}\times\mathbf{b}=\mathbf{0}$ for all $\mathbf{a},\mathbf{b}\in G.$

On the sides $ AB,BC,CD,DA $ of the parallelogram $ ABCD, $ consider the points $ M,N,P, $ respectively, $ Q, $ such that $ \overrightarrow{MN} +\overrightarrow{QP} =\overrightarrow{AC} . $ Show that $ \overrightarrow{PN} +\overrightarrow{QM} = \overrightarrow{DB} . $

Some points with the integer coordinates are marked on the coordinate plane. Given a set of nonzero vectors. It is known, that if you apply the beginnings of those vectors to the arbitrary marked point, than there will be more marked ends of the vectors, than not marked. Prove that there is infinite number of marked points.

Let $ABCD$ and $A'B'C'D'$ be two arbitrary squares in the plane that are oriented in the same direction. Prove that the quadrilateral formed by the midpoints of $AA',BB',CC',DD'$ is a square.

Let $ \Delta ABC$ be a triangle with $M$ the middle of the side $[BC]$. On the line $BC$, to the left and to the right of the point $M,$ at the same distance from $M,$ let us consider $d_1$ and $d_2,$ which are perpendicular to the line BC. The perpendicular line from $M$ to $AB$ intersects $d_1$ in $P,$ and the perpendicular line from $M$ to $AC$ intersects $d_2$ in $Q.$ Prove that \[AM\perp PQ.\] [b]Author: Marin Bancoș Regional Mathematical Contest GRIGORE MOISIL, Romania, Baia Mare, 2012, 9th grade[/b]

Let $n$ be a positive integer and let $V$ be a $(2n-1)$-dimensional vector space over the two-element field. Prove that for arbitrary vectors $v_1,\dots,v_{4n-1} \in V,$ there exists a sequence $1\leq i_1<\dots<i_{2n}\leq 4n-1$ of indices such that $v_{i_1}+\dots+v_{i_{2n}}=0.$

Let $A,B\in\mathcal{M}_{2}(\mathbb{R})$ (real $2\times 2$ matrices), that satisfy $A^{2}+B^{2}=AB$. Prove that $(AB-BA)^{2}=O_{2}$.

A river flows between two houses $ A$ and $ B$, the houses standing some distances away from the banks. Where should a bridge be built on the river so that a person going from $ A$ to $ B$, using the bridge to cross the river may do so by the shortest path? Assume that the banks of the river are straight and parallel, and the bridge must be perpendicular to the banks.

Let $ k$ be an integer, $ k \geq 2$, and let $ p_{1},\ p_{2},\ \ldots,\ p_{k}$ be positive reals with $ p_{1} \plus{} p_{2} \plus{} \ldots \plus{} p_{k} \equal{} 1$. Suppose we have a collection $ \left(A_{1,1},\ A_{1,2},\ \ldots,\ A_{1,k}\right)$, $ \left(A_{2,1},\ A_{2,2},\ \ldots,\ A_{2,k}\right)$, $ \ldots$, $ \left(A_{m,1},\ A_{1,2},\ \ldots,\ A_{m,k}\right)$ of $ k$-tuples of finite sets satisfying the following two properties: (i) for every $ i$ and every $ j \neq j^{\prime}$, $ A_{i,j}\cap A_{i,j^{\prime}} \equal{} \emptyset$, and (ii) for every $ i\neq i^{\prime}$ there exist $ j\neq j^{\prime}$ for which $ A_{i,j} \cap A_{i^{\prime},j^{\prime}}\neq\emptyset$. Prove that \[ \sum_{b \equal{} 1}^{m}{\prod_{a \equal{} 1}^{k}{p_{a}^{|A_{b,a}|}}} \leq 1. \]

There are $n$ distinct points in the plane. Given a circle in the plane containing at least one of the points in its interior. At each step one moves the center of the circle to the barycenter of all the points in the interior of the circle. Prove that this moving process terminates in the finite number of steps. what does barycenter of n distinct points mean?

On the sides (excluding its endpoints) $ AB,BC,CD,DA $ of a parallelogram consider the points $ M,N,P,Q, $ respectively, such that $ \overrightarrow{AP} +\overrightarrow{AN} +\overrightarrow{CQ} +\overrightarrow{CM} = 0. $ Show that $ QN, PM,AC $ are concurrent. [i]Adrian Ivan[/i]

Two triangles $ ABC$and $ A'B'C'$ are positioned in the space such that the length of every side of $ \triangle ABC$ is not less than $ a$, and the length of every side of $ \triangle A'B'C'$ is not less than $ a'$. Prove that one can select a vertex of $ \triangle ABC$ and a vertex of $ \triangle A'B'C'$ so that the distance between the two selected vertices is not less than $ \sqrt {\frac {a^2 \plus{} a'^2}{3}}$.