In convex hexagon $ABCDEF$, $AB \parallel DE$, $BC \parallel EF$, $CD \parallel FA$, and \[ AB+DE = BC+EF = CD+FA. \] The midpoints of sides $AB$, $BC$, $DE$, $EF$ are $A_1$, $B_1$, $D_1$, $E_1$, and segments $A_1D_1$ and $B_1E_1$ meet at $O$. Prove that $\angle D_1OE_1 = \frac12 \angle DEF$.

Let $n,k$ be positive integers such that $n\ge k$. $n$ lamps are placed on a circle, which are all off. In any step we can change the state of $k$ consecutive lamps. In the following three cases, how many states of lamps are there in all $2^n$ possible states that can be obtained from the initial state by a certain series of operations? i)$k$ is a prime number greater than $2$; ii) $k$ is odd; iii) $k$ is even.

In a quadrilateral $ABCD$ let $E$ be the intersection of the two diagonals, I the center of the parallelogram whose vertices are the midpoints of the four sides of the quadrilateral, and K the center of the parallelogram whose sides pass through the points. divide the four sides of the quadrilateral into three equal parts (see illustration ). [img]https://cdn.artofproblemsolving.com/attachments/1/c/8f2617103edd8361b8deebbee13c6180fa848b.png[/img] a) Prove that $\overrightarrow{EK} =\frac43 \overrightarrow{EI}$. b) Prove that $$\lambda_A \overrightarrow{KA} +\lambda_B \overrightarrow{KB} + \lambda_C \overrightarrow{KC} + \lambda_D \overrightarrow{KD} = \overrightarrow{0}$$ , where $$\lambda_A=1+\frac{S(ADB)}{S(ABCD)},\lambda_B=1+\frac{S(BCA)}{S(ABCD)},\lambda_C=1+\frac{S(CDB)}{S(ABCD)},\lambda_D=1+\frac{S(DAC)}{S(ABCD)}$$ , where $S$ is the area symbol.

A cube with side length $ 1$ is sliced by a plane that passes through two diagonally opposite vertices $ A$ and $ C$ and the midpoints $ B$ and $ D$ of two opposite edges not containing $ A$ and $ C$, ac shown. What is the area of quadrilateral $ ABCD$? [asy]import three; size(200); defaultpen(fontsize(8)+linewidth(0.7)); currentprojection=obliqueX; dotfactor=4; draw((0.5,0,0)--(0,0,0)--(0,0,1)--(0,0,0)--(0,1,0),linetype("4 4")); draw((0.5,0,1)--(0,0,1)--(0,1,1)--(0.5,1,1)--(0.5,0,1)--(0.5,0,0)--(0.5,1,0)--(0.5,1,1)); draw((0.5,1,0)--(0,1,0)--(0,1,1)); dot((0.5,0,0)); label("$A$",(0.5,0,0),WSW); dot((0,1,1)); label("$C$",(0,1,1),NE); dot((0.5,1,0.5)); label("$D$",(0.5,1,0.5),ESE); dot((0,0,0.5)); label("$B$",(0,0,0.5),NW);[/asy]$ \textbf{(A)}\ \frac {\sqrt6}{2} \qquad \textbf{(B)}\ \frac {5}{4} \qquad \textbf{(C)}\ \sqrt2 \qquad \textbf{(D)}\ \frac {3}{2} \qquad \textbf{(E)}\ \sqrt3$

Let $ v_1, v_2, \ldots, v_{1989}$ be a set of coplanar vectors with $ |v_r| \leq 1$ for $ 1 \leq r \leq 1989.$ Show that it is possible to find $ \epsilon_r$, $1 \leq r \leq 1989,$ each equal to $ \pm 1,$ such that \[ \left | \sum^{1989}_{r\equal{}1} \epsilon_r v_r \right | \leq \sqrt{3}.\]

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.

Let $Q$ be an $n$-by-$n$ real orthogonal matrix, and let $u\in \mathbb{R}^n$ be a unit column vector (that is, $u^Tu=1$). Let $P=I-2uu^T$, where $I$ is the $n$-by-$n$ identity matrix. Show that if $1$ is not an eigenvalue of $Q$, then $1$ is an eigenvalue of $PQ$.

Find the number of ordered triples of positive integers $(a, b, c)$ such that $abc$ divides $(ab + 1)(bc + 1)(ca + 1)$.

For $m\ge 3,$ a list of $\binom m3$ real numbers $a_{ijk}$ $(1\le i<j<k\le m)$ is said to be [i]area definite[/i] for $\mathbb{R}^n$ if the inequality \[\sum_{1\le i<j<k\le m}a_{ijk}\cdot\text{Area}(\triangle A_iA_jA_k)\ge0\] holds for every choice of $m$ points $A_1,\dots,A_m$ in $\mathbb{R}^n.$ For example, the list of four numbers $a_{123}=a_{124}=a_{134}=1, a_{234}=-1$ is area definite for $\mathbb{R}^2.$ Prove that if a list of $\binom m3$ numbers is area definite for $\mathbb{R}^2,$ then it is area definite for $\mathbb{R}^3.$

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) $K_1,K_2,..., K_n$ are the feet of the perpendiculars from an arbitrary point $M$ inside a given regular $n$-gon to its sides (or sides produced). Prove that the sum $\overrightarrow{MK_1} + \overrightarrow{MK_2} + ... + \overrightarrow{MK_n}$ equals $\frac{n}{2}\overrightarrow{MO}$, where $O$ is the centre of the $n$-gon. (b) Prove that the sum of the vectors whose origin is an arbitrary point $M$ inside a given regular tetrahedron and whose endpoints are the feet of the perpendiculars from $M$ to the faces of the tetrahedron equals $\frac43 \overrightarrow{MO}$, where $O$ is the centre of the tetrahedron. (VV Prasolov, Moscow)

Let $\Delta$ be a convex figure in a plane $\mathcal P$. Given a point $A\in\mathcal P$, to each pair $(M,N)$ of points in $\Delta$ we associate the point $m\in\mathcal P$ such that $\overrightarrow{Am}=\frac{\overrightarrow{MN}}2$ and denote by $\delta_A(\Delta)$ the set of all so obtained points $m$. (a) i. Prove that $\delta_A(\Delta)$ is centrally symmetric. ii. Under which conditions is $\delta_A(\Delta)=\Delta$? iii. Let $B,C$ be points in $\mathcal P$. Find a transformation which sends $\delta_B(\Delta)$ to $\delta_C(\Delta)$. (b) Determine $\delta_A(\Delta)$ if i. $\Delta$ is a set in the plane determined by two parallel lines. ii. $\Delta$ is bounded by a triangle. iii. $\Delta$ is a semi-disk. (c) Prove that in the cases $b.2$ and $b.3$ the lengths of the boundaries of $\Delta$ and $\delta_A(\Delta)$ are equal.

In a scalene triangle $ ABC, H$ and $ M$ are the orthocenter an centroid respectively. Consider the triangle formed by the lines through $ A,B$ and $ C$ perpendicular to $ AM,BM$ and $ CM$ respectively. Prove that the centroid of this triangle lies on the line $ MH$.

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 \[\min_{i\ne j}|F(x_i,x_j)|\le\frac{C}{n}.\]

The National Foundation of Happiness (NFoH) wants to estimate the happiness of people of country. NFoH selected $n$ random persons, and on every morning asked from each of them whether she is happy or not. On any two distinct days, exactly half of the persons gave the same answer. Show that after $k$ days, there were at most $n-\frac{n}{k}$ persons whose “yes” answers equals their “no” answers.