MathDB

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)

.
[hide=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)

.
[hide=Part 2] In 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

.
[hide=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

\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?
[hide=Part 4] In 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.

Problem Statement