Preliminary Part (a) Let $M$ be a point in the plane of a triangle $ABC$. Prove that $$\overrightarrow{MA}\cdot\overrightarrow{BC}+\overrightarrow{MB}\cdot\overrightarrow{CA}+\overrightarrow{MC}\cdot\overrightarrow{AB}=0$$Deduce that the altitudes of triangle $ABC$ are concurrent at a point $H$, called the orthocenter. (b) Let $\Omega$ be the circumcenter of a triangle $ABC$ and $H$ be the point given by $\overrightarrow{\Omega H}=\overrightarrow{\Omega A}+\overrightarrow{\Omega B}+\overrightarrow{\Omega C}$. Show that $H$ is the orthocenter of $ABC$. For every set $X$ of points in the plane, we denote by $\mathcal H(X)$ the set of orthocenters of all triangles with vertices in $X$. We call a planar set $X$ orthocentric if it does not contain any line and contains the entire $\mathcal H(X)$. Part 1(a) Find all orthocentric sets of three points. (b) Find all orthocentric sets of four points. (c) Let $X$ be a set of four points on a circle and let $Y=\mathcal H(x)$. i. Prove that $Y$ is the image of $X$ under an isometry. ii. Determine $\mathcal H(Y)$. (d) i. If $\Gamma$ is a nondegenerate circle, determine $\mathcal H(\Gamma)$. ii. If $D$ is a nondegenerate disk, determine $\mathcal H(D)$. Part 2In this part, $R$ is a positive number, $n$ an integer not smaller than $2$, and $X$ the set of vertices of a regular $2n$-gon inscribed in the circle with center $O$ and radius $R$. Consider the set $\mathfrak T$ of triangles with the vertices in $X$. An element of $\mathfrak T$ is chosen at random. (a) What is the probability of choosing a right-angled triangle? (b) What is the probability of choosing an acute triangle? (c) Let $L$ be the squared distance from $O$ to the orthocenter of the chosen triangle. Find the expected value of $L$. Part 3(a) Let $a,b,c$ be real numbers with $a(b-c)\ne0$ and $A,B,C$ be the points $(0,a)$, $(b,0)$, $(c,0)$, respectively. Compute the coordinates of the orthocenter $D$ of $\triangle ABC$. (b) Let $X$ be the union of a line $\delta$ and a point $M$ outside $\delta$. Determine $\mathcal H(X)$. Prove that $\mathcal H(x)\cup X$ is an orthocentric set. (c) Let $X$ be an orthocentric set contained in the union of the coordinate axes $x$ and $y$ and containing at least three points distinct from $O$. i. Show that $X$ contains at least three points on axis $y$ with nonzero $x$-coordinates of the same sign. ii. Show that $X$ contains at least three points on axis $y$ with positive $x$-coordinates. (d) i. Find all finite orthocentric sets of at most five points contained in the union of the axes $x$ and $y$. ii. Let $X$ be an orthocentric set contained in the union of the axes $x$ and $y$ and consisting of at least six points. Prove that there exist two sequences $(x_n)$, $(x’_n)$ of nonzero real numbers such that, for each $n$, the points $(x_n,0)$ and $(x’_n,0)$ are in $X$, and $$\lim_{n\to\infty}=\infty,\qquad\lim_{n\to\infty}x’^n=0.$$Can the set $X$ be finite? Part 4In this part we are concerned with constructing some remarkable orthocentric sets. (a) Let $k$ be a nonzero real number and $Y$ be the hyperbola $xy=k$. i. Let $A,B,C,D$ be distinct points of $Y$, with the respective abscissas $a,b,c,d$. Prove that $AB$ and $CD$ are orthogonal if and only if $abcd=-k^2$. ii. With the above notation, find the orthocenter of triangle $ABC$. iii. Prove that $Y$ is orthocentric. Throughout this part, $q$ denotes a nonzero integer and $X$ the set of points $(x,y)$ satisfying $x^2+qxy-y^2=1$. (a) i. Prove that the equation $t^2-qt-1$ has two distinct real roots and show that these roots are irrational. Throughout this part, $r$ and $r’$ denote these roots and $s$ the similitude defined by $z\mapsto(1-ri)z$ in terms of complex numbers. i. Prove that $s(X)$ is the hyperbola given by $xy = k$ for some real $k$, and find $k$. Deduce that $X$ is an orthocentric set. (b) Let $G$ be the set of integer points of $X$ and $\Gamma$ be the set of $x$-coordinates of the elements of $s(G)$. i. Show that $\Gamma$ is the set of numbers of the form $x+ry$, where $x,y$ are integers and $(x+ry)(x+r’y)=1$. ii. Show that $-1\in\Gamma$ and $r^2\in\Gamma$. iii. Prove that the product of two elements in $\Gamma$ and the inverse of an element of $\Gamma$ are in $\Gamma$. Show that $\Gamma$ is infinite. (c) Conclude that the set $G$ of integer points of $X$ is an infinite orthocentric set.